Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and...

95
Conference matrices and graphs of order 26 Citation for published version (APA): Paulus, A. J. L. (1973). Conference matrices and graphs of order 26. (EUT report. WSK, Dept. of Mathematics and Computing Science; Vol. 73-WSK-06). Eindhoven University of Technology. Document status and date: Published: 01/01/1973 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 28. Jul. 2020

Transcript of Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and...

Page 1: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

Conference matrices and graphs of order 26

Citation for published version (APA):Paulus, A. J. L. (1973). Conference matrices and graphs of order 26. (EUT report. WSK, Dept. of Mathematicsand Computing Science; Vol. 73-WSK-06). Eindhoven University of Technology.

Document status and date:Published: 01/01/1973

Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverne

Take down policyIf you believe that this document breaches copyright please contact us at:[email protected] details and we will investigate your claim.

Download date: 28. Jul. 2020

Page 2: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

A.J.L. Paulus

by

TECHNOLOGICAL UNIVERSITY EINDHOVEN

THE NETHERLANDS

DEPARTMENT OF MATHEMATICS

September 1973

T.H.-Report 73-WSK-06

Conference matrices and graphs of order 26

TECHNISCHE HOGESCHOOL EINDHOVEN

NdDERLAND

ONDERAFDELING DER WISKUNDE

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Abstract

The strongly regular graphs of orders 26 and 25, derivable from the

four known equivalence classes of conference matrices of order 26 are de­

termined by means of a computersearch. For one of these four equivalence

classes the results are not in the literature. Several numerical data are

obtained, for instance concerning cliques and automorphism groups. The

Algol procedures and resulting matrices are included.

AMS subject classification: 05B20, 05C30

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Chapter 4. Cdnference matrices of order 26 23

Chapter 3. Backtracking 19

Chapter 5. Algol procudures 41

41

41

44

46

46

47

pages

5. I. Introduction

5.2. The procedure Eigenvector

5.3. The procedure Cliquestructure

5.4. The procedures Decrep and Invdecrep

5.5. The procedure permutatie standaard

5.6. The Algol text

2.1. General definitions and theorems 3

2.2. Equivalence relations and automorphisms 4

2.3. Cliquestructure 9

2.4. Strong .graphs with two eigenvalues J3

2.5. Regular solutions for At satisfying (A-p II) (A-PZI) =0 17

3. I. Introduction 19

3.2. Definitions 19

3.3. The fundamental property 19

3.4. The pseudo Algol text of the backtracking algorithm 20

3.5. Remark 20

4~1 Introduction 23

4.2. The existence of four equivalence classes for C26 24

4.3. A representation of C26

27

4.4. Regular conference matrices of order 26 30

4.5. Cliques 35

4.6. Strongly regular graphs of order 26 37

4. 7. Remark 39

Chapter 1. Introduction

Contents

Chapter 2. Strongly regular graphs 3

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Chapter 6. Tables of resulting matrices

References

60

89

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I. Introduction

n + P - PI 0k ~

for some reals PI > Pz. If, in addition, there exists an integer Po such

that AJ = POJ, then G is said to be strongly regular.

A k-clique in a graph G is a complete sub graph of G of order k.

Chapter Z deals with strong graphs and strongly regular graphs, espe­

cially those graphs G, having only two distinct eigenvalues PI and PZ' which

implies n - 1 + PIP Z = O. If, in addition, PI and pZ are integers, then the

graph G may be regular, say AJ = POJ, with Po € {PI,P Z}, and then a neces­

sary condition for the existence of a k-clique is (theorem Z.4.4, cf. [3J)

A strong graph G of order n ~s a graph on n vertices, whose (-1,0,1)

adjacency matrix A of size n x n satisfies

Any two graphs GI

and GZ

are isomorphic whenever their adjacency ma­

trices Al and AZ satisfy pTA)P = AZ for some permutation matrix P of order

n. Any two graphs GI

and GZ

are called equivalent whenever their adjacency

matrices Al and AZ satisfy pTAIP = DA2

D for some permutation matrix P of

order n and some diagonal matrix D of order n, with diagonal elements +1

or -1.

In chapter 5 several algorithms are described to determine all pairwise

nonisomorphic regular graphs within the equivalence class of a graph, whose

integer eigenvalues satisfy n - I + PIP Z = O. One of these algorithms (sec­

tion 5.5) is due to F.C. Bussemaker (private communication). Most of these

algorithms are based on the backtracking algorithm, which is described in

chapter 3.

Chapter 4 deals with the investigation of the four known equivalence

classes of symmetric conference matrices of order Z6 (strong graphs of order

26 with the two eigenvalues 5 and -5). We denote these classes by LSI, StI,

LSII and StII. From these four classes 10 pairwise nonisomorphic regular

conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor­

phic strongly regular graphs of order 25 (theorem 4.6.1) can be derived.

For the distribution of these graphs among the four classes we refer to the

drawing on page O. All matrices are listed in chapter 6.

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For LSI, StI and LSII the results correspond to the results of

S.S. Shrikhande and Vasanti N. Bhat ([IOJ, [IIJ). The results for Stll are

new; those of [IIJ are not correct.

StIl contains one regular conference matrix with a trivial automor­

phism group, and with the property that all vertices are in exactly 70

4-cliques (theorem 4.5.1), which is remarkable in view of the small auto­

morphism group. Furthermore StIl contains two regular conference matrices

with an automorphism group of order 2 (one of these two matrices has the

same property as the first one) and two regular conference matrices with

an automorphism group of order 6.

The 10 regular conference matrices are equivalent to their complement

(theorems 4.4.2, 4.4.3 and 4.4.4). The strongly regular graph of order 25,

derived from LSI is isomorphic to its complement; the other 14 strongly

regular graphs of order 25 are nonisomorphic to their complement (theorem

4.6.2).

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{{x,y}IQ I =Ct , and ra

G is strong whenever (A - P1I)(A - PZI) = (n - I + PIPZ)JG is regular whenever AJ = POJ

G is strongly regular whenever G is strong and regular.

A. . = -I for {i,j} E r1,J

A.. = ,I for {i,j} E 1t(2\f1,J

Q' c Q, lit' I = n', and r' c {{ X , y} E f I x E ~' AyE It'} •

A. . = 0 for i E It1,1

(po is an integer, PI and pZ are reals, PI > PZ)'

For the proof of the following two theorems we refer to the sections 2 and

3 in [8J.

1 is the unit matrix (sometimes denoted by 1 , to indicate the order).n

J 1S the matrix, all of whose elements equal I, i is the vector, all of

whose components equal I.

o(M) 1S the spectrum of the square matrix M.

If/G is a graph with adjacency matrix A, then

K(a,8) is the graph (Ita U 1t8,f) of order n = a + 8, where

G = (It,r) is complete (void) whenever r = 1t(2) (f = 0).The adjacency matrix A of G = (It,f) is the square matrix of order n, defin­

ed as follows:

n := {I,Z, ••• ,n}, the elements of ~ are called vertices.

~(k) is the set of all k-subsets of Q.

If r c Q(2) then G := (~,r) is a graph of order n, and then G* := (Q,Q(Z)\f)

lS the complement of G.

G' := (Q',r') is a subgraph of G = (Q,f), of order n' ~ n, whenever

2. Strongly regular graphs

Z.I. General definitions and theorems

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2.1.1. Theorem. If A 1S the adjacency matrix of a graph G, satisfying

(A - PjI)(A - P2I) = (n - I + PIPZ)J F 0, PI > 0, then there exists

an integer PO' satisfying (PO - PI)(P o - P2) = n(n - 1+ PIP Z),

such that AJ = POJ, that is, G is strongly regular.

a(A) = {P O,P I ,P 2} with the multiplicities I, ]..II and ]..12' say.

If ]..II F ~Z then PI and Pz are odd integers. If ]..II = ]..IZ then Po 0

and PI = -P Z = In.

2.1.2. Theorem. If A is the adjacency matrix of a graph G, satisfying

(A - PII)(A - PZI) = 0, PI > 0, then a(A) = {P I ,P 2} with the mul­

tiplicities ]..II and ]..12' say.

If A is not the adjacency matrix of K(a,S) or its complement, then

~I = ]..12 = !n implies PI = -P Z = In-I and ~I F ]..IZ implies that PI

and Pz are odd integers.

Z.2. Equivalence relations andautomorphisms

The Kronecker delta O.. is defined by1']

o. . = a for 1 F j and O. . = I •1,] 1,?-

~ is the set of all square matrices of order n.n

Sn is the set of all permutations of n (so Sn corresponds to Sn' the symme-

tric group of order n!).

• 2. I. Definition • P := {p € '/ll 3 V.. n (p. . = o. C»}n n 1T€Sn 1,J € 1,J 1,1T J

t. := {E € 111 31T€Sn

'V. • (E .. =+0. C»}n n 1,J€n 1,] - 1,1T J

~ := {D E ·m I \:j. . (D. . = + o.. )} .n n 1.,JErl 1.,J - 1,]

It is easy to verify that, under the usual matrix multiplication, e is ao n

group, and that [j) and SO are subgroups of t. , of orders n! and Zn, respec-n n n

tively.

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DZ and, ~n= I, hence D}

for i,j E Q •

elements, D,D2='Y @&1.

n n

= 0.0,0'(,)0 (') (0)1 J TI 1 1T ~ ,TI J

Define P E: (JJ and D E~ by P, , := 0, (') and D, , := 0 (,)0, , forn n ~,J 1,TI J ~,J n ~ 1,J

i,j E n, then E PD, which follows from

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Now let D' be another element of 50 , defined by D~ .: = o! 0, . for i ,j E n,n ~,J ~ ~,J

where o! = + 1 for i E Q, then~

Theorem. &> is a normal subgroup oft •n n

Proof. Let E be an element oft. ,defined by E, ,:=0.0. (') for i,j E: n,n ~,J ~ ~,n J

where 0i = + 1 for i E n, and TI E Sn'

I .(0.0, k)(o (k) 0)(0:° 0 )(0 (»(o 0 ,) =k n n ~~, 1T ,Jt, Jt, Jt"m m,1T n n n,J~Jt"m,nE..

Therefore ETD'E E~ , which implies the theorem.n

Theorem. E =:P @ ~ •n n n

Proof. From the proof of 2.2.2 it follows that for any E E £n there is a

P E ~ and a D E 5> such that E = PD.n n

Now sup~ose P1D 1 = PZDZ for P1,P2 E ~n and D],DZ E~n' Then P~PZDZ = DI

,

hence P1PZ = D1DZ'

Since PrPZ only has nonnegative

addition, PI = PZ' Therefore gn

Clearly It I (n:)Zn.n

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Now we are able to define the usual equivalence relations between graphs,

in terms of their adjacency matrices.

Z.Z.4. Definition. If AI and AZ are the adjacency matrices of GI and GZ' then GIand GZ are said to be isomorphic (G I ; GZ) whenever

T3PE~ (P AlP = AZ)' and GI and GZ are said to be equivalent

. n T(G 1 ~ GZ) whenever 3

EEt(E AlE = AZ)'

n

For the adjacency matrices Al and AZ these relations are denoted in the same

way: AI ;' AZ and AI - AZ'

It is easy to verify that these relations are indeed equivalence relations.

In the theory of graphs the automorphism group Aut(G) of the graph G = (n,r)~s defined as follows.

2.2.5. Definition. If G = (n,r) is a graph on n, then

For the adjacency matrices of graphs we have

2.2.6. Definition. !fA is the adj acency matrix of a graph G, then

J.>(A) := {P E ~ pTAP = A}n

~(A) := {E E c. ETAE = A} ,n

~(A) := {D E ~ DAD ~ A} .n

It is easy to see that C(A) is a group, that ~(A) is a subgroup of teA) and

that ~(A) and Aut(G) are related by:

P E ~(A) if and only if there is a TI E Aut(G) such that

P. . = 8. (.) for i ,j En.1.,J ~,7T J

$(A) is not necessarily a group. Nevertheless we have

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z. Z. 7. Theorem. /t(A) I Ij) (A) I lao (A) I•

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Proof. Define the relation ~ on C(A) byP

E1 ~P EZ whenever E2E~ E a:>(A) , for E] ,EZ E~ (A) •

So we have

p is an equivalence relation on ~(A), hence~(A) partitions teA) into

equivalence classes (cosets of~(A)), each having I~(A) I elements. From

2.2.3 it follows that for any E E t there is a unique D E93 and P E:J->n n nsuch that E = PD.

Now let E1 and EZ be elements of e(A) and

and EZ = PZDZ' where P1,PZ E~n and D1,DZsuch that PP1D 1 = PZDZ' which implies

suppose E1 p EZ' while E] = P1D 1E£I • Then there is aPE c;]:>(A)

n

Since p~pTPZ only has nonnegative elements, D]D2 = I, hence D1 = DZ• There­

fore any equivalence class of teA) under the relation -p has a unique re­

presentation in~ •n T T_

If E E teA), and E = PD, then DP APD = A, hence DAD = P AP, or DAD = D.

Therefore this representation inJD is an element of~(A). Conversely, anyn

D E~(A) yields an equivalence class in ~(A), so we have proved IE (A) 1 == p:> (A) II~(A) I.

2.Z.8. Theorem. If A] and AZ are the adjacency matrices of the graphs G] and GZ'

then

G} ;: GZ implies

G] GZ implies

= 19(Az) I and

IE: (AZ) I·

Proof. We prove the second statement, the first statement can be proved in

the same way.

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If GI ~ GZ' then there LS an E E C such that ETA E = AZ' or, which is theT

n . )

same, A) = EAZE •

If E) E E(A) then

T T T T T ETA EE E}EAZE E]E = E E]AIE}E = AZ ,}

hence E) € e(A1) implies ETE)E € ~(AZ).

Together with ETE)E = ETEZE iff E) = EZ it follows that 1~(A2)1 ~ le(A])I.

In the same way it follows that It(AI)1 ~ IC(AZ)I , hence equality holds.

Z.Z.9. Corollary. If A is the adjacency matrix of a graph G, then the number of

distinct graphs within the class of graphs, which are isomorphic

( . 1 ) 1 n: ( . 1 (n:)Zn)equ~va ent to G, equa s ·1~(A)r respect~ve y 1e(A)T .

Remark. A class of equivalent graphs corresponds to a two-graph (see [9J).

Let A be the adjacency matrix of a representative G of such a class. Then

{I,-I} is a normal subgroup of C. (A) ({I ,-I} is the centre of ~ ), and there-n

fore {I,-I} yields a factorgroup of equivalence classes of ~(A), correspond-

ing to the automorphism group of the two-graph, representing the equivalence

class of G.

In the sequel the following theorem will be useQ.

2.2.)0. Theorem. If (A - p]I)(A - PZI) = (n - ] + p)PZ)J then

(pTAP - P)I) (pTAP - PZI) (n - I + P1PZ)J for any P €(i)n. If, in

addition, n - ] + P1P Z = 0 then (ETAE - PII) (ETAE - PZI) = 0 for

any E E C. •n

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In section 2.3 necessary conditions for graphs, to be isomorphic, will be

discussed. As far as we can see these conditions are not sufficient. In

order to be able to decide whether two graphs are or are not isomorphic,

F.e. Bussemaker made an algorithm (section 5.5) to determine a unique re­

presentative of a class of isomorphic graphs. The existence of such a

unique representative will be clear by the following definitions.

Z.Z. II. Definition. If A is the adjacency matrix of a graph G, then the vector ~(A)

of length n - is defined by

b. (A) :=1

n

Lj=i+l

Zn-j-I(I - A.• )1,]

for 1 = I ,Z, ••• ,n-I •

2.2.12. Definition. If u and v are vectors of length m, then u <0 v whenever

31<k< «uk < vk) A 0J I<'<k-1 (u. = v.») •- _m -J - J J

The class of graphs, isomorphic to G, is finite. Therefore there is a graph

in the class, with adjacency matrix S, such that for the adjacency matrix A

of any other graph in the class ~(A) <0 ~(8). This matrix is called the

standard matrix of a class of isomorphic graphs. If, for 1 = 1,Z, 8. is the1.

standard matrix corresponding to the adjacency matrix A. of a graph G., then1. 1

it follows that G1 ~ G2 if and only if 8 1 = 8Z'

2.3. Cliquestructure

An important question is, under which conditions two graphs are iso­

morphic. In general it is easier to prove that two graphs are nonisomorphic

than that they are isomorphic. This follows from the fact, that the, so

called, cliquestructures of two isomorphic graphs are identical, and that

it is possible to compute certain cliquenumbers, which only depend on this

cliquestrl.lcture.

Conversely it is not known whether these numbers determine a class of iso­

morphic graphs uniquely, especially if these graphs are strong, or strongly

regular.

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2.3.1. Definition. A k-clique of a graph G is a complete subgraph of G, of order k.

It is possible to calculate the total number of k-cliques in a graph; ob­

vious ly this number is an invariant of the class of graphs , isomorphic to

the given one.

Furthermore, it is possible to calculate the number of k-cliques, contain­

ing a certain vertex, and, doing this for all vertices, this set of numbers

has to be the same for two isomorphic graphs. The same can be done for k­

cliques, containing a given pair of vertices, etc.

A nice representation of the cliquestructure of a graph G is given by the

matrices K(k) (G), defined as follows.

