Review Article Distance Degree Regular Graphs and Distance...

Review ArticleDistance Degree Regular Graphs and Distance Degree InjectiveGraphs An Overview

Medha Itagi Huilgol

Department of Mathematics Bangalore University Central College Campus Bangalore 560001 India

Correspondence should be addressed to Medha Itagi Huilgol medhabubernetin

Received 29 June 2014 Revised 25 October 2014 Accepted 28 October 2014 Published 8 December 2014

Academic Editor Luisa Gargano

Copyright copy 2014 Medha Itagi HuilgolThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The distance 119889(V 119906) from a vertex V of119866 to a vertex 119906 is the length of shortest V to 119906 pathThe eccentricity 119890(V) of V is the distance to afarthest vertex from V If 119889(V 119906) = 119890(V) (119906 = V) we say that 119906 is an eccentric vertex of VThe radius rad(119866) is theminimum eccentricityof the vertices whereas the diameter diam(119866) is themaximum eccentricity A vertex V is a central vertex if 119890(V) = rad(119866) and a vertexis a peripheral vertex if 119890(V) = diam(119866) A graph is self-centered if every vertex has the same eccentricity that is rad(119866) = diam(119866)The distance degree sequence (dds) of a vertex V in a graph 119866 = (119881 119864) is a list of the number of vertices at distance 1 2 119890(V) inthat order where 119890(V) denotes the eccentricity of V in119866 Thus the sequence (119889

1198940 1198891198941 1198891198942 119889

119894119895 ) is the distance degree sequence

of the vertex V119894in 119866 where 119889

119894119895denotes the number of vertices at distance 119895 from V

119894 The concept of distance degree regular (DDR)

graphs was introduced by Bloom et al as the graphs for which all vertices have the same distance degree sequence By definitiona DDR graph must be a regular graph but a regular graph may not be DDR A graph is distance degree injective (DDI) graph ifno two vertices have the same distance degree sequence DDI graphs are highly irregular in comparison with the DDR graphs Inthis paper we present an exhaustive review of the two concepts of DDR and DDI graphs The paper starts with an insight into alldistance related sequences and their applications All the related open problems are listed

1 Introduction

The study of sequences in Graph Theory is not new Asequence for a graph acts as an invariant that contains alist of numbers rather than a single number A sequencecan be handled and studied as easily as a single numericalinvariant but a sequence carries more information about thegraph it represents There are many sequences representing agraph in literature namely the degree sequence the eccentricsequence the distance degree sequence the status sequencethe path degree sequence and so forth A sequence S is saidto be graphical if there is a graph which realizes S Degreesequences of graphs were the first ones to be studied as thequestion of realizability of any sequence for a graphwas a fun-damental one An existential characterization was given byErdos and Gallai [1] Then the constructive characterizationwas found independently by Havel [2] and later by Hakimi[3] that is now referred to as theHavel andHakimi algorithmThe eccentric sequences were the next and the first in the class

of distance related sequences to be studied for undirectedgraphs Some fundamental results in this direction are dueto Lesniak-Foster [4] Ostrand [5] Behzad and Simpson[6] which primarily deal with the conditions for graphicaleccentric sequences The minimal eccentric sequences weremainly studied by Nandakumar [7] But for digraphs theeccentric sequences were studied quite lateWe can find somepapers in 2008 due to Gimbert and Lopez [8]

Next distance based sequences were the path degreesequences and distance degree sequences These were studiedby Randic [9] for the purpose of distinguishing chemicalisomers by their graph structure Path degree sequence of agraph has its application in describing atomic environmentsand in various classification schemes for molecules

We will now define all the terms that give graph theoreticexpressions for the above discussed cases

For all undefined terms we refer to [10]Let 119866 = (119881 119864) denote a graph with the set of vertices

119881 whose cardinality is the order 119901 and two element subsets

Hindawi Publishing CorporationJournal of Discrete MathematicsVolume 2014 Article ID 358792 12 pageshttpdxdoiorg1011552014358792

2 Journal of Discrete Mathematics

of 119881 known as edges forming 119864 whose cardinality is size 119902Unless mentioned otherwise in this article by a graph wemean an undirected finite graph without multiple edges andself-loops

The distance 119889(V 119906) from a vertex V of 119866 to a vertex 119906 isthe length of shortest V to 119906 path The degree of a vertex V isthe number of vertices at distance one

The sequence of numbers of vertices having degree 0 12 3 is called the degree sequence which is the list of thedegrees of vertices of 119866 in nondecreasing order

The eccentricity 119890(V) of V is the distance to a farthest vertexfrom V

The minimum of the eccentricities is the radius rad(119866)and the maximum diameter of 119866 diam(119866)

A graph119866 is said to be self-centered if all vertices have thesame eccentricity

If 119889(V 119906) = 119890(V) (119906 = V) we say that 119906 is an eccentricvertex of V

The eccentric sequence of a connected graph 119866 is a list ofthe eccentricities of its vertices in nondecreasing order

The distance and path degree sequences of a vertexare generalizations of the degree of a vertex The distancedegree sequence (dds) of a vertex V in a graph 119866 is a listof the number of vertices at distance 1 2 119890(V) in thatorder where 119890(V) denotes the eccentricity of V in 119866 Thusthe sequence (119889

1198940 1198891198941 1198891198942 119889

119894119895 ) is the distance degree

sequence of the vertex V119894in 119866 where 119889

119894119895denotes the number

of vertices at distance 119895 from V119894 The 119901-tuple of distance

degree sequences of the vertices of 119866 with entries arrangedin lexicographic order is the distance degree sequence (DDS)of 119866 Similarly we define the path degree sequence (pds) ofV119894as the sequence (119901

1198940 1