2.3.2. Definition. If G is a graph on ~, and k ~s an integer ~ 2, then the symmet­

ric matrix K(k) (G) of order n is defined by (K(k) (G» .. :=1,]

:= total number of k-cliques in G, all containing the vertices

i and j.

These, so called, cliquematrices have some properties, formulated in the

following two theorems.

2.3.3. Theorem. If A is the adjacency matrix of G, then A..~,J

for i :f j,

(k - I) (K (k) (G». . ,~,~

and

where ck(G) ~s the total number of k-cliques ~n G.

Proof. By elementary calculations.

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2.3.4. Theorem. If A is the adjacency matrix of a strongly regular graph G, satis­

fying (A - P1I)(A - PZI) = (n - 1+ P IP2)J and AJ = POJ, then

o for A.. =~,J

(K(3) (G))i,j = fen - 3 - PO) +t(l - PI) (I - PZ) for Ai,j = -I

TI;(n-I-po)(2(n-3-po) + (I - PI)(I-P Z)) for

~ = j .

Proof. If the vertices i and j are nonadjacent (A.. = I), then they are in~,J

no 3-clique. If the vertices i and j are adjacent (A.. = -I), then let xI~ ,J

be the number of vertices, adjacent to i and to j, x2 the number of vertices,

adjacent to i and nonadjacent to j, x3 the number of vertices, nonadjacent

to i and adjacent to j, and x4 the number of vertices, nonadjacent to i and

to j. From the equations for A it follows, that

-XI - x2 + x3 + x4 = + Po

-Xl + x2 - x3 + x4 = + Po

This implies XI = ~(n - 3 - PO) + 1(1 - PI)(I

The number of vertices, adjacent to vertex it

2.3.3 it follows that

(3)- P2) = (K (G)).. •

~,]

equals ~(n - ] - PO); from

(K(3)(G)) .. = _1(n - ] - PO)(2(n - 3 - PO) + (l - PI)(I - P2)) •

~,~ 16

This completes the proof.

If A] and AZ are the adjacency matrices of the graphs G1 and GZ' and G] ~ GZ'

then there exists aPE jJ , such that pTA P = A2 , and then, for any k ~ Z,n ]

= K(k) (G )Z

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Of course the idea of computing cliquematrices can be generalized by calculat­

ing the numbers of cliques, containing three, or even more, given vertices.

But, for our purposes, it seems to be true that the matrices K(2)(G),K(3)(G),

K(4)(G), ••• already yield sufficient information about nonisomorphic graphs.

Now we return to k-cliques in a graph.

In the f~llowing incidence vectors of cliques are used; they are defined as

follows.

2.3.5. Definition. If G is a graph on Q, and Q' is the vertex set of a k-clique of

G (so IQ'I = k), then the incidence vector v of this clique is

a vector of length n, satisfying v. = I for i E Q' and v. = 0 ~ ~

for i € Q\Q'.

A set of k-cliques is called linearly independent iff their incidence vectors

are linearly independent.

We recall a lemma about eigenvalues:

2.3.6. Lemma. creal + SJ ) = {a + Sn,a} with multiplicities 1 and n-l. n n

Proof. a(J ) = {n,O} with multiplicities 1 and n-l. n

If the matrix H diagonalizes J , then it also diagonalizes aI + SJ , which n n n implies the lemma.

For strong graphs it is possible to indicate an upper bound for the order of

a clique, in terms of the order and the eigenvalues of its adjacency matrix.

2.3.7. Theorem. If A is the adjacency matrix of a strong graph G, satisfying

(A - P1I)(A - P2I) = (n - 1 + P1P2)J, and G contains a k-clique,

then the following inequality must hold:

Proof. If G contains a k-clique, then there exists aPE ~ such that n

BJ t D i n-k

k

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- 13 -

From 2.2.10 it follows that

[0-0 jlI-J

0-: I rJ(J-p )I-J

o-:J (n - I + 0 10 2) [: :J2

=BT

BT

which implies

«1 - p )1 - J)«1 - T (n - 1 + P IP2)J ,I p 2)1 - J) + BB

or, uS1ng J2 = kJ,

Now any eigenvalue of BBT is nonnegative, since BBTv

AVTV = vTBBTv = (BT~)T(BT~) ~ 0 •

~pplying 2.3.6 it follows that

\~ implies

which implies the theorem.

2.3.8. Definition. A maximal clique in a strong. graph is a clique, whose order k is

maximal with respect to (k-I +PI)(k- I +P 2) ~ ken - I + PIP Z),

2.4. Strong graphs with two eigenvalues

If the adjacency matrix A of a strong graph G has two distinct eigen­

values PI and P2 ' say, then it follows from 2.1. I that n - I + P j P2 = 0,

hence A satisfies (A - PII)(A - P2I) = O.

In this section we will only consider strong graphs with two, integer,

eigenvalues PI and P2 ' satisfying

PI> I, p 2 < - I and n -. I + PIP 2 O.

First we give a result on + matrices.

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- 14 -

2.4.1. Theorem. If B is a matrix of size m x n, with elements ~ J, satisfying

BBT

= aI + 6J, while m > and n > 0, then a + 6 n, a + Sm ~ 0,

equality holding iff BTl £' and a + 8(m - J) > O.2

If, in addition, Bl = rl then n(a + 8m) ~ r m, equality holdi~g iffBT. rm .

.1 = n .1'

Proof. a ~ (BTi)T(BT1) = ITBBTl = am + 6m2 = m(a + Sm). Hence a + 8m ~ 0,

equality holding iff BTl = ~. Multiplying any column of B by -J does not

h h . B T f BT -- [.I' IRTJ ,c ange t e equat~on B = aI + 8J. There ore we may assume

where RT is a matrix of size n x (m - 1), satisfying

+ 8J •

Hence a + S = n, Ri= 61 and RRT = aT + SJ.

Applying the inequality of Cauchy-Schwarz to the vectors .1 and RT1, both of

length n, we have <lTR,i) 2 s: (iT1) (lTRRT1 ), which implies

2226 (m - 1) s: n(a(m - 1) + S(m - 1) ) ,

hence

n(a + 6(m - I» ~ S2(m - 1) ~ ° .If a + S(m - 1) = ° then 62

= 0, hence S = a = 0, which is impossible since

a + S = n > 0, and therefore a + S(m - 1) > O.

If Bi = rl, then we may apply the inequality of Cauchy-Schwarz to the vec­

tors i and BTl, both of length n, and then we have

2 2 2r m s: n(am + 6m) ,

hence n(a + Sm) ~ r 2m, equality holding iff

it is easy to verify that this multiple has

This completes the proof.

T, , 1 . 1B .1 ~s a mu tiP ermto be equal to -­n

of .1, and then

From 2.3.7 it follows that a maximal clique in G is of order 1 - P2'

2.4.2. Theorem. The incidence vector of a maximal clique ~n G is an eigenvector of

A with eigenvalue P2•

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- I') -

Proof. From 2.2.10 we may assume that

[

1-JA =

BT

I-p2

From (A - PII)(A - PZ1) = 0 it follows that BBT

= (PI - 1)(1 - PZ)1+ (I-PZ)J.

Applying theorem Z.4. 1 with

a: = (p 1 - 1) (1 - P2)' S = (I - PI) and m = (I - p 2 ) ,

it follows that a: + Sm = 0, hence BTl = O.

The incidence vector of the maximal clique in this case is

Tv ) .

Therefore

_( l

I-p n-I+pZ Z

Av =

2.4.3. Theorem. IfG has a maximal clique, such that

[1-J Jt I-p

A = Z

BT t n-l+PZ

then oeD) = {PI + P2 - I, PI' Pz} with multiplicities -P2' ~I and

]JZ' where

~ I =p 2 (I + P2) (p 2 - I)

PI - PZand 11 Z

2+ PI - P 2 - P I )-

PI - PZ

Proof. From (A - PII)(A - PZ1) = 0 it follows that

a) BBT = (p I - I) (I - p 2) I + (I - PI) J

b) BD = (PI + P2 - I)B (since JB = 0, cf. 2.4.2)T

c) B B + (D - PII)(D - ( 21) = o.

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From c) it follows that BTBD + (D - PI1)(D - PZ1)D = O. Together with b)Tthis leads to (PI + Pz - I)B B + (D - PII)(D - PZ1)D = 0, and therefore

(D - P11)(D - PZ1)(D - (PI + Pz - 1)1) = 0, hence oeD) C {PI +PZ-I,PI,P Z}.

Applying Z.3.6 it follows that O(BBT) {(PI - 1)(1 - PZ)' O} with multi­

plicities -P z and I. Therefore O(BTB) = {(Pt - 1)(1 - PZ)' O} with multi­

plicities -P Z and PZ(Z - PI). Now suppose H-IDH is diagonal, so

From c) it follows that H- 1BTBH must be diagonal too. Suppose

H- IBTBH = d· ( \ \ \ 0 0 0) h \ ( 1) ( I )1 ag 1\, 1\ , •••:-.:' t' ,: .. , )' were f\ = P I - - P2 •

-pZ times PZ(Z-P I ) times

Since 0i € oeD) C {PI + Pz - I, PI' pz} it follows that 0i = PI + Pz - 1 for

i = 1,2, ••• ,-oZ' and <\ € {PI,P Z} for i .. -P Z+I,-P Z+Z, ••• ,p 2(I-P t).

The multiplicity of (PI + P2 - I) is equal to -PZ. Let the multiplicities of

PI and Pz be ~I and ~Z' respectively, then

~I + ~Z - Pz = PZ(I - PI) (= order of D) and

This implies the theorem.

Since PI and Pz are integers, it may be possible that G 1S regular, that is

Ai = pol with Po € {p I,P Z}. Regularity of G yields a new restriction on the

order of a k-clique in G, in the case that Ai = PIle

Z.4.4. Theorem. If G contains a k-clique, while Ai = poi, then

Preof. We may assume that--

e-J

:]t k

A =BT t n-k

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From (A -.Jp),I) (A - P2I) = 0 we have BBT;: (p - 1)(1 - p )I + (2-p -P -k)J ) 2· I 2 ' and from Al ;: poi we have Bl = (PO + k - 1)1 •

Applying the inequality of Cauchy-Schwarz to the vectors 1 and BTl, both of

length n-k, we have

2 2 2 k (PO + k - I) s (n - k)(k(P I - 1)(1 - P2)+k (2 - PI - P2 - k».

For Po this yields k n and for Po P2 this yields k I = PI' s s - PZ' + PI

Hence

Po - Pz PI - Po n + PI - Po k s n (I - P ) + =

PI - P2 + PI PI - P2 2 + PI

Remark. Theorem Z.4.4 is a special case of a formula by P. Delsarte [3J for

strongly regular graphs in general:

k $

By substituting n - I + P1P2 = 0 this formula yields theorem 2.4.4.

2.4.5. Theorem. If (A - p)I)(A - P2 I ) ;: 0, PI and P2 having the multiplicities ~l

and ~2' say, and N is a matrix of size ~) x n and of rank ~I' . f' ANT NT h sat~s y~ng = PI J t en

Proof. NT(NNT)-lN u the projector on the eigenspace of At cot:responding

to the eigenvalue Pl'

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2.5. Regular solutions for A, satisfying (A - PII)(A - PZI) = O.

In this section G will always be a graph on ~, whose adjacency matrix A

satisfies (A - PII)(A - P2I) = 0, PI and P2 being integers. We are interest­

ed in finding all pairwise nonisomorphic regular graphs within the equiva­

lence class of G.

If Ai pi with P E {PI,P Z}, then for any P E tyn we have

pTAPi = pTAl pT. .= P 1. = P.J.

Together with the fact that f) is a normal subgroup of c.. it follows that itn n

1S sufficient to determine the regular matrices 1n the set

{DAD IDE ~ } •n

2.5.1. Definition. ~* := {Dj IDE,f) }.n - n

If DADi = pi, then ADi = pDi. hence (A - pI)d = Q, where d = Di E~:.

SO the determination of the regular matrices in the set {DAD IDE ~ } cor­n

responds to the determination of the set

{d E if) * I (A - pI) d = O} •- n - -

This can be realized by use of the procedure Eigenvector (section 5.2).

Among these regular solutions there may occur isomorphic pairs. However,

for each regular solution we can determine the corresponding standard matrix,

and this leads to the determination of all nonisomorphic regular solutions.

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. kfunction f

k: W + {O,I} is defined in such a way

kE W the statement

- 19 -

3. Backtracking

3. I. Introduction

In combinatorial problems it may happen that a large number of possibi­

lities has to be tested on certain properties. Then the question arises,

whether there exists a procedure to do this in an orderly, quick way. Some­

times this procedure may be given by the, so called, backtracking algorithm,

of which a general description is given in this chapter.

3.2. Definitions

w is the finite set of the v distinct e~ements wl

,w2

, ••• ,wv ' where v> 1.

Wk

is the set of k-sequences of W, so W = {(xI' ••• ,xk) I xi E W for

i = 1,2, ••• ,k}.

Clearly Iwkl = vk •

Let n be an integer> I.

For any k E {I,2, ••• ,n} a

that for any (xI""'~)

~s true.

The problem now ~s to determine V := {(xI' ••• ,x ) E Wn I f (xI' ••• ,x ) = I}.n n n n

3.3. The fundamental property

kIf (a I ,.oo ,~) E Wand fk(a

I,oo. ,a

k) .. 0 for a certain k E {I, ••• ,n},

then for all (xI, ••• ,xn) E Wn with xi = ai for ~ i ~ k it follows from the

definition of f l , ••• ,fn that fn(xl, ••• ,xn) = O.

This property is used in the backtracking algorithm, since, if

fk(al""'~) = 0 for a certain k E {I, ••• ,n}, then we can immediately indi-n n-k. h h" . h Vcate a subset of W of order v , wh~ch as not ~ng ~n common w~t •

n

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- 20 -

3.4. The pseudo Algol text of the backtracking algorithm

We introduce the function index: W~ {I,2 t ••• t v} by

index(tl1.) := jJ

for j = 1,2, ••• ,v. Note that the index of the element ~ E W is not neces­

sarily k. Then we can show the structure of the backtracking algorithm by

the following pseudo Algol text.

k := I; £ := I; goto L2;

Ll: if £ = v then

begin if k = ~ goto ready;

3.5. Remark.

L2:

ready:

k := k - I; £ := index(~); goto LI

end;

£:=£+1;

~ :=w.R,;

if fk(x 1, •.• ,~) = 0~ goto Ll;

if k = n~ begin print(x1, ••• ,xn ); goto Ll end;

k :=k+ I; £:= I; gotoL2;

In general only W, nand f are known, and then it 1S the task of then

programmer to find functions f ,f2 , ••• ,f 1 such that1 n-

These functions are efficient if for many n-sequences (xl' •.• ,xn ) E Wn\Vn ,

fk(xI""'~) = 0 for small values of k. If they are efficient then the

algorithm apparently works very quickly. On the other hand the enumeration

of efficient functions will usually be more complicated than the enumera­

tion of less efficient functions. Again it is the task of the user of the

algorithm to find a compromise between these two difficulties.

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- 21 -

Finally there is still another consideration to be careful in apply­

ing the backtracking algorithm. This can be best illustrated by an example.

Suppose one wants to determine all partitions of an integer N into at most

n distinct integers > O. This can be realized in two ways by applying the

backtracking algorithm as follows.

ix. ::; N, for1

kfk(xl""'~) = I whenever L

i=1::;k::;n-1.

Frob lem I : W = {O,}}; f (x I ' ••• , x )n n

1 whenevern

Li=1

ix. =1

N, and

(n+l-i)x.:-:;; N, for 1:-:;; k::; n-l.1

nProblem II: W = {O,I}; f (xl' ••• ,x ) = I whenever L (n + 1- i)x. =

n n i= I 1k

fk(xl, ••• ,xk) = I whenever Ii= 1

N, and

The two applications only differ in the structure of the functions fl, .•• ,fn ,

although their enumerations are of the same degree of difficulty, and, in

addition, the solutions of problem II are the reversed solutions of problem

I, but this is not essential.

During the action of the algorithm, as soon as a function f k 1S enume­

rated, an integer is increased by I. At the end of the action the value of

this integer gives an indication for the execution time.

For n = N = 5 the action of the algorithm in both cases is given by the

trees designed on page 22.

Apparently application II seems to be more efficient than application I. This

is confirmed for n = N = 20, where the number of enumerations of fk is about

3500 for problem I, respectively 500 for problem II. Probably this difference

will become relatively greater for larger values of nand N.

Reference [1].

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4. Conference matrices of order 26

4.1. Introduction

A conference matrix C of order n is a square matrix with diagonal ele­

ments 0 and other elements +1 or -I, satisfying

TCC = (n - 1)1 •

In [4J it has been proved that there are essentially no other than symmetric

and skew conference matrices, and that a necessary condition for the exis­

tence of a symmetric (skew) conference matrix of order n is:

n = 2(mod 4) (resp. n = O(mod 4» •

In [7J, section 5, it has been proved that a necessary condition for the

existence of a symmetric conference matrix of order n is, that n is the sum

of two squares of integers.

If C is a symmetric conference matrix of order n, then

CCT = C

2 = (n - 1)1, hence (C - In-I I)(C + In-1 I) = 0 ,

which implies that C is the adjacency matrix of a strong graph, correspond­

ing to the graphs of sections 2.4 and 2.5.

This implies that a necessary condition for C to be regular is, that In-1

is an integer, so, in view of all these conditions, n = 2,10,26,50,82, •••

n = 2 ~s a trivial case.

For n = 10 it is easy to verify that there exists only one equivalence

class, and that the Peterson graph and its complement are the only regular

graphs in this class.

The next case is n = 26, and this case will be investigated in the

following sections, where several (numerical) results are obtained by means

of a computer search. The following programs are involved:

integer procedure Eigenvector,

integer procedure Cliquestructure,

procedure Decrep and procedure Invdecrep,

procedure permutatie standaard •

The description of these algorithms is given in chapter 5.

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--------- ,- :U~ -

4.2. The existence of four equivalence classes for C26

4.2.1. Definition. A >teiner triple system is a set of b triples out of a set V of

velements, such that each pair of distinct elements of V occurs

in exactly one triple.

If the elements of V and the triples are labeled (v1' ••• ,vv'

resp. b1

, ••• ,bb)' then the incidence matrix N (of size v x b) of

the Steiner triple system is defined by N.. = 1 for v. E b. and~ ,J ~ J

N. . = 0 for v. i. b .•1-,J ~ J

For t~ proof of the following theorem we refer to [6J.

4.2.2. Th~rem. A Steiner triple system on v symbols only exists for v ­

(mod 6), and then the incidence matrix N satisfies

NNT= !(v - 3)1 + J, NJ = !(v - I)J and IN = 3J, while b =

or 3

1"6v(v - I).

4.2.'. Theorem. If N is the incidence matrix of a Steiner triple system on v sym­

bols, then A := 51 + J - 2NTN is the adjacency matrix of a strongly

regular graph, satisfying

(A - 5I)(A - 81 + vI) = ~(v - 13)(v - 18)J ,

AJ i(v - 3)(v - 16)J •

for i # j. Using 4.2.2

2(NTN) .. = 6 - 6 = 0, s~nce NTJ = 3J.~,~ T

A.. = 1- 2(N N) .. = + 1, since the definition of a~,J ~,J

Steiner triple system implies that any two triples of the system have at

most one element in common, hence (NTN) .. = 0 or~, J

it is easy to verify the equations for the matrix A in the theorem, hence A

Proof. A.. = 6 ­1.,1.

For i ~ j we have

1S the adjacency matrix of a strongly regular graph.

For v = 13 we have

(A - SI)(A + 51) = 0, AJ = -5J ,

hen~e A is a regular sy~netric conference matrix of order 26.

I

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- 25 -

There are L nonisomorphic Steiner triple systems on 13 symbols (see [6]).

They yield 2 conference matrices, which will turn out to be nonequivalent

(see section 4.4).

4.2.4. Definition. A Latin square L of order n is a square matrix of order n t every

row and every column being a permutation of the n distinct sym­

bols 1,2, ••• ,n.

2 2 4.2.5. Definition. A Latin square graph L3(n) of order n is the graph whose n . h 2 . f' f d vert~ces are ten entr~es 0 a Lat1n square L 0 or er n t

two vertices Li,j and Lk ,£' for (i,j) F (k,£), being adjacent

whenever (i = k) v (j = £) v (L .. = Lk n)' 1,J ,x.

4.2.6. Theorem. Any Latin square graph L3 (n) is strongly regular with

(A - 51) (A + 2nI - 51) = (n - 4)(n - 6)J and AJ == (n - I)(n - 5)J,

A being the adjacency matrix of L3 (n).

Proof. If L is a Latin square of order n, and L. . and L are adj acent, l,J K,£

then the number of vertices (entries of L), adjacent to L .. and nonadjacent l,J .

to Lk ,£ equals 2(n - 2), and therefore

A2 == n2 - 2 - 4(2(n - 2» = n2 - 8n + 14 • -~i.j ,~,£

If Li,j and Lk,i are nonadjacent, then the number of vertices, adjacent to

L .. and nonadjacent to L_ i' equals 3(n - 3), hence l,J K,

2 AL .. ,L n

1,] le,x. = n

2 - 2 - 4(3(n - 3»

Therefore A2 = aA + 61 + yJ, where

= n2 - 12n + 34 •

S + Y = n2

- I, -a + y = n2 - 8n + 14, a + y = n2 - 12n +34 ,

which implies (A - 51)(A + 2nI - 51) = (n - 6)(n - 4)J.

The number of vertices, adjacent to any vertex L .. equals 3(n - I), hence ~,J

AJ = (n2 - I - 2(3(n - I»)J = (n - 5)(n - I)J.

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For n = 5 we have (A - 5I)(A + 51) = -J and AJ O.

Now define the matrix C of size 26 x 26 by

then

,v

Hence C is a conference matrix of order 26.

There are 2 nonequivalent Latin squares of order 5 (see [6J). They yield 2

conference matrices of order 26, which will turn out to be nonequivalent to

eachother and to the conference matrices, derived from the two Steiner triple

systems (see section 4.4).

Now let N be the matrix [N1INZ] of size 13 x 26, where N1 and N2 are

circulants of order 13,

N1 = circul(l ,0,1,0,0,0,0,0,1,0,0,0,0)

N2 = circul(0,0,0,O,O,O,1,1,0,O,1,O,0) ,

then N is the incidence matrix of a Steiner triple system on 13 symbols.

Furthermore, let the matrix N* of size 13 x 26 be defined by

*(N ). . : = I - (N). .~,J ~,J

for

(i,j) E: {(1,6),(I,ZI),(1,24),O,25),(6,6),(6,21),(6,24),(6,25)} ,

*(N ). . : = (N). . in any other case ,1,J ~,J

*then N is the incidence matrix of another Steiner triple system, nonisomor-

phic to the first one (which will follow from section 4.4). The equivalence

class of conference matrices, derived from N, is called StI, and the class,

*derived from N , is called StII.

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- 27 -

The Latin squares

2 3 4 5 1 2 3 4 5

5 2 3 4 2 4 5 3

Ll := 4 5 2 3 md L2 := 3 4 5 2

3 4 5 2 4 5 2 3

2 3 4 5 5 3 2 4

yield two equivalence classes of conference matrices of order 26, called

LSI, respectively LSII.

4.2.7. Theorem. StI, StII, LSI and LSII are distinct equivalence classes.

Proof. From section 4.4 it will follow that the regular conference matrices

within any class are nonisomorphic to the regular conference matrices of any

other class, which implies the theorem.

Remark. Up till now we do not know whether there exist other equivalence

classes. F.C. Bussemaker is trying to determine all pairwise nonisomorphic

strongly regular graphs of order 25, whose adjacency matrices S satisfy

2 S = 251 - J md SJ = 0 ,

by means of a computersearch. In view of section 2.6, we will be able to

solve this problem, as soon as this computersearch will be finished.

From M.B.L.E. Reserach Laboratory in Brussels we received with thanks

the drawing on the next page, which represents a conference matrix out of

LSI.

4.3. A representation of CZ6

We prove that the representatives of the four equivalence classes t as

constructed in 4.2, cm be represented as in theorem 2.4.5, in such a way

that the 13 rows of the matrix N. are linearly independent incidence vectors ~

of 6-cliques (which are maximal cliques).

First we give a

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- 30 -

Now let A be the adjacency matrix of L3

(5), then

C := l: .T]-1.

A

is a conference matrix of order 26, and N :~ [1 I MSJ is the matrix of size

13 x 26, whose rows are linearly independent incidence vectors of 6-cliques

~n C, hence maximal cliques, which implies that the incidence vectors are

eigenvectors of C (cf. theorem 2.4.2), and then we may apply theorem 2.4.5,

hence C 51 - 10NT(NNT)-I N•

If N is the incidence matrix of a Steiner triple system on 13 symbols,

then the 13 rows of N are linearly independent incidence vectors of 6-cli­

ques in the corresponding conference matrix of order 26.

Again we may apply theorem 2.4.5.

4.4. Regular conference matrices of order 26

It will turn out (theorems 4.4.2, 4.4.3 and 4.4.4), that any conference

matrix out of any of the four equivalence classes 8tl, StII, LSI and LSII

is equivalent to its complement. If C ~s a conference matrix out of one of

these classes, then the determination of {2. E .e;6 I C2. ~ 5~} yields the

complement of the regular conference matrices, which can be found by the de­

termination of {2. E ID;6 I Cd = -5d}. For this reason we may restrict our­

selves to the determination of {2. E ID;6 I C2. = -52.}.

The procedure Eigenvector (section 5.2) just determines

for some integer k, which is determined by the procedure itself.

The result is

I(~ E ~;6 I Cd = -5d 1\ dk = 1}I =

14 for StI ,

14 for StU,

130 for LSI ,

30 for LSIl ,

by taking for C the matrices as con~tructed in section 4.2.

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If Cd = -S~ and P E J(C). then CP~ = -SP~, and Since Pd is not neces­

sarily equal to~. it may happen that two distinct vectors ~ yield isomor­

phic regular conference matrices. Therefore for any of the )4 regular solu­

tions of StI and StII the corresponding standard matrix is computed, and

then, by comparing these standard matrices, all nonisomorphic regular solu­

tions are found.

This could also be done for LSI and LSI1. However, in View of the exe­

cution time, needed for the computation of any standard matrix (for a regu­

lar solution out of LSI for instance 6-7 minutes), another strategy is pur­

sued:

By use of the procedure permutatie standaard it is possible not only to com­

pute standard matrices, but also several automorphisms of these standard

matrices, and of the original matrices. Now suppose Cd = -S~ and P E eJ.>(C) ,

then CPd = -SPd. Therefore it is possible to reduce

by use of some elements of ry(C) without loosing any class of isomorphic re­

gular conference matrices.

Indeed, if C~) = -S~I and C~2 = -5~2 and P E ~(C) such that ~I = P~2' while

D)i = 21 and Dzi = ~Z (D 1,DZ E~Z6)' then D1eD 1 and DZCDZ are isomorphic

regular conference matrices.

After this reduction, for the remaining regular solutions the corresponding

standard matrices are computed and compared to eachother.

In this way we find the following

4.4.1. Theorem. There exist Z classes of isomorphic regular conference matrices in

StI, with standard matrices Stl,1 and Stl,Z, 5 classes of isomor­

phic regular conference matrices in StII. with standard matrices

St2,1, St2,2, StZ,3, StZ,4 and StZ,S, 1 class of isomorphic regu·

lar conference matrices in LSI, with standard matrix LSI,I and Z

classes of isomorphic regular conference matrices in LSI1, with

standard matrices L82,1 and L52,2.

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for i,j E {1 ,2, ••• ,25}.oa. ,ca.~ J

Now let c be a nonsquare in GF(25). Then ca2 is a nons quare for any

a E GF(25)\{oL

Define P E 'J.l25

by P. . ;=~,]

Then

T(P SP). .

~,J

x(ca. - ca.) =~ J

x(c)x(a. - a.) =~ J

= -x(a. - a.)~ J

-8 ..~,]

If

then E E ~26 and ETCE = -C, which implies the theorem.

4.4.3. Theorem. Any conference matrix out of StI is equivalent to its complement.

Proof. (from J. M. Goethals [private conununication J) •

Take N = [NIIN2J as in section 4.2. This yields the conference matrix

where

A circul(O,1 ,-1,1,1,-1,-1,-1,-1,1,1,-1,-1) and

B circul(l, 1,-1,1,-1,-1,-1,-1,-1,1,-1,-1,-1)

Define P E ~13 by Pi,i+l = 1 for i = 1,2, ••• ,12 and P 13 ,1 = 1, then

12I and I pJ

j=OJ (pO ;= I) •

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Then we have

and

Now define Q E3J13

by

Q. . : = 82 , 1 . for I ~ i ~ 7 and I ~ j ~ 13 and~,J ~- ,J

Qi,j := 82i- 14 ,j for 8 ~ ~ ~ 13 and I ~ J ~ 13 •

, f 11 h QTAQ d 2T 2 T f' e b~t 0 ows t at = -A an Q BQ = B • Now de ~ne E E ~26 Y

E :- ~2 -:2J

12

Lj=O

pJ = J

Tthen E CE -C, which implies the theorem.

4.~.4. Theorem. Any conference matrix out of StII or LSII is equivalent to its com­

plement.

Proof. Let C be an element out of LSII, and suppose C~ = 5~, where ~ E~;6'

Define D E ~26 by Di = ~, then the standard matrix, corresponding to -DCD

turns out to be equal to LS2, 1 or to LS,2,2. This can be done by a computer­

search, using the procedures Eigenvector and permutatie standaard. Therefore

any conference matrix out of LSII is equivalent to its complement.

The same can be done for StII, which implies the theorem.

The 10 standard matrices A are listed in chapter 6, together with their

representations ~(S).

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4.5. Cliques

A suitable test on the correctness of the procedure permutatie stan­

daard is the computation of the matrices K(k) for k ~ 4, since, for a re­

gular conference matrix C of order 26, it follows from 2.3.4 that

rfor C.

~,J

(K (3) (C». . for C. = -1~,] 1,J

60 for i = j .From 2.4.4 it follows that the order of a clique in C26 cannot exceed 6.

Therefore we compute K(k) for 4 ~ k ~ 6, by use of the procedure Clique­

structure (section 5.3). It will turn out that, for the 10 standard ma­

trices of 4.4 to be nonisomorphic, it is sufficient to consider the main

diagonal of K(k) for each of these 10 matrices. (In fact K(5) yields suf­

ficient information.)

We calculate the frequency of any value of the diagonal elements of K(k) ,

and, in addition, the total number of k-cliques, which is equal to26 (k)I K. . /k (see section 2.3).

i=l ~,~

Then we have the table designed on page 36.

St2,1 is an interesting case, which is formulated in the following

4.5.1. Theorem. There exists a regular conference matrix of order 26 with no other

automorphism than the identity, having the property that all ver­

tices are in exactly 70 cliques of order 4.

Proof. See the table on page Jb, and the table 1n section 4.4, where the nu­

merical results are listed.

There is just one more conference matrix (St2,3) having the property that

all vertices are in the same number of cliques of order 4, but this matrix

has an automorphism group of order 2. Other matrices, of which one could

expect to be "more regular" (greater automorphism groups), lack this proper-

ty.

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If one considers the whole matrix K(k) (instead of the main diagonal)

then one can attach to each of the 26 vertices some more cliquenumbers.

Using the fact that P E ~(A) implies P E P(K(k) (A», and applying this to

St2, I, we can conclude that the order of Y (St2, I) cannot exceed 4, since

there are just two distinct pairs of vertices in St2,I, each pair having

the property that both vertices have the same set of cliquenumbers. From

these four candidate automorphisms it can easily be verified that only the

identity isa real automorphism of St2,1. This illustrates the fact, that

the clique matrices K(k) do not yield sufficient information about a graph,

a fact which is endorsed by the following section too.

4.6. Strongly regular graphs of order 25

Let C be a conference matrix of order 26, and define D E~26 by

D I ,I : = I and D. . : = CI . for 2 ~ i ~ 26 •1.,1. ,1.

Then (DCD). I1,

Therefore

(DCD) I . = I for 2 ~ i ~ 26, hence vertex I lS isolated.,1

fo .TJDCD- ~ ~

where S is a symmetric matrix of order 25, wit4 0 on its diagonal and +

elsewhere~ satisfying

S2 = 251 - J and SJ "" O.

Hence S is the adjacency matrix of a strongly regular graph with parameters

(eigenvalues) PI = 5, Pz = -5 and Po O.T

If C' = P CP for any P E ~Z6~ then the same procedure yields a matrix S'

which is not necessarily isomorphic to S:

Isolating two distinct vertices may yield two nonisomorphic strongly

regular graphs of order 25.

On the other hand, if P E <?(C), and 1T E S{) such that P. . = <5. (.)' then" 1,] 1,n J

isolating i and 7T(i) yields two isomorphic 25-matrices.

A complete survey of all nonisomorphic strongly regular graphs of order

25, derivable from the four equivalence classes StI, StII, LSI and LSII, is

found by a computersearch.

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- 38 -

first several automorphisms of a representative of a class are determined

by the procedure permutatie standaard. This yields a restriction on the

number of vertices which have to be isolated. Then the remaining vertices

are isolated, and for any 25-matrix the corresponding standard matrix is

computed. Comparing these standard matrices it is possible to find all

nonisomorphic solutions for 25-matrices, derived from the four equivalence

classes of 4.2.

From this computersearch we have

4.6.1. Theorem. The number of pairwise nonisomorphic strongly regular graphs of

order 25, derivable from any of the four equivalence classes of

4.2 equals 1 for LSI, 2 for StI, 4 for LSII and 8 for StII.

For all of these 15 graphs the corresponding standard matrices S and their

representation ~(S) are listed in chapter 6.

If S is such a strongly regular graph of order 25, then SJ = 0, hence

it may be possible that S ~ -So This is formulated in the following

4.6.2. Theorem. The strongly regular graph of order 25, derivable from LSI is iso­

morphic to its complement.

The three sets of strongly regular graphs of order 25, derivable

from StI, LSII and StII are closed against complementation, but

none of these graphs is isomorphic to its complement.

Proof. For any of the 15 standard matrices S, the standard matrix, corres­

ponding to -S, is co~;uted. The results of this computation imply the theorem.

As 1n section 4.5, the cliquematrices K(k) are determined; in V1ew of

theorem 2.3.7 we may restrict ourselves to k ~ 5, and in view of theorem

2.3.4 we may restrict ourselves to k ~ 4, hence 4 $ k $ 5. Then we have the

following clique table (concerning the diagonal elements of K(4) and K(5)):

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total number of vertices, total number

which are in x cliques of order k. of k-c11ques

k 4 5 4 5

x 10 11 12 13 14 15 16 0 1 3

81 25 25 75 15

82 15 10 10 15 90 3

53 15 10 10 15 90 3

84 3 3 18 1 12 12 1 73 3

85 12 12 1 25 79 15

86 3 3 18 1 12 12 1 73 3

87 12 12 , 12 12 1 79 3

s8 2 15 8 10 15 89 3

89 2 15 8 10 15 89 3

810 2 2 17 4 10 15 87 3

811 1 6 9 3 6 10 15 83 3

812 1 17 7 10 15 89 3

813 2 2 17 7 10 15 87 3

814 1 17 7 10 15 89 3

815 1 6 9 3 6 10 15 83 3

From this table it follows that each of the following pairs may be ~somor­

phic:

{S2,S3}, {S4,S6}, {S8,S9}, {SIO,SI3}, {SII,SIS} and {SI2,SI4} •

By considering the composition of the rows of the cliquematrices K(4) it

follows that, except for the pairs {S2,S3} and {SI2,SI4}, all pairs are

nonisomorphic. F.e. Bussemaker proved that S2 and 83, respectively 812 and

814 are indeed nonisomorphic.

4.7. Remark.

The results in this chapter mainly correspond to the results of

5.S. Shrikhande and Vasanti N. Bhat ([ 10J and [IIJ), except for the equiva­

lence class StII.

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The graphs of order 26 in [IOJ are related with the conference matri­

ces of this chapter as follows:

G*' = LS2 I2 '

H) , I = H2, 1 = St ) , Z •

Apparently HI,I should have been nonisomorphic to HZ,I. However, it was

not; indeed, upon closer examination S.S. Shrikhande informed us of the fact

that the two Steiner triple systems D} and DZ «(IIJ, pages 13,14) are iso­

morphic, which implies the relation HI,I ; HZ,I.

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5. Algol procedures

5. I. Introduction

In this chapter several procedures on the investigation of graphs are

described, and an Algol text of these procedures is given.

The procedure Eigenvector is described in section 5.2; this procedure

1S used to find all regular graphs within a given equivalence class of

graphs.

Section 5.3 deals with the cliquestructure of a given graph. Several

characteristic matrices can be determined by the procedure Cliquestructure.

The procedures Decrep and Invdecrep are given in section 5.4. They deal

with the representation of a graph.

The procedure permutatie standaard is given in section 5.5. For the ad­

jacency matrix of a graph the corresponding standard matrix is determined

by this procedure.

Finally, in section 5.6, the Algol text of the foregoing procedures 1S

given.

5.2. The procedure Eigenvector

If A is a square matrix of order n, with integer elements, and if p is

an integer eigenvalue of A, with multiplicity ~, say, then the procedure

determines an integer k, satisfying 1 ~ k ~ n, and then all eigenvectors d

of A, satisfying Ad p~, d[kJ = I and d[iJ = +1 or -1 for 1 ~ i ~ n, are

determined.

5.2.1. Formal parameters

integer n

integer array A:

integer ro

<expression>; denotes the order of the matrix A.

array of dimensions [1:n,l:n]; contains the elements of

the matrix A. At call, the elements of A are integers; on

exit of the procedure, A has the form [~ ~J, the elements

of A and R being integers; A is a diagonal matrix of order

n - ~.

<expression>; denotes the integer eigenvalue p of A.

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Boolean pr

Boolean pu

Eigenvector

5.2.2. Explanation

- 42 -

<Boolean expression>; if pr =~ then the computed e~gen­

vectors of A are printed, and, in addition, their total

number.

<Boolean expression>; if pu =~ then the computed eigen­

vectors of A are punched, and, in addition, the integer

zero is punched, to indicate the end of the tape.

on exit of the procedure the value of the functiondesigna­

tor is equal to the multiplicity ~ of p.

If A is a square matrix of order n, with integer elements, and if p is

an integer eigenvalue of A with multiplicity ~, say, then A - pI has rank

n - ~. Therefore one can take n - ~ linearly independent rows of A - pI, which

form a matrix B of size (n - ~) x n and rank n - ~, and then we have

(A - pI).! == .Q. iff Bx == a

for any vector ~ of length n.

Clearly B only has integer elements.

Now it is possible to choose n - ~ linearly independent columns in B, so there

exists a matrix Q} E ~n such that BQ} == [XIYJ, where X is a nonsingular ma­

trix of size (n - ~) x (n - ~), and Y is a matrix of size (n - ~) x ~. Of

course X and Y only have integer elements. Therefore X-I only has rational

elements, which implies

where X-Iy only has rational elements. Now one can choose a diagonal matrix

A of order n - ~ with positive diagonal elements, such that R' :== AX-1y only

has integer elements. (R' is a matrix of size (n - ~) x ~). This A is uni­

quely determined by requiring

gcd (A. ., R~ l' R~ 2"'" R~ ) == I (greatest common divisor)1,1 1, 1, 1,~

-} Ifer I 5 1 ~ n - ~. Then we have AX BQj

== [A R'J.

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Finally one can choose the matrix Q3

E p~ such that R := R'Q3 satisfies the

inequality

n-~ n-~

I IR. ·1 ~ L IR. R, Ii=1 ~ ,J i=1 ~,

for ~ j < R, :S j.l.

If

-I Ithen AX BQIQZ = [A RJ.

By taking P := Q;Q~ it follows that AX-IB = [AIRJP.

Since A and X-I are nonsingular we have

Bx = 0 iff [A IRJP~ = .2., or

(A - pI)~ = .2. iff [AIRJP~ = Q '

where A is a diagonal matrix of order n - J.l, R is a matrix of s~ze

(n - ~) x J.l and P E Ij) , satisfyingn

a) A. . are positive integers for 1 ~ i ~ n - j.l.l,~

b) R. . are integers for 1 ~ i ~ n - j.l and I ~ j ~ ~.1.']

c) gcd(J\ .. , R. I, ••• ,R. ) = 1 for 1 ~ i ~ n - j.l.~,l 1., ~,~

n-~ n-~

d) L IR. . I ~ L IR. R, I for 1 ~ j < Q. ~ ~.i=1 1,J i=1 ~,

Now letJ:)* be the set of vectors of length n with components equal to +1 orn-I. Clearly we have5Y* = {Pd I d E.i)*} for any P;;:tJ> • So, if we want to

n - - n ndetermine

i'I' (p) : = {d E JJ* j Ad = pd} ,n - n - -

then it is sufficient to determine

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since 8,' (p) =n

If [A IRJ~ = 0

- 44 -

{pTdl d E 8, (p)}.- - n

*for any d E.tJ , then- n

A•. d. +1.,1. ~

]l

Lj=l

R.• d • = 0~ ,J n-]l+J

for 1 sis n - )1, which implies

t ]J

I R. .d . -A. .d. - I R. .dn-)1+jj=1 l.,] n-J.l+J 1,l. 1. j=t+ I 1.,]

for 1 s i s n - ]J and for S t S lJ, hence

t

I Lj=l

)l

Lj=t+1

/R.. 1~.J

(since A. . > 0)~,~

for lsi ~ n - ]l and for I ~ t S lJ.

Since the right hand side of this inequality is independent of ~, this can

be used to determine ~ (p) by use of the backtracking algorithm (cf. chap-n

ter 3).

[A1RJ~ = Q implies [AIRJ(-~) = Q, and therefore we restrict ourselves to

*~ (p) : =n

{d E JJ*n

[AIRJd = 0 A d = I} •- - -n-lJ+1

Together with ~'(p) = {pTd IdE ~ (p)} this explains the presence of then - n

integer k at the beginning of this section.

Remark. Restrictions a) and b) are made to avoid rounding errors during the

computation, restriction c) is made to avoid large numbers during the com­

putation, and restriction d) is made to accelerate the backtracking process

(cf. section 3.5).

5.3. The procedure Cliquestructure

If A is the (-1,0,1) adjacency matrix of a graph G of order n, then,

for any pair of vertices (i,j) of G, the procedure determines the number of

m-cliques, containing this pair. These numbers are placed in the array K.

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5.3. I. Formal parameters

integer n

integer array A

integer m

integer array K

Cliquestructure

5.3.2. Explanation

<expression>; denotes the order of the matrices A and K.

array of dimensions [I :n, 1:n]; contains the elements of

the square matrix A. At call, A is the (-1,0,1) adjacency

matrix of a graph G of order n.

<expression>; denotes the order of the cliques which have

to be counted.

array of dimensions [l:n,l:n]; contains the elements of

the square matrix K. On exit of the procedure, K[i,j] is

equal to the number of m-cliques in the graph G, contain­

ing the vertices i and J.

on exit of the procedure the value of the functiondesigna­

tor is equal to the total number of ~cliques in the

graph G.

The procedure 1S based on the backtracking algorithm (see chapter 3).

First all elements of K are made equal to zero. If the vertices of the

graph G are enumerated by 1,Z, •.• ,n, then a general step in the algorithm

is as follows.

Let Vk = (x 1,xZ ' ••• ,~) with XI < Xz < ••• < ~ be the vertex set of a com­

plete sub graph (k-clique) in G, of order k < m. We try to extend Vk by a

vertex out of the candidate set

Ck + 1 = {~+ 1, ~ + 2, ••• ,n}.

To that end the consecutive vertices x of Ck

+1

are tested on the condition

A[x. ,x] = -1 for all j E {l ,2, ... ,k}. (A is the (-1,0, I) adjacency matrixJ

of G.)

As soon as k becomes equal to m, the elements of K, corresponding to this

~clique, are increased by one:

K[ x. , x. J := K[ x. ,x.] := K[ x. , x. J + IJ 1. 1.J 1.J

tor I ~ 1. S j ~ m.

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From section 2.3 we know that the total number of m-cliques in G equals

nI K[i,iJ, which can be enumerated at the end of the procedure.

m i= I

A corresponding algorithm has been published ~n [2J, chapter II.

5.4. The procedures Decrep and Invdecrep

If A is the (-1,0,1) adjacency matrix of a graph G of order n, if b ~s

a vector of length n - 1, and if A and b satisfy the equality

nb[iJ = I 2n- j - 1(1 - A[i,jJ)

j=i+1for 1 ~ i ~ n - I,

then the procedure Deerep determines b if A is given and the procedure

Invdecrep determines A if b .is given. (See section 2.2.11.)

5.4. I. Formal parameters

integer n

integer array A

integer array k

<expression>; denotes the order of A.

array of dimensions [1:n,l:nJ; contains the elements of

the symmetric matrix A. At call of the procedure Decrep

and on exit of the procedure Inv3ecrep, A is the (-1,0,1)

adjacency matrix of a graph.

array of dimension [l:n-IJ; contains the components of the

vector £; at call of the procedure Invdecrep and on exit

of the procedure Decrep, £ is the decimal representation

of the (-1,0,1) adjacency matrix A of a graph.

5.5. The procedure permutatie standaard

If A is the (-1,0,1) adjacency matrix of a graph G of order n, then

this procedure determines the standard matrix S, corresponding to G. (See

section 2.2.12). The permutation matrix, which transforms A into S, is re­

presented by the vector g.

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- 47-

5.5. I. Formal parameters

integer orde

integer array matrix

<expression> ; denotes the order n of the matrices

A and s· is called by value.,array of dimensions [1:n,l:n); contains the ele-

ments of the symmetric matrix A; at call, A is the

integer array standaard

(-1,0,1) adjacency matrix of a graph G of order n.

array of dimensions [1:n,l:n); contains the ele­

ments of the symmetric matrix S; on exit of the

procedure, the upper triangular elements of S are

the elements of the standard matrix, corresponding

to A.

integer array permutatie: array of dimension [I:n); contains the components

of the vector ~; on exit of the procedure, ~ re­

presents the permutation matrix which transforms

A into S.

5.5.2. Explanation

The procedure is based on the backtracking algorithm (chapter 3). For

details about the structure of the procedure we refer to a report by

F.e. Bussemaker, which is to appear. We just want to remark that the local

array perm can be used to generate several automorphisms of the standard

matrix (private communication with F.C. Bussemaker).

5.6. The Algol text

An Algol text of the foregoing procedures is given on the following

pages.

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- 48 -

inteGer Procedure Eie;envector (n ,1\ , ro , pr , pu);

inteber n, raj inLeger arra;r Aj Boolean pr, pu;

begin comment

If A is 1'1. square matrix of order n with integer elements only, Hnd if

ro is an integer eigenvalue of A. then this procedure detennines an

integer k, s~tisfying 1 < k < n. am then all eigenvectors d of 1\.- -satisfying Ad .. rod. d[k] "" 1 and d[ i J = +1 or -1 for 1 ~ i :::: n, are

deteI'1llined, If 'Or = true then these eigenvectors are printed, am, in- -

addition. their total number, If pu =~ then they are punched, and,

in additio::l, the integer zero is PtUlChed to indicate the end of the

tape, On exit of the procedure, the value of the f\mctiondesig~tor

Eigenvector is equal to the multiplicity of the eigemmlue ro;

inte6er r.i.j.k,l,xl.x2,x3,x4,x5;

inte&er array p,d[ 1: n], B1.B2[ 1: n. 1: n];

integer procedure gcd(a, b ); value a I b; integer a I b;

begin comment

If' a x b =I 0 then the f'unctiornsignator gcd becomes equal to

the greatest conmon divisor of' a ani b, Ifax b = 0 then it

becomes equal to a_bs(n) + nbs(b)j

integer x,yj

x := abaCa); y := abs(b);

1.1: 11 x y y "" 0~ goto 12;

.!! x < y~ y := y - x x (y ~ x)~ x := x - y x (x ~ y)j

goto V;

12 : gcd: "" X + Y

~ gcd;

orocedure rowpennutation(a.,b)j value n,b; inte~r a,bj

~egin con:ment

The rows a and b at the :l!lltrix P, are exchanged;

integer j, x;

for j := 1 step 1 until n do- -begin x := A[a.,j]; A[a.,j] := A[b,j); A[b,j) :"" x errl.

~ rowpermuta.tion;

procedure cOlumnpermutation(a.,b)j value a,b; inte5er a,b;

'be fi in COl!illent

7he colcunns a and b of the matrix A a.re exchanged. This exchP-nge

1s stored up in the arra.y p[l:n);

Page 55: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 49 -

integer i, x;

!s?!: i :::: 1 step 1 until n 22begin x := /\[1.8.]; A[1.fl.] := A[i,b]; A[i,b] :'" x end;

x ::::: p[ f1]; p[ a] :::.: p[ b ]; p[ b] : == x

!E!.!! columnpermutatlonj

!2£ i := 1 ste.e 1 until n !!2be6in p[i] := I; A[i,iJ := A[i,1] - ro end;

r == 0;

L1: xl:= 0;

!2r j := r + 1

begin 1".2£ i :""

begin x?

s.te;e 1 until n ~

r + 1 stet> 1 until n do--- -::= s,bs (A [ i, j ]) ;

:= - 11, [r. j 1 end;--

12:

if '/2 > 0 then- -begin if x2::: ~ begin xl ;= 1; 1= := ij 5otJO 12 ~;

1f xl ::: 0 V xl :> x2~ begin xl .= x2; k := i end

end'-'!! xl > ()~ gete 12

em'-'&oto L3;

r ;: r + 1;

if k:f r then rowpermutation(k.r)j- -g j .. r ~ columnpermutation(j.r);

if P.[r,r] < 0 then-.. ----.

befjin!2!. j := r stev 1~ n ~ A[r, .j]

for i := 1 sten 1 '~til n do.......... -----.........-.... ......begin 1! i '" r ~

begin x2 := A[i.r];

if x2 =f 0 then- -begin x3 := gcd(xl,x2);

x4 := xl/X); x5 := x2 / X)j

l! i < r ~ A[l.i] :== A[i.i] x x4;

.f2!: j := r st:,e,E 1 unt i1 n ~

A[ i, j] := x4 )/ A[ i, jJ - x5 y [, [r, j ] ;

x3 := if i < r then A[ L i] else 0;- - -for J := r + 1 steu 1 until n do- - -

Page 56: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 50 -

begin xh := fI [ i, j ] ;

if x4 ~ 0 then x3 :"" gcd(x),x1+)- -1:.! x3 > 1~

begin 1! i < r ~ A(i,i] := A[i,i] / x3;

.f2t j ::: r + 1 st,ep 1~ n ~

P[i,j] := .t.,[i,jJ / ::<:3

end-end-em

goto U;

L3: xl:= r + 1; x2 := r + 2;

!2t j :== xl s,te,p 1 ~ntil n ~ d( j) := OJ

~ i 1 step 1~ r £.<?!2!. j ;= xl step 1 until n S2. (1.[ j] := d[ j] + ahs(A[i.j]);

for i :== n steu - 1 until x2 do- -- -begin x3 . "" d[ 1]; x4 := i;

!2!: j : == i - 1 step - 1 unt 11 xl 2ebegin.!! x3 ';> el[ ,j]~ 'begin x3 := d[ j); x4 := j ~ end;

if' x4 :f i then- -bcgip d[xl.l-] := d[i]; d[lJ := x3; columnpermutation(i •.xl..:..) ~

end;

~ i :=

f2r .1. :~

!2!: { .:::c

step

steD-sten-.

Wltil n do d[D[i]] := ij- .until n ~ p[ i] := dL;"J;

until r E2 B2[ l,n] := Mi, 1];

!2!: J .::: n step - 1 until .x2 ~ !.2!: i := 1 step

B2[i,j - 1] := B2[i,j] + abs(A[i,j]);

k := 0; d[ xl J := 1;

for i := 1 steT> 1 until r do B1[i,xl] := A[ l,x'];- --- -i! .0\1~ '\:)e5in RUNOUI'; RUNUCT em;

if xl = n then- -~iJl !2!. i : = 1 ste,2 1 until r E£.

qe€jin .x3 := A( LnJ;

1! abs (x3) ;f 132 ( i, n]~ &oto 1.6;

d[i] := if x3 < 0 then 1 else - 1- --end'-'

until r do- -

Page 57: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 51 -

it pr then- -beeain!2t 1 :- 1 8~e;rz 1 until n S2 FDCT(2,O,d[p[1]]};

N.I£R

if pu then- -begin!2!: i := 1 steE 1 until n S2 FIXP(2,O,d.[p[i]]);

PUNLCR

em;

k := 1; ~ 16

em'-'J := x2; d[ j] := 1; €joto L5;

I1+: if d[ j] - - 1 then- -begin .!! J = x2~ goto 16;

J := j - 1; soto I.hend'-'d[ j] := - 1;

L5: x3:= d[ j]j

if j = n then- -be€jin !2t i := 1 ste;rz 1 UlIt11 r ~

begin x4 := B1[i.j - 1] + x3 x A[1,.)];

.!! fibs(x4) =J B2[ i, j]~ e;oto Wjd[ i] : ... if x4 < 0 then 1 el.se - 1- --

end;

if' pr then- -beCS1n!2!: i :::;: 1 step 1 until n 22 FDCT(2,O,d[p[1]]);

NLCR

if pu then- -be61n!2t 1 :- 1 step 1 until n S!2. FDCP(2,O,d[p[i]]);

PUNLCR

end'-'k := k + 1; soto rJ.t

em else--bea1n !2!. i : = 1 ste;rz 1 until r ~

b,tz&11,!::<4 := Bl[ t, j - l} + x3 X A[ 1, j];

.!! abs(x4) > B2[i, jJ~~ 14;Bl[i.J] := x4

Page 58: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 52 -

end;

J : III j + 1; d[ J] := 1; goto L5

~l;

r£J : if pr then- -be&~ CARRIAGE(2); PRINl'I'EXT({ number of eigenvectors equals:i-);

ABSFlXT (8,0, k); CARRIAGE (2);

PRINl'TEJ..'T(~multiplicity of roequals:f.); ABSFlXT(2,O,n - r)eM·-'if pu then- -be&1nRUllJTJr; ABSFlXP(2.0,O); PUNLCR; RtmOtJr; RUNDUr end;

Eigenvector := n - r

end Eigenvector;-

Page 59: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 53 -

inte§er Erocedure Cl1questructure(n,A,m,K);

integer n,m; inte§er array A,K;

be§in £anment

If A is the (-1,0,1) adjacency matrix at a graph G at order n, then

the procedure canputes the number at m-cliques in G, containing a

given pair at vertices. Doing this for all pairs, these. numbers are

placed in the array K;

inte§er i,j,k,l,c; integer arral cl[l ~];

for i := 1 steE 1 until n do for j := i steE 1 until n do K[i,j] :=0;- - - - ----- ----k := 1; 1 := 0;

L1 : if 1 = n then- -begin!!. k = 1~ goto 12;

k := k - 1; 1 := cl[k]; gato L1

end;-1 := 1 + 1;

!.2!' j := k - 1 step - ,~ 1 ~

begin!! A[cl[j ],11 = 1~ gato Ll ~;

cl[k] := 1;

if'.k =m then- -be§1n ~,I := 1 steE 1 until m~

begIn c := cl[i];

!.S:. j := i step 1 untI! m 22 K[c,cl£j]] := K[c,cl[j]] + 1

em;-§oto Ll

end;-12:

k := k + 1; §ato Ll;

c := K[l,1]; ,

~.i := 2 steE 1~ n ~

be§in c := c + K[i,i];

~ j := i - 1 steE -

end;---Cl1questructure := c / m

end Cliquestructure ;-

Page 60: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 54 -

Er2cedure Decrep(n.A,b)j integer nj integer arral A,b;

be5in inteSjer i, jj

!2!: i : = n - 1 ste:e - 1 unt11 1 ~

b[i] := if A[t,i + 1] = 1 then 0 else 1;- --!2!: j ::: 2 ste:e 1 until n 2:2. !2! i: = j - 2 steR - 1

b[i] := 2 x b[t] + (if A[i,j] = 1 then 0 else 1)- --em Decrep;-

until 1 do-

J2?::ocedu;re Illvdecrep(n, A,b); integer nj integer array A, b;

begin inte&er i, j, xi integer anal. c[ 1: n - 1].;

A[n,n] := OJ

~ i ; = n - 1 step - 1 unt11 1 do

begin A[ i, i J := OJ c[ i] := b[i] end;

~ J := n steJ2 - 1 until 2 E-2 !2!: i := J - 1 until 1 S£.begin x := crt] ~ 2;

A[ 1, j] := if 2 x x = c[ i] then 1 else - 1.;- --c[t] := x

!2!: i := 2 step 1 until n 9a. !2r. j := i - 1 steR - 1 until 1 E:2.A[ L j] := A[ j, i]

end Invdecrepj-

Page 61: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

.!'l'Ocedur;,.pemutatie stan;1.a.a.rd, (orde, nf3.trix, permutatle, standaard);~ orde;

integer orde; inteW.r ~~;:,: nTttrix, pemutatle, standaard;

beESin .~::tE~t i, J,ordel ,teverspringeni; Boolean eqw;tl,vergelijken;

.!.!l\;"'\?ier an'ay perm[ 1: orde ] •groep[ 1: orde, 0: orde J;

&2c.e~we;, permutatie onderzoek (i,groepstal,groep); value i,groepstal;

inteil~r i,groepstal; integer arra.;:,: groep;

begin integer dim,groroe,grt8.l,hvl ,hv2,hv3,hv4 ,hv5, j,k,kandtal, 1, nummer,nwkatal,Qudkatal,

rij.teller,tellerl,teller2j

integer array array1 , array2, kani, st[ 1: orde ], gr[ 1: orde, 0: orde h Boolean array niettrans[ 1: orde );

l?£ocedure kaniidaatsselectle (oudka:tal, nwkatal, kand, o:rde, array, label); value o:rdej

intef5er ou.dka.tal, nwkatal,orde; inteeaer arra;:,: kam,array; label label;

begin irrteeaer hv, j,l{..maxmintal,mintal;

naxmintal := -1;

!2!: j := 1 step 1 until oudkatal £2­begin hv := kT:ind[ j ]; mintal := 0;

!2r. k := 1 step 1 until orde ~ .:!! ll1ltrix[hv,array[k]] = -1~ minta.l := mintal + 1;

1! mintal > nfixm1ntal~ begin nwkatal : = 1; ktind[ 1 J :'" hv; ~intal :=mintal~

~ !! mintal '" ~)CID.intal ~ be&in nwkatal := nwkatal + t; !{8nd[mlkatal] := hv~

em;-1! mrkate.l = 1~ goto label;

oudkate.l := mrka.tal

end kandldaatsselectie;-nwkate.l := oudkatal := groep[ 1,0];

!2r. J := 1 step 1 until oud.k1ital !!2 kam[ J] := arrayl[ j] := groep[ 1, j];

\Jl\Jl

Page 62: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

if oudkatal =+ 1 then- -begin.!! oud1<l:ita.l =+ 2~ lqiItl.1daatsselect1e (Olldkatal,nwkntal,kand, ou.dk1ltal,ar.rayl ,kandtest);

!2r J := 2 step 1 until groepstal ~

begin dim := groep[ j,o]; !£!: k := 1 step 1 until dim ~ arrayl[k) := groep[ j,k]j

kandidaa.taselect1e (oudkatal, nwka:tal, 103.nd,dim, a.rrayl , kandtest)

end·~

nwk!l:tal : = 0;

!2!. j := 1 step 1 until oud.katal ~ niettrans[ J] : = true;

for j := 1 sten 1 until oudkate.l do if niettre.11$[ jJ then~. ~ -..--. --begin nwkntal := nwkJ1tal + 1; hV1 := kani[mrka.tal] := kani[ J]; l1V'2 := hvl - 1;

!s: k := j + 1 step 1 until o~ta.l ~ .!! niettrans[ k]~

begin hv3 := kand[k]j hv4.:= hv3 - ,;

!2!: 1 := 1 ,step 1 until hV2, hvl + 1 step 1 until hv4,

hv3 + 1 step 1 until o:t'de !!2..!! llJ9.trix[hv1,l] .. Ilfltr1x[hv3,1]~~ doorj

niettrs.ns [ k] : = false.;

door: em.-end-

em;-kandtest: kS.nitB.l := nwkatal;

!2!: l1UllII1er := 1 step 1 until kanitB.l 22be6in perm[i J := rij := kand[~r];

grtal := OJ

!2t j ::: .1 step 1 until groepstal 22begin teller1 := teller2 :'" 0; grorde := groep[ j,Ol;

!2.t k := 1 ste;e 1 until grorde 22begin hv1 := groep[ j,kJ; h'v2 := n»3.tr1x[rij,hv1];

.!! l1V'2 • -1~ begin tellerl : = tellerl + 1 j arrayl [tellerl J := hvl ~

\Jl0'

Page 63: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

1 j gr[ grtal, 0] := teller1 j

Wltil tellerl do gr[grtal,k] := arrayl[k]-

~ g hv2 = 1~ be6,in teller2 := teller2 + 1; array2[teller2] := ltv1~

em"-'if' teller1 =4 0 then- -begin grtal := grtal +

!2t k ::; 1 ste:e 1

el¥i;

if' teller2 =+ 0 then- -begin e;rtal := grtal + 1; gr[grtal,O] := teller2;

!2!: k := 1 step 1 until teller2 ~ gr[grtal,k] := array2[kJ

end-el¥1"~

teller := i;

!2:: J := 1 step 1 until grtal ~

begin grorde := gr[j,Ol; hvl := Ultrix[rij,gr[j,l Jh

!2!. k := 1- ste:e 1 until grorde 9:2 begin teller := teller + 1; st[ teller] : = ltv1 el¥1-V1......

el¥1"='if' vergelijken then- -begin !2!: j := i + 1 ste:e 1 unt11 orde S2

begin hv1 := st[ j ]; hv2: = star.rlaard[ i, j J;if hvl < hv2 then- -begin !2t k := j step 1 until orde 2£ standaard( i,k] := st[k];

~ k := 1 ste;p 1 until i !!2 permutatie[k] := perm[k];

vergelijken := false; ~ af\lerken

end-~ .!! hvl > hv2~ begin equal := false; goto klaar !:!!!

em."-'equal : = true-

Page 64: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

eni-else begin 1'2r. j := i + 1 step 1 until orde S£. stB.ndaard.[i,j] := st[ J); perlllltatie[i] := riJ end;

at\re'!:'ken: if 1 = arde - 2 then- -begin!! 1 vergel1Jken~

begin lwl := gr[1 , 1 J; hv2 :=.!! grtal = 1~ gr[ 1,2]~ gr[ 2,1 ];

permutatie[ ordel] := hvlj permuts:t1e[ orde] := hv2;

statdae.'rd.[ ordel ,orde J := Dfltr1x[hvl,hv2];

vergelijken := true; equal := false- -em else--begin k := 0; 1 :== 1; !2!: k := k + 1 while perm[k] = permutatie[k) ~ 1 := k + 1;

teverspringeni := 1; .!! i > 1~~ einde

em.;

~ klear

end;-if i = orde - grtal then- -beSin hvl := grtal - 1;

!2t J :"" 1 step 1 until hvl ~

beSin hv2 := i + Ji hv3 := gr[ j, 1].; perm[hv2) := hv3;

!2!: k := j + 1 st2J2 1 until grtal S£ st[i+k) := DJatrix[hv3,gr[k,1));

if verge1iJken then- _.beSin !2£ k := hV'2 + 1 step 1 until arde £2

begin hv4 := st[k]; hv5 := stan:leard[hv2,k];

if hv4 < hv5 then- -beain !2.t 1 := k step 1 until orne ~ staniBard[hv2,l] := st[ 1];

for 1 := 1 ste;e 1 until hv2 ~ permutatie[ 1] := perm[ 1);

vergelijken := false-end.-

U1(Xl

Page 65: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

~ !! hv4 > hv5 then begin equal := false; goto klaar ~

!:.!!!end-else-begin!2!: k :== hv2 + 1 ste~ 1 until orde 9£ stamaard[hv2,kJ := st[k)j

permutatie[ hv2] := hv3

em-~;

if 1 vergeli jken then- -be&in permuta:tie[ orde] : = gr[ grtal, 1]; vergelijken := true; equal : = false S!!else-~§1n k :=0; 1 := 1; !2r. k := k + 1 while perm[k] = perDlltat1e[k] ~ 1 :=

teverspringen1 := 1; .!! i > 1~~ einde~~ klaar

end-em-else penuutat1eonierzoek(1+1,grtB.l,gr); .'-!! equal~ begin.!f. 1 > teverspringeni~ §ot.o einie em;

klaar: end;

einie:

end permutat1e omerzoek;-

k + 1;

vr-0

ordel := orde - 1;

!2!: 1 : = 1 step 1 unt11 orde S2begin matrix[1,1] := 0; for J := i + 1 step 1 Wltll orde 2£ lIIitr1x[ j,1J

groep[ 1 , 0] := orde; !2!: j := 1 ste;e 1 unt11 orde 2-.2 groep[ 1, j] := j;

vergelljken := false;

permutatieonderzoek(l,l,groep)

em. penuutatie staniaard.;-

:= _true 1, .1] em;-

Page 66: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 60 -

6. Tables of resulting matrices

Table I contains the decimal representations (see 2.2. II) of the 10

regular conference matrices of order 26, corresponding to section 4.4. The

tables 2,3, ••• ,11 give the connection lists of these matrices (for any ver­

tex the adjacent vertices are given). The tables 12 and 13 contain the de­

cimal representations of the standard matrices of the 15 strongly regular

graphs of order 25, corresponding to section 4.6. (SI ~s derived from LSI;

S2 and S3 are derived from StI; 84, 85, 86 and S7 are derived from LSI1;

88,89, ••• ,SI5 are derived from StIl). The tables 14,15, ••• ,28 give the con­

nection lists of these matrices.

Page 67: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

tablE" :decimal representations

1D1 ,1 St1,1 St1,2 LS2,1 LS2,2 St2,l St2,2 St2,3 St2,4 St2,5

33553408 33553408 33553408 33553408 33553408 33553408 33553408 33553408 33553!.~08 33553'-1-08

16712688 16712688 16712688 16712688 16712688 16712688 16712688 16712688 16712688 167126888176399 8116399 8176399 8176399 8176399 8176399 8116399 8176399 8176399 81763993354831 33511 831 3684590 3322094 3684590 3684590 3684590 3684590 335'-1-831 332209u1101052 1093308 435421 1126045 435421 435421 435421 435/+21 1093308 1126045

1:>179 23923 218795 23923 218795 218795 219803 218195 23923 23923255938 45331n 61298 510171 118134 11813!.;· 63346 118131+ 453347 510171251733 120300 121269 178997 91113 211735 106405 97173 120300 118997

47845 97178 96092 7941')7 11253)+ 23929 94044 110441 97118 3042230170 46933 48067 2861 !.\- 7485 44723 59875 7767 46933 6141611870 13883 7771 30172 28581 28617 19579 30551.\- 7831 137397597 11614 13623 8106 7891 14140 12605 1609 11614 80143479 2173 3527 3390 8042 6844 5575 8102 3693 3687 0-

3691 3499 2174 3813 4038 3765 2778 3499 3516 -2503377 1957 1964 183 700 363 1646 365 437 125694 726 697 857 123 215 922 183 634 915187 407 365 429 442 485 343 318 315 318119 61 158 118 103 218 174 227 61 229

61 110 59 121+ 117 90 59 115 126 9362 55 55 23 51 37 55 27 55 396 25 21 26 9 29 28 28 9 269 10 10 9 , !+ ,h Q 12 6 "....

3,.. /

3 7 3,..

5 6 ./l) i) 0 ':)

3 1 1 3 ~ 3 1 3 1 10 1 1 1 0 1 1 1 1 1

Page 68: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 6:l -

table 2

com:ec1.;ion list n:,' LS1, 1

1 2 3 4 c; 6 7 8 9 10 11 12 13 14 15 16/

2 1 3 4 L; 6 7 8 0 10 17 18 19 ~o 21 22/

3 1 2 4 5 6 7 8 11 12 17 18 23 24 25 26

4 1 2 3 5 6 0 10 13 1l~ 19 20 23 24 25 26~'

') 1 2 3 4 6 11 12 15 16 19 20 21 22 23 24

6 1 2 3 4 5 13 14 15 16 17 18 21 22 2<::' 26./

'7 1 2 3 9 10 11 12 13 1~ 17 18 19 21 ?3 25

8 1 2 3 9 10 11 12 14 16 17 18 20 22 24 26

9 1 2 4 '7 [3 11 13 14 15 17 19 20 21 24 26

10 2 4 '7 8 12 13 14 16 18 19 20 22 23 ,.,rC-,.,.!

" 1 3 5 7 8 0 13 15 16 17 20 22 23 2u 2~"

12 1 3 5 '7 8 10 1l~ 15 16 18 10 21 23 24 26~

13 1 4 6 '7 c 10 11 15 16 18 10 22 24 25 26/

14 4 6 8 9 10 12 1'5 16 17 20 21 23 25 26

15 1 c; 6 '7 ° 11 12 13 14 18 20 21 22 23 26./

,6 5 6 8 1r 11 12 13 14 17 19 21 22 24 25

17 2 3 6 '7 8 0 " 14 16 19 21 22 23 25 26-'

18 2 3 6 '7 8 10 12 13 15 20 21 22 24 2r; 26/

19 2 4 5 '7 9 10 12 13 16 17 21 22 23 24 26

20 2 4 5 8 9 10 11 14 15 18 21 22 23 24 25

21 2 c; 6 7 9 12 ,,~ 15 16 17 18 19 20 24 25/

22 2 5 6 8 10 11 13 15 16 17 18 19 20 23 26

23 3 4 5 7 10 11 12 14 15 17 1° 20 22 25 26/

24 3 4 5 8 9 11 12 13 16 18 19 20 21 25 26

25 3 4 6 '7 10 11 13 14 16 17 18 20 21 23 24

26 3 4 6 8 9 12 13 14 15 17 18 19 22 23 24

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- 63 -

t8"bl~ 3

connertion list of St1,1

1 ';> 3 4 ') 6 7 8 9 10 11 12 13 14 1';; 16,.., 1 3 4 ') 6 7 8 () 10 17 18 19 ?O 21 22c

3 1 2 4 c; 6 7 8 11 12 17 18 23 24 25 ,;>f>-'

4 1 2 3 :5 6 9 10 13 14 19 20 23 24 25 26

5 1 2 3 4 6 11 13 15 16 17 19 21 22 23 24

6 1 2 3 4 5 12 14 15 16 18 20 21 22 25 26

7 1 2 3 8 9 11 12 13 15 17 19 20 21 25 26

8 1 2 3 7 10 11 12 14 16 18 19 20 21 CJ3 24

0 1 2 4 7 10 12 13 14 15 17 18 10 22 23 25/ .'

10 1 2 4 8 9 11 13 14 16 17 18 20 22 24 26

11 1 3 c; 7 8 10 13 14 16 17 21 22 23 25 26-'

12 1 3 6 7 8 0 13 15 16 18 20 22 23 24 25/

13 1 4 5 7 c 10 11 12 15 20 21 22 ?3 24 26

14 1 'l 6 8 9 10 11 1') 16 18 19 21 23 2~, 26

1 ~ 5 6 7 ('\ 12 13 14 16 17 18 19 21 24 26

16 5 6 8 10 11 12 14 15 17 19 20 22 24 2~;

17 2 3 5 7 9 10 11 15 16 18 19 22 24 25 26

18 2 3 6 8 9 10 12 14 F 17 21 22 23 24 26

19 2 4 c:; 7 8 9 14 15 16 17 20 21 23 24- 25-'

20 ? !l 6 7 8 10 12 13 16 19 21 22 24 25 26

21 2 5 6 7 8 11 13 14 15 18 10 20 22 ?3 26/

22 2 c 6 0 10 11 12 13 16 17 18 ?O 21 23 25.J -'

23 3 4 5 8 9 11 12 13 14 18 19 21 22 24 25

24 3 4 5 8 10 12 13 15 16 17 18 19 20 23 26

25 3 4 (, 7 9 11 12 14 16 17 19 20 22 ?3 26

26 3 4 f, ... 10 11 13 14 15 17 18 20 21 24 2C;!

Page 70: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 64 -

table h

cor~ect1on list of St1,2

3

4

7

8()

10

11

1314

16

1'7•

20

21

22

2

1,,1

1,1

1

1

1

2

2

2

2

2

3

3

3

3

2

2

':>L

2

2

2

2

3

31.1

4

4

3

34

4

4

4

4

4

4

4

3

3

3

3

56

6

56

76

6

7

56

7

b

:7 6 7 8 9 10 11 12 13 14 15 16

5 6 7 8 9 10 17 18 19 20 21 22

5 6 7 8 11 12 17 ,8 23 2h 25 2(,

5 6 7 13 14 '5 19 20 21 23 24 254 8 0 11 13 16 19 20 2? ?3 24 26

4 0 10 1 ::> 14 16 17 19 21 23 25 26

4 11 12 13 1S 16 17 18 20 21 22 25

5 10 11 12 ,4 1~ 18 19 21 22 24 26

6 10 12 13 14 16 17 18 20 22 23 24

A a " 13 ,4 , 5 17 18 , 9 20 2~ 26

7 8 , (i 14 , 5 , 6 , 7 20 22 ?3 2C) 26

7 P 9 13 ,4 ,6 18 21 22 2~ 2) 26

7 9 10 12 1) 16 10 19 20 24 25 26

8 9 10 11 12 15 20 21 22 23 24 2~

8 10 11 13 14 16 17 18 19 21 23 24

7 9 1 1 12 13 1 5 17 19 21 22 23 26

'/ 9 10 11 1:> 16 18 20 21 23 24 26

8 9 10 12 13 1) 17 19 22 23 24 2)

6 8 10 13 15 16 'I 8 2'1 22 23 2.:;' 26

7 9 10 11 '13 14 17 21 22 24 2:;' 26

7 8 12 14 15 16 17 19 20 22 24 26

8 :; 11 12 14 16 18 19 20 21 23 2)

u 9 11 14 1;, 16 17 18 19 22 24 2.:;

8 9 12 13 14 1 I) 17 18 ro 21 23 26

7 10 11 1? 13 14 1: 19 20 22 23 26

b 10 11 12 13 16 17 19 20 21 ::>1, 25

Page 71: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 6S -

??

23 24 2')

22 23 2l~ 2C;

22 25 26

23 24 25 26

12 13

17 11)

1R 20 21

q 10 11

12 1')

1L

9 10 17 18 19 20 21

12 17 18

13 14 19 ?O ;:>1

15 16

15 '6

e 11

8

8

13

9 11

7

7

7

12 13

n 10 11

a " 12

6

9 11

c 10 11

C/

15 16

e 12 13 ,4

14 '5 16

6 10 13

c; 12 14

q 10 11

8 10 13

6

6

19 20 22 23 25 26

1 ~ 17 18 21 22 24 26

17 20 21 23 25 26

17 18 19 20 ;-,4 25

, 6 18 19 20 22 ?3 24

17 18 19 21 23 25

15 1h 18 21 22 23 24 25

1'; 1£: 17 19 20 ~'1 24 26

'3 ,4 19 21 22 24 25 2612 13 14 17 18 20 22 ?3 26

12 14 16 '8 19 21 23 24 26

12 13 16 17 20 21 22 24 ?C,

12 14 1 ~) 17 20 21 22 23 24

14 1tl 18 19 22 24 25 26

91213141" 17'8192:' 2325

13 , ') , 6 18 19 20 21 23 26

12 13 16 17 , 9 21 22 25 26

13 ,4 15 17 18 , 9 20 25, 26

9 '0 '2 13 15 18 20 21 23 24 26

9 14 , 5 16 17 20 22 23 24 25

6

P 10 12

8

8 11

, 0 11

4

4

7

7 '0 "p

9 10 11

9 10

77

7

7

8

7

8

8

"I,

8 10 11

7

7

8

8 10 11

7

7

56

6

6

c;

5

G

6

6

6

6

6

3

u

6

3

4

4

3

3

3

'2

5

,..,L

4

4

2

2

2

5

4

4

4

t:)

2

3

3

2

3

3

3

4

1+

4

2

"c

2

2

1

1

.)

3

3

3

,1

1

1

1

1

1

1

1,

3

7

6

4

131U

15

16

17

'18

e9

10

11

12

1':20

21

22

2324

Page 72: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 66 -

t(lble 6

conpPction list of LS~,2

2

34c;

6

7Bo

'01 1

12

'3

15

16

17

18

19

?O

21

22

::'6

1

1

1

1

1

1

1

::'

2

2

2

2

3

3

3

3

3

2

2

2

2

2

2

2

3

34

4

4

5

3

34

4

4

4

4

4

5

4

44

3

3

3

3

3

56

5

6

56

76

6..,5

56

'7

5

5

6

6

5 6 7 8 9 10 11 12 13 14 15 16

) 6 7 8 9 10 17 ,8 19 20 21 22

5 6 7 8 " 12 17 18 23 24 25 26

5 6 7 13 14 15 19 20 21 23 24 25

4 8 9 11 13 16 19 20 22 23 2h 26

4 9 10 12 14 16 17 19 21 23 25 26

4 10 11 12 15 16 18 20 21 22 24 25

;; 10 12 13 14 15 17 18 20 22 23 26

6 10 11 13 14 16 17 18 19 22 24 25

.., e 9 14 15 16 18 21 22 23 24 26

'7 0 12 13 1 ') 16 17 18 10 21 21+ 26

7 8 11 14 1 :; ,6 17 19 20 22 2> 26

8 9 11 14 15 16 17 18 20 21 23 25

8 0 '0 12 13 15 18 19 20 2h pc) ~)6

8 10 " 12 13 14 17 , 9 21 22 23 211

'7 9 1(> 11 12 13 20 21 22 23 25 26

8 9 " '2 13 15 18 , 9 21 22 2"3 2:'8 9 10 11 13 14 17 20 21 24 25 26

6 9 11 12 14 15 17 20 21 2P 24 26

7 8 12 13 14 16 ,8 , 9 21 22 ')5 26

7 10 11 13 15 , 6 , 7 , 8 , 9 ?O 23 26

8 9 '0 12 , 5 ,6 17 , 9 20 23 24 25

6 8 '0 13 15 ,6 17 2' 22 24 2:1 ?6

7 9 '0 1 1 14 1 5 18 19 22 23 25 26

7 0 12 , 3 14 16 17 18 20 22 23 24

8 '0 11 12 14 16 18 19 20 21 23 24

Page 73: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 67 -

table 7

connection list of St2,1

1 2 3 4 5 6 7 8 9 10 11 12 13 ,It 15 ,6

2 1 3 4 5 6 7 8 9 10 17 18 19 20 21 2?

3 1 2 4 5 6 7 8 11 12 17 18 23 24 25 26

4 1 2 3 5 6 7 13 14 15 19 20 21 23 24 I"Cc".'

5 1 2 3 4 8 9 11 13 16 19 20 22 23 24 26

6 1 2 3 4 9 10 12 14 16 17 19 21 23 25 26

7 1 2 3 4 10 11 12 1') 16 18 20 21 22 24 21;. /

8 1 2 3 5 a 10 13 14 15 17 18 22 24 25 26-'

a 2 5 6 8 12 14 15 16 18 20 21 22 ?3 26-'

10 1 2 6 7 8 11 13 15 16 17 19 21 22 25 26

11 1 3 5 7 10 12 13 15 16 17 18 19 ro 23 26

12 1 3 6 7 9 11 13 14 16 17 18 21 22 23 24

13 1 4 co, 8 10 " 12 14 15 17 19 21 22 23 24'"

14 1 4 6 8 9 12 13 15 16 17 18 19 20 24 r,cc.. ,)

15 1 4 7 8 9 10 11 13 14 18 20 21 23 25 26

16 1 5 6 7 9 10 11 12 14 19 20 22 24 25 26

17 2 3 6 8 10 " 12 13 14 18 19 20 21 24 26

18 2 3 7 8 9 11 12 14 15 17 19 20 22 23 2r .j

19 2 4 5 6 10 " 13 14 16 17 18 20 22 ?3 25

20 2 4 5 7 a 11 14 15 16 17 18 19 21 24 26

21 2 4 6 7 9 10 12 13 15 17 20 22 23 24 26

22 2 5 7 8 9 10 12 13 16 18 19 21 23 24 25

23 3 4 5 6 9 11 12 13 15 18 19 21 22 25 2(-.

24 3 4 c:; 7 8 12 13 14 16 17 20 21 22 25 26'"

25 3 4 6 7 8 10 14 15 16 18 19 22 23 24 26

26 3 5 6 8 9 10 11 15 16 17 20 21 23 24 25

Page 74: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 68 -

table 8

connection list of St2,2

1 2 3 4 5 6 7 8 Q 10 11 12 13 14 1c; 16~'

2 , 3 4 5 6 7 8 9 10 17 18 10 20 ?1 22.-

3 1 2 4 5 6 7 8 11 12 17 18 23 24 25 26

4 . 1 2 3 5 6 7 13 14 15 19 20 21 23 24 25

5 1 '2 3 4 8 9 " '3 16 19 20 22 73 24 ?f,

6 1 '2 3 4 9 10 12 14 15 17 10 22 23 25 26"

7 , 2 3 4 , 1 12 13 14 ,6 17 18 20 21 22 25

8 2 3 5 10 11 14 15 16 17 ,8 19 21 24 26

9 2 5,.,

10 12 13 15 ,6 '7 18 20 22 23 24tJ

10 1 f", 6 8 9 " 12 13 15 18 19 20 21 2"' 26c. .-

11 1 3 '3 7 8 10 12 15 16 20 21 22 23 25 2612 , 3 6 7 q 10

" 13 14 18 21 22 23 24 26

13 1 4 5 7 9 10 12 14 ,6 ,8 19 20 24 25, 26

14 1 4 6 1 8 12 13 15 16 17 10 21 22 24 26.-

15 1 4 6 8 9 10 11 14 16 17 20 21 23 24 25

16 1 c; 7 8 9 11 13 14 15 17 18 19 22 23 25,/

17 2 3 6 7 e Q 14 15 16 18 20 22 24 2~J 26-'

18 2 3 7 8 9 10 12 13 16 17 19 21 23 24 25

19 2 4 5 6 8 10 13 14 16 18 21 22 23 ~r::. 26-,/

20 2 4 5 ... 9 10 11 13 15 17 21 22 24 25 26,21 2 4 7 8 10 " 12 14 15 18 19 20 22 23 24

22 2 I:; 6 7 9 11 12 14 16 17 1° 20 21 23 26,/ /

23 3 4 5 6 9 11 12 15 16 18 19 21 22 24 ?;

24 3 I~ 5 8 9 12 13 14 15 17 18 20 21 23 26

2<: 3 4 6 7 10 " 13 15 16 17 18 19 20 23 26,/

26 3 5 6 8 10 11 12 13 14 17 '9 20 22 24 25

Page 75: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 69 -

table 9

connection list of St2,3

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

2 1 3 4 5 6 7 8 9 10 17 18 19 20 21 22

3 2 4 5 6 7 8 11 12 17 18 23 24 25 26

4 1 2 3 <; 6 7 13 14 15 19 20 21 23 24 2r,,; -,/

t: 1 2 3 -4 8 n 11 13 16 19 20 22 23 24 26.I /

6 1 2 3 4 9 10 12 14 16 17 19 21 23 25 26

7 1 2 3 4 10 " 12 , 5 16 18 20 21 22 24 1')<',t:... /

8 1 2 3 e- 10 12 13 14 15 17 18 19 22 24 26'"

9 1 2 t:: 6 10 11 13 15 16 17 18 20 21 23 26"

10 1 2 6 7 8 9 14 15 1f 17 20 22 24 2;' 26

"1 3 ~ 7 0 12 13 1t: 16 17 18 20 22 ?3 r)C

/'c.,.,

12 , 3 6 7 A 11 14 1) 16 18 19 21 22 23 26

13 4 c:: e 0 " '4 15 16 17 18 19 21 2L 25/'

,4 1 4 f- A 10 11 12 13 1::: 17 , n 20 22 23 r,c',J ,; c)

1<:. 1 h 7 8 9 10 12 13 14 18 20 21 23 2!~ 26,;

16 5 6 7 9 10 11 12 13 19 21 22 24 25 26

17 " 3 6 8 c 10 11 13 14 18 21 22 23 24 25c/'

18 '" 3 7 8 9 " 12 13 15 17 19 20 21 25 26c

19 2 4 5 f, 8 12 13 14 16 18 20 21 22 25 26

20 2 4 5 7 9 10 11 14 15 ,8 10 22 23 ~c: 26,/ :- ~

21 2 4 6 7 9 12 13 15 16 17 18 19 22 23 21+

22 2 5 7 8 10 11 12 14 16 17 19 20 21 23 24

23 3 u 5 6 0 11 12 14 15 17 20 21 22 24 26

24 3 4 5 7 8 10 13 15 16 17 21 22 23 25 26

25 3 4 6 7 10 11 13 14 16 17 18 19 20 24 26

26 3 5 6 8 9 10 12 15 16 18 10 20 23 24 25,/

Page 76: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 70 -

ta.ble 10

connection list of St2,4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

2 1 3 4 5 6 7 8 q 10 17 18 19 20 21 22

3 1 2 4 5 6 7 8 11 12 17 18 23 24 25 26

4 1 2 3 5 6 9 10 13 14 19 20 23 24 25 26

5 1 2 3 4 6 11 13 15 16 17 19 21 22 23 24

6 1 2 3 4 5 12 14 15 16 18 P.O 21 22 25 26

7 1 2 3 8 9 11 12 13 15 17 19 20 21 25 26

8 1 2 3 7 10 11 12 14 16 18 10 20 21 23 24-'I

9 1 2 4 7 10 12 13 14 15 17 18 19 22 23 25

10 1 2 4 8 9 11 13 14 16 17 18 20 22 24 26

11 1 3 5 7 8 10 14 15 16 17 19 22 24 25 2612 1 3 6 7 8 9 13 15 16 18 20 22 23 24 2~-,

13 1 4 5 7 9 10 12 15 16 17 20 21 23 24 26

14 1 4 6 8 9 10 11 15 16 18 10 21 23 25 26

15 1 ',) 6 7 9 11 12 13 14 18 19 21 22 24 26

16 1 5 6 8 10 11 12 13 14 17 20 21 ~" 23 25cc..~

17 2 3 5 7 9 10 11 13 16 18 21 22 23 25 26

18 2 3 6 8 9 10 12 14 15 17 21 22 23 24 26

19 2 4 5 7 8 9 11 14 15 20 21 22 23 24 25

20 2 4 ,,":'.,. 7 8 10 12 13 16 19 21 22 24 25 26\.-i

21 .... I:; 6 7 8 13 14 15 16 17 18 19 20 23 26:::. /

22 2 5 6 0 10 11 12 15 16 17 18 10 20 24 25-" -'

23 3 4 5 8 9 12 13 14 16 17 18 19 21 24 25

24 3 4 5 8 10 11 12 13 15 18 19 20 22 23 26

25 3 4 6 ., 9 11 12 14 16 17 19 20 22 23 26

26 .3 4 6 ., 10 11 13 14 15 17 18 20 21 24 25

Page 77: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 71 -

table 11

connection list of St2,5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 1c, 16.-'

2 3 4 5 6 7 8 9 10 17 18 19 20 ?1 22

3 1 2 4 5 6 7 8 11 12 17 18 23 ?4 25 26

4 1 2 3 5 6 9 11 13 14 19 20 21 23 24 25C; 1 2 3 4 6 10 13 15 16 17 19 22 23 24 26-6 1 2 3 4 5 12 14 15 16 18 20 21 22 25 26

7 1 2 3 8 9 10 11 12 15 19 20 22 23 25 26

8 1 2 3 7 9 , 1 13 14 15 17 18 21 22 24 26

9 2 4 7 8 12 13 14 16 17 19 20 22 24 25

10 1 2 5 7 11 12 13 15 16 17 18 19 20 21 23

11 1 3 4 7 8 10 13 14 16 18 19 ?1 23 25 26

12 1 3 6 7 9 10 14 15 16 17 18 20 23 24 25

13 4 5 8 9 10 11 15 16 17 20 21 24 25 26

14 1 4 6 8 9 11 12 15 16 18 19 21 fV'I 23 24.C

15 1 5 6 7 8 10 12 13 14 20 21 22 23 24 26

16 1 5 6 9 10 11 12 13 14 17 18 19 22 25 26

17 2 3 5 8 9 10 12 13 16 18 21 22 23 24 2c;..-

18 2 3 6 8 10 11 12 14 ,6 17 19 20 21 24 26

19 2 4 5 7 9 10 11 14 16 18 20 22 23 24 26

20 2 4 6 7 9 10 1? 13 15 18 19 21 24 25 26

21 2 4 6 8 10 11 13 1l~ 15 17 18 20 22 23 25

22 2 5 6 7 8 9 14 15 16 17 19 21 23 25 26

23 3 4 5 7 10 11 12 14 15 17 19 21 22 24 25

24 3 4 5 8 9 12 13 14 15 17 18 19 20 23 26

25 3 4 6 7 9 11 12 13 16 11 20 21 22 23 26

26 3 5 6 1 8 11 13 15 16 18 19 20 22 24 25

Page 78: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

table ,,,)

dectm:.i. l'2presentations

~2 .~? SL~ 85 0<" ;:)7,.)....) ,:) )

1 ;'3120 1677]120 16773120 16773120 16773120 16773120 1:)77312081 })495 8130496 8130496 8130496 8130496 8130496 3130496331+')11 SA, 3932223 3932223 3803198 33451,68 3932223 393222310,;8:.) i,7 233016 233016 3>')204, 1098547 233016 233016

12495 15811 r3 158118 157093 121+95 15811 e, 158118240522 87381 87381 86870 436906 87381 87331213949 45771 451'71 55523 87L~02 45771 45771

!j.;?5h 1 117067 117319 80971 108121 104803 10382222e, !J. 371'92 38307 25305 231 h1 42198 41.92911.>705 9'-'.45 .9668 32028 26021 17293 1864314002 14961 15450 . 399 5782 15132 155371238 5")60 5932 7088 6486 5810 6764t'")"jCi, 3598 2737 1639 2)~57 3177 1818C.(".-) ! .....

1 <'7"'7 921 1+74 q48 161 '+ 1622 NIt ... I t

50); 3f)~·~ 87L, 845 106 61'7 653167 242 101 426 211 2'75 291

91 218 141 19 113 241 21.~2

25 21 114 108 13 108 10538 35 22 21 7 26 2810 22 22 23 15 25 25

5 9 12 5 10 10 103 3 3

,..3 14- 4I)

3 3 1 0 2 3 30 0 1 1 0 1 1

Page 79: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

table 11

dec imkt j~o.presentations

~)p. 39 510 ::;11 512 313 314 51 ')

157' 120 16Tr3120 16773120 15T73120 16773120 16Tr3120 16773120 161'(312081301'96 81301+96 8130496 8130496 8130496 8130496 3130496 81304963932:'.23 3803198 3932223 3932223 3932223 3932223 3932223 3932223

23.3tJ S 362041 233016 233016 233010 23301 ,) 2330i ( 233016"I :;e118 157093 158118 158118 158118 158118 15811 e 15811887381 86870 87381 87381 87381 87381 87381 87381);5771 92363 45771 ·5771 45771 !l5Tr'l 45771 457"71

117319 47763 117319 117067 117319 117067 117319 11706731'356 3"T28'r 42387 37782 42381 37797 38286 377979619 15972 ll-588 9~l-45 4594 9430 8689 9430

13938 13405 15465 ·5885 15450 15891 15465 14940T324 874 5810 5560 5804 5560 5938 55602° C '7 3480 28hIt 2674 2865 2668 27 1 6 3619 .....,l-lj i

W180) 1230 921 753 921 732 921 1583I346 781 87Ll 876 8"r4 882 87[, 370

241 339 86 1LI-2 101 163 85 236163 170 142 218 163 233 147 2;J116 113 113 21 92 21 108 2111.j. 52 37 35 22 14 37 1426 28 28 22 22 14 28 11.1-

5 9 5 9 12 13 5 13.-2 ':), 3 3 1 3 1t)

--'

3 2 3 1 2 2 21 1 0 ". 1 1 1> I

Page 80: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 74 -

table 14

connection list of 81

1 2 3 4 5 6 7 8 9 10 11 12 13

2 1 3 4 5 6 7 14 15 16 17 18 19

3 1 2 4 5 8 9 14 15 20 21 22 234 1 2 3 5 10 11 16 17 20 21 24 25

5 1 2 3 4 12 13 18 19 22 23 24 25

6 1 2 8 9 10 12 14 16 17 18 22 24

7 1 2 8 9 11 13 15 16 17 19 23 25

8 1 3 6 .., 10 12 15 16 20 22 23 25

9 1 3 6 .., 11 13 14 ,.., 21 22 23 24

10 1 4 6 8 12 13 14 '''' 19 20 21 25

"1 4 .., 9 12 13 15 16 18 20 21 24

12 1 5 6 8 10 11 15 18 19 21 23 24

13 1 5 .., 9 10 11 14 18 19 20 22 25

14 2 3 6 9 10 13 16 18 19 20 21 23

15 2 3 .., 8 11 12 17 18 19 20 21 22

16 2 4 6 .., 8 11 14 18 20 23 24 25

17 2 4 6 7 9 10 15 19 21 22 24 25

18 2 5 6 11 12 13 14 15 16 21 22 25

19 2 5 .., 10 12 13 14 15 17 20 23 24

20 3 4 8 10 11 13 14 15 16 19 22 24

21 3 4 9 10 11 12 14 15 ,.., 18 23 25

22 3 5 6 8 9 13 15 17 18 20 24 25

23 3 5 .., 8 9 12 14 16 19 21 24 25

24 4 5 6 9 11 12 16 17 19 20 22 2325 4 5 7 8 10 13 16 17 18 21 22 23

Page 81: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 75 -

table 15

connection list of 82

1 2 3 4 5 6 7 8 9 10 11 12 13

2 1 3 4 5 6 7 14 15 16 17 18 19

3 1 2 4 5 6 7 20 21 22 23 24 25

4 1 2 3 8 9 10 14 15 16 20 21 22

5 1 2 3 8 11 12 14 17 18 20 23 24

6 1 2 3 9 11 13 15 17 19 21 23 25

7 1 2 3 10 12 13 16 18 19 22 24 25

8 1 4 5 9 10 11 14 17 19 22 24 25

9 1 4 6 8 10 13 16 17 18 21 23 24

10 1 4 7 8 Q 12 15 18 19 20 23 25.-

11 1 c; 6 8 12 13 14 16 19 20 21 25./

12 1 5 7 10 11 13 15 17 18 20 21 22

13 6 7 0 11 12 14 15 16 22 23 24./

14 2 4 5 8 11 13 15 16 18 22 23 25

15 ·2 4 6 10 12 13 14 17 19 20 22 23

16 2 4 7 9 11 13 14 18 19 20 21 24

17 2 5 6 8 9 12 15 18 19 21 22 24

18 2 5 7 9 10 12 14 16 17 21 23 25

19 2 6 7 8 10 11 15 16 17 20 24 25

20 3 4 5 10 11 12 15 16 19 21 23 24

21 3 4 6 9 11 12 16 17 18 20 22 25

22 3 4 7 8 12 13 14 15 17 21 24 25

23 3 5 6 0 10 13 14 15 18 20 24 25./

24 3 5 7 8 9 13 16 17 19 20 22 2325 3 6 7 8 10 11 14 18 19 21 22 23

Page 82: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 76 -

table 16

connection list of 83

2 3 4 5 6 7 8 9 10 11 12 13

2 1 3 4 5 6 7 14 15 16 17 18 19

3 1 2 4 5 6 7 20 21 22 23 24 25

4 1 2 3 8 9 10 14 15 16 20 21 22

5 1 2 3 8 11 12 14 17 18 20 23 24

6 1 2 3 9 11 13 15 17 19 21 23 25

7 1 2 3 10 12 13 16 18 19 22 24 25

8 1 4 5 9 10 11 14 16 19 23 24 25

9 1 4 6 8 10 13 15 17 18 20 24 25

10 , 4 7 8 9 12 17 18 19 21 22 2311 1 5 6 8 12 13 14 15 19 21 22 24

12 1 5 7 10 11 13 15 16 17 20 22 23

13 1 6 7 9 11 12 14 16 18 20 21 25

14 2 4 5 8 " 13 16 17 18 21 22 25

15 2 4 6 9 11 12 16 17 19 20 22 24

16 2 4 7 8 12 13 14 15 19 20 23 25

17 2 5 6 9 10 12 14 15 18 22 23 25

18 2 5 7 9 10 13 14 17 19 20 21 24

19 2 6 7 8 10 11 15 16 18 21 23 24

20 3 4 5 9 12 13 15 16 18 21 23 24

21 3 4 6 10 " 13 14 18 19 20 22 2322 3 4 7 10 11 12 14 15 17 21 24 25

23 3 5 6 8 10 12 16 17 19 20 21 25

24 3 5 7 8 9 11 15 18 19 20 22 25

25 3 6 7 8 9 13 14 16 17 2:? 23 24

Page 83: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 77 -

table 17

connection list of s4

1 2 3 4 5 6 7 8 9 10 11 12 13

2 1 3 4 5 6 7 14 15 16 17 18 19

3 1 2 4 5 6 8 14 20 21 22 23 24

4 1 2 3 7 9 10 15 16 20 21 22 25

5 1 2 3 8 11 12 15 17 18 20 23 25

6 1 2 3 9 11 13 16 17 19 21 23 24

7 , 2 4 10 11 13 14 18 19 20 24 25

8 1 3 5 9 12 13 14 15 19 22 24 25

9 1 4 6 8 11 12 16 18 19 21 22 25

10 1 4 7 11 12 13 14 15 17 21 22 23

", 5 6 7 9 10 17 18 22 23 24 25

12 1 5 8 9 10 13 14 16 17 18 20 21

13 1 6 7 8 10 12 15 16 19 20 23 24

14 2 3 7 8 10 12 17 18 19 21 22 24

15 2 4 5 8 10 13 16 17 19 22 23 25

16 2 4 6 9 12 13 15 17 18 20 22 24

17 2 5 6 10 11 12 14 15 16 21 24 25

18 2 5 7 9 11 12 14 16 19 20 22 23

19 2 6 7 8 9 13 ,4 15 18 21 23 25

20 3 4 5 7 12 13 16 18 21 23 24 25

21 3 4 6 9 10 12 14 17 19 20 23 25

22 3 4 8 9 10 11 14 15 16 18 23 24

23 3 5 6 10 11 13 15 18 19 20 21 22

24 3 6 7 8 11 13 14 16 17 20 22 25

25 4 5 7 8 9 11 15 17 19 20 21 24

Page 84: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 78 -

table 18

connection list of 85

1 2 3 4 5 6 7 8 9 10 11 12 13

2 1 3 4 5 6 7 14 15 16 17 18 19

3 1 2 4 5 8 9 14 15 20 21 22 234 1 2 3 5 10 11 16 17 20 21 24 25

5 1 2 3 4 12 13 18 19 22 23 24 25

6 1 2 7 8 10 12 14 16 18 20 22 24

7 1 2 6 9 11 13 15 17 19 20 22 24

8 1 3 6 9 10 12 15 16 19 21 22 25

9 1 3 7 8 11 13 14 16 19 20 23 25

10 1 4 6 8 11 12 15 17 18 20 23 25

" 1 4 7 9 10 13 1 ~; 16 18 21 23 24

12 1 5 6 8 10 13 14 17 19 21 23 24

13 1 5 7 9 11 12 14 17 18 21 22 25

14 2 3 6 9 12 13 16 17 18 20 21 2315 2 3 7 8 10 , 1 17 18 19 21 22 2316 2 4 6 8 9 11 14 18 19 21 24 25

17 2 4 7 10 12 13 14 15 19 20 21 2518 2 5 6 10 11 13 14 15 16 22 23 25

19 2 5 7 8 9 12 15 ,6 17 23 24 25

20 3 4 6 7 9 10 ,4 17 22 23 24 25

21 3 4 8 11 12 13 14 15 16 17 22 24

22 3 5 6 7 8 13 15 18 20 21 24 25

23 3 5 9 10 11 12 14 15 18 19 20 24

24 4 5 6 7 11 12 16 19 20 21 2~ 2325 4 5 8 9 10 13 16 17 18 19 20 22

Page 85: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 79 -

table 19

connection list of 86

1 2 3 4 5 6 7 8 9 10 11 12 13

2 1 3 4 5 6 7 14 15 16 17 18 19

3 1 2 4 5 6 7 20 21 22 23 24 25

4 1 2 3 8 9 10 14 15 16 20 21 22

5 1 2 3 8 11 12 14 17 18 20 23 24

6 1 2 3 9 11 13 15 17 19 21 23 25

7 1 2 3 10 12 13 16 18 19 22 24 25

8 1 4 5 9 10 13 14 17 19 20 24 25

9 1 4 6 8 10 12 15 18 19 21 23 24

10 1 4 7 8 9 11 16 17 18 22 23 25

11 1 5 6 10 12 13 14 16 17 21 22 2312 1 5 7 9 11 13 15 16 18 20 21 24

13 1 6 7 8 11 12 14 15 19 20 22 25

14 2 4 5 8 11 13 15 16 19 22 23 24

15 2 4 6 9 12 13 14 16 18 20 23 25

16 2 4 7 10 11 12 14 15 17 21 24 25

17 2 5 6 8 10 11 16 18 19 20 21 25

18 2 5 7 9 10 12 15 17 19 20 22 23

19 2 6 7 8 9 13 14 17 18 21 22 24

20 3 4 5 8 12 13 15 17 18 21 22 25

21 3 4 6 9 11 12 16 17 19 20 22 24

22 3 4 7 10 11 13 14 18 19 20 21 23

23 3 5 6 9 10 11 14 15 18 22 24 25

24 3 5 7 8 9 12 14 16 19 21 23 25

25 3 6 7 8 10 13 15 16 17 20 23 24

Page 86: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 80 -

table 20

connection list of 87

2 3 4 5 6 7 8 9 10 11 12 13

2 1 3 4 5 6 7 14 15 16 17 18 19

3 1 2 4 5 6 7 20 21 22 23 24 25

4 1 2 3 8 9 10 14 15 16 20 21 22

5 1 2 3 8 11 12 14 17 18 20 23 24

6 1 2 3 9 11 13 15 17 19 21 23 25

7 1 2 3 10 12 13 16 18 19 22 24 25

8 1 4 5 9 10 13 15 17 18 22 23 24

9 1 4 6 8 10 12 16 17 19 20 23 25

10 1 4 7 8 9 11 14 18 19 21 24 25

11 1 5 6 10 12 13 14 15 18 20 21 25

12 1 5 7 9 11 13 14 16 19 20 22 23

13 1 6 7 8 11 12 15 16 17 21 22 24

14 2 4 5 10 11 12 15 16 19 21 23 24

15 2 4 6 8 11 13 14 16 18 22 23 25

16 2 4 7 9 12 13 14 15 17 20 24 25

17 2 5 6 8 9 13 16 18 19 20 21 24

18 2 5 7 8 10 11 15 17 19 20 22 25

19 2 6 7 9 10 12 14 17 18 21 22 2320 3 4 5 9 11 12 16 17 18 21 22 25

21 3 4 6 10 11 13 14 17 19 20 22 24

22 3 4 7 8 12 13 15 18 19 20 21 23

23 3 5 6 8 9 12 14 15 19 22 24 25

24 3 5 7 8 10 13 14 16 17 21 23 25

25 3 6 7 9 10 11 15 16 18 20 23 24

Page 87: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 81 -

table 21

connection list of s8

1 2 3 4 5 6 7 8 9 10 11 12 13

2 3 4 5 6 7 14 15 16 17 18 19

3 2 4 5 6 7 20 21 22 23 24 25

4 1 2 3 8 9 10 14 15 16 20 21 22

5 1 2 3 8 11 12 14 17 18 20 23 21+

6 1 2 3 9 11 13 15 17 19 21 23 25

7 1 2 3 10 12 13 16 18 19 22 24 25

8 1 4 5 9 10 11 14 16 19 23 24 25

9 1 4 6 8 10 13 17 18 19 20 22 2310 1 4 7 8 9 12 15 17 18 21 24 25

11 1 5 6 8 12 13 15 16 19 20 21 24

12 1 5 7 10 11 13 14 15 18 21 22 2313 1 6 7 9 11 12 14 16 17 20 22 25

14 2 4 5 8 12 13 15 16 17 22 23 25

15 2 4 6 10 11 12 14 17 19 21 22 24

16 2 4 7 8 11 13 14 18 19 20 21 25

17 2 5 6 9 10 13 14 15 18 20 24 25

18 2 5 7 9 10 12 16 17 19 20 21 2319 2 6 7 8 0 11 15 16 18 22 23 24.;

20 3 4 5 9 11 13 16 17 18 21 22 24

21 3 4 6 10 11 12 15 16 18 20 23 25

22 3 4 7 9 12 13 14 15 19 20 23 24

23 3 5 6 8 9 12 14 18 19 21 22 25

21+ 3 5 7 8 10 11 15 17 19 20 22 25

25 3 6 7 8 10 13 14 16 17 21 23 24

Page 88: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- ~2 -

table 22

connection list of 89

1 2 3 4 5 6 7 8 9 10 11 12 13

2 1 3 4 5 6 7 14 15 16 17 18 19

3 1 2 4 5 6 8 14 20 21 22 23 24

4 1 2 3 7 9 10 15 16 20 21 22 25

5 1 2 3 8 11 12 15 17 18 20 23 25

6 1 2 3 9 11 13 16 17 19 21 23 24

7 1 2 4 9 11 12 ,4 18 19 22 24 25

8 1 3 5 10 12 13 14 16 18 21 24 25

9 1 4 6 7 10 13 17 18 20 23 24 25

10 1 4 8 9 12 13 14 15 16 19 20 23

", 5 6 7 12 13 15 19 21 22 23 25

12 1 5 7 8 10 11 16 17 19 20 22 24

13 1 6 8 9 10 11 14 15 17 18 21 22

14 2 3 7 8 10 13 15 18 19 22 23 24

15 2 4 5 10 11 13 14 16 17 22 23 25

16 2 4 6 8 10 12 15 17 19 21 24 25

17 2 5 6 9 12 13 15 16 18 20 22 24

18 2 5 7 8 9 13 14 17 19 20 21 25

19 2 6 7 10 11 12 14 16 18 20 21 2320 3 4 5 9 10 12 17 18 19 21 22 2321 3 4 6 8 11 13 16 18 19 20 22 25

22 3 4 7 11 12 13 14 15 17 20 21 24

23 3 5 6 9 10 11 14 15 19 20 24 25

24 3 6 7 8 9 12 14 16 17 22 23 25

25 4 5 7 8 9 11 15 16 ,8 21 23 24

Page 89: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- tlJ -

table ?3

connection list of 810

1 2 3 4 5 6 7 8 9 10 11 12 13

2 1 3 4 5 6 7 14 15 16 17 18 19

3 1 2 4. 5 6 7 20 21 22 23 24 25

4 1 2 3 8 9 10 14 15 16 20 21 22

5 1 2 3 8 11 12 14 17 18 20 23 24

6 1 '2 3 9 11 13 15 17 19 21 23 25

7 1 2 3 10 12 13 16 18 19 22 24 25

8 1 4 5 9 10 11 14 16 19 23 24 25

9 1 4 6 8 10 12 15 17 18 21 24 25

10 1 4 7 8 9 13 17 18 19 20 2? 23

" 1 5 6 8 12 13 14 15 19 20 22 25

12 1 5 7 9 11 13 15 16 18 20 21 24

13 1 6 7 10 11 12 14 16 17 21 22 2314 2 4 5 8 11 13 16 17 18 21 22 25

15 2 4 6 9 11 12 16 17 19 20 22 24

16 2 4 7 8 12 13 14 15 19 21 23 24

17 2 5 6 9 10 13 14 15 18 22 23 24

18 2 5 7 0 10 12 14 17 19 20 21 25/

19 2 6 7 8 10 11 15 16 18 20 23 25

20 3 4 5 10 11 12 15 18 19 21 22 2321 3 4 6 9 1? 13 14 16 18 20 23 25

22 3 4 7 10 11 13 14 15 17 20 24 25

23 3 5 6 8 10 13 16 17 19 20 21 24

24 3 5 7 8 9 12 15 16 17 22 23 25

25 3 6 .., 8 9 11 14 18 19 21 22 24,

Page 90: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 84 -

table 24

connection list of 811

1 2 3 4 5 6 7 8 9 10 11 12 13

2 1 3 4 5 6 7 14 15 16 17 18 19

3 1 2 4 5 6 7 20 21 22 23 24 25

4 1 2 3 8 9 10 14 15 16 20 21 22

5 1 2 3 8 11 12 14 17 18 20 23 24

6 1 2 3 9 11 13 15 17 19 21 23 25

7 1 2 3 10 12 13 16 18 19 22 24 25

8 1 4 5 9 10 11 14 17 19 22 24 25

9 1 4 6 8 10 13 16 17 18 21 23 24

10 1 4 7 8 9 12 15 18 19 20 23 25

11 1 5 6 8 12 13 14 15 16 22 23 25

12 1 5 7 10 11 13 15 17 18 20 21 22

13 , 6 7 9 11 12 ,4 16 19 20 21 24

14 2 4 5 8 11 13 16 18 19 20 21 25

15 2 4 6 10 l' 12 16 17 10 20 22 23./

16 2 4 7 9 11 13 14 15 18 22 23 24

17 2 5 6 8 9 12 15 18 19 21 22 24

18 2 5 7 9 10 12 14 16 17 21 23 2510 2 6 7 8 10 13 14 15 17 20 24 25.-

20 3 4 5 10 12 13 14 15 19 21 23 24

21 3 4 6 9 12 13 14 17 18 20 22 25

22 3 4 7 8 11 12 15 16 17 21 24 25

23 3 5 6 9 10 11 15 16 18 20 24 2524 3 5 7 8 9 13 16 17 19 20 22 2325 3 6 7 8 10 11 14 18 19 21 22 23

Page 91: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 85 -

ta.ble 25

connection list of 812

1 2 3 4 5 6 7 8 9 10 11 12 13

2 1 3 4 5 6 7 14 15 16 17 18 19

3 1 2 4 5 6 7 20 21 22 ~ 24 25

4 1 2 3 8 9 10 14 15 16 20 21 22

5 1 2 3 8 11 12 14 17 18 20 23 24

6 1 2 3 9 11 13 15 17 19 21 23 25

7 1 2 3 10 12 13 16 18 19 22 24 25

8 1 4 5 9 10 11 14 16 19 23 24 25

9 1 4 6 8 10 12 15 17 18 22 23 25

10 1 4 7 8 9 13 17 18 19 20 21 24

11 1 5 6 8 12 13 14 15 19 21 22 24

12 1 5 7 9 11 13 15 16 18 20 22 ~

13 1 6 7 10 11 12 14 16 17 20 21 25

14 2 4 5 8 11 13 16 17 18 21 22 25

15 2 4 6 9 11 12 16 17 19 20 22 24

16 2 4 7 8 12 13 14 15 19 20 23 25

17 2 5 6 9 10 13 14 15 18 20 24 25

18 2 5 7 9 10 12 14 17 19 21 22 2319 2 6 7 8 10 11 15 16 18 21 23 24

20 3 4 5 10 12 13 15 16 17 21 23 24

21 3 4 6 10 11 13 14 18 19 20 22 ~

22 3 4 7 9 11 12 14 15 18 21 24 25

23 3 5 6 8 9 12 16 18 19 20 21 25

24 3 5 7 8 10 11 15 17 19 20 22 25

25 3 6 7 8 9 13 14 16 17 22 23 24

Page 92: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 86 -

table 26

connection list of 813

1 2 3 4 5 6 7 8 9 10 11 12 'l32 1 3 4 5 6 7 14 15 16 17 18 19

3 1 2 4 5 6 7 20 21 22 23 24 254 1 2 3 8 9 10 14 15 16 20 21 2?

5 1 2 3 8 11 12 14 17 18 20 23 246 1 2 3 9 11 13 15 17 19 21 23 25

7 1 2 3 10 12 13 16 18 19 22 24 258 1 4 5 9 10 11 14 17 19 22 24 25

9 1 4 6 8 10 13 16 17 18 20 23 2510 1 4 7 8 9 12 15 18 19 21 23 2411 1 5 6 8 12 13 14 15 16 21 24 2512 1 5 7 10 11 13 15 17 18 20 21 22

13 1 6 7 9 11 12 14 16 19 20 22 2314 2 4 5 8 11 13 16 18 19 21 22 2315 2 4 6 10 11 12 16 17 19 20 21 2416 2 4 7 9 11 13 14 15 18 20 24 2517 2 5 6 8 9 12 15 18 19 20 22 2518 2 5 7 9 10 12 14 16 17 21 23 2519 2 6 7 8 10 13 14 15 17 22 23 2420 3 4 5 9 12 13 15 16 17 22 23 2421 3 4 6 10 11 12 14 15 18 22 23 2522 3 4 7 8 12 13 14 17 19 20 21 25

23 3 5 6 9 10 13 14 18 19 20 21 2424 3 5 7 8 10 11 15 16 19 20 23 2525 3 6 7 8 9 11 16 17 18 21 22 24

Page 93: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 87 -

table 27connection list of 814

1 2 3 4 5 6 7 8 9 10 11 12 13

2 1 3 4 5 6 7 14 15 16 17 18 19

3 1 2 4 5 6 7 20 21 22 23 24 25

4 1 2 3 8 9 10 14 15 16 20 21 22

5 1 2 3 8 11 12 14 17 18 20 23 24

6 1 2 3 9 11 13 15 17 19 21 23 25

7 2 3 10 12 13 16 18 19 22 24 25

8 1 4 5 9 10 11 14 16 19 23 24 25

9 1 ~ 6 8 10 13 15 17 18 22 23 24

10 1 4 7 8 9 12 17 18 19 20 21 25

11 1 5 6 8 12 13 14 15 19 20 22 25

12 1 5 7 10 11 13 15 16 17 20 21 24

13 1 6 7 9 11 12 14 16 18 21 22 2314 2 4 5 8 11 13 16 17 18 21 22 25

15 2 4 6 9 11 12 16 17 19 20 22 24

16 2 4 7 8 12 13 14 15 19 21 23 24

17 2 5 6 9 10 12 14 15 18 21 24 25

18 2 5 7 9 10 13 14 17 19 20 22 23

19 2 6 7 8 10 11 15 16 18 20 23 25

20 3 '+ 5 10 11 12 15 18 19 21 22 23

21 3 4 6 10 12 13 14 16 17 20 23 25

22 3 4 7 9 11 13 14 15 18 20 24 25

23 3 5 6 8 9 13 16 18 19 20 21 24

24 3 5 7 8 9 12 15 ,6 17 22 23 25

25 3 6 7 8 10 11 14 17 19 21 22 24

Page 94: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 88 -

table 28

connection list of 815

2 3 4 5 6 7 8 9 10 11 12 132 1 3 4 5 6 7 14 15 16 17 18 19

3 1 2 4 5 6 7 20 21 22 23 24 25

4 1 2 3 8 9 10 14 15 16 20 21 22.8 14 18 245 1 2 3 11 12 17 20 23

6 1 2 3 9 11 13 15 17 19 21 23 25

7 1 2 3 10 12 13 16 18 19 22 24 25

8 1 4 5 9 10 11 14 17 19 22 24 250 1 4 6 8 10 13 16 17 18 20 23 25

10 1 4 7 8 9 12 15 18 19 21 23 24

11 1 5 6 8 12 13 14 16 19 21 22 2312 1 5 7 10 11 13 15 17 18 20 21 22

13 1 6 7 9 11 12 14 15 16 20 24 25

14 2 4 5 8 11 13 15 16 18 21 24 25

15 2 4 6 10 12 13 14 17 19 20 21 24

16 2 4 7 9 11 13 14 18 19 20 22 23

17 2 5 6 8 9 12 15 18 19 20 22 25

18 2 5 7 9 10 12 14 16 17 21 23 25

19 2 6 7 8 10 11 15 16 17 22 23 242() 3 4 5 9 12 13 15 16 17 22 23 24

21 3 4 6 10 11 12 14 15 18 22 23 25

22 3 4 7 8 11 12 16 17 19 20 21 25

23 3 5 6 9 10 11 16 18 19 20 21 24

24 3 5 7 8 10 13 14 15 19 20 23 25

25 3 6 7 8 9 13 14 17 18 21 22 24

Page 95: Conference matrices and graphs of order 26 · conference matrices of order 26 (theorem 4.4. I) and 15 pairwise nonisomor phic strongly regular graphs of order 25 (theorem 4.6.1) can

- 89 -

References

I. N.G. de Bruijn, Non-numerieke computertoepassingen, in J.J. Seidel, Compu­

terwiskunde, Utrecht (1970), 75-94.

2. F.C. Bussemaker and J.J. Seidel. Symmetric Hadamard matrices of order 36,

T.H.-Report 70-WSK-02, Eindhoven (l970).

3. P. Delsarte, An algebraic approach to the association schemes of coding

theory, Ph. D. Thesis, Universite Catholique de Louvain (1973).

4. P. Delsarte, J.M. Goethals and J.J. Seidel, Orthogonal matrices with zero

diagonal II, Canad. J. Math. 23 (1971), 816-832.

5. J.M. Goethals and J.J. Seidel, Orthogonal matrices with zero diagonal,

Canad. J. Math. ~ (1967), 1001-1010.

6. M. Hall, Jr., Combinatorial Theory (Blaisdell 1967).

7. J.H. van Lint and J.J. Seidel, Equilateral point sets in elliptic geometry,

Kon. Ned. Akad. Wetensch. Arnst. Proc. A, ~ (= Indag. Math. 28)

(1966), 335-348.

8. J.J. Seidel, Strongly regular graphs with (1,-1,0) adjacency matrix having

eigenvalue 3, Linear Algebra and Its Applications l (1968), 281­

298.

9. J.J. Seidel, On two-graphs and ShultIs characterization of symplectic and

orthogonal geometries over GF(2), T.H.-Report 73-WSK-02, Eindhoven

(1973).

10. 8.S. Shrikhaode and Vasanti N. Bhat, Graphs derivable from L3 (5) graphs,

Sankhya, series A, 33 (1971), 315-350.

11. 8.S. Shrikhande and Vasanti N. Bhat, Nonisomorphic solutions of pseudo­

(3,5,2) and pseudo-(3,6,3) graphs. Annals of the New York Academy

\)f ~~ciences (Int. Coni. Comb. Math.), 175, 331-350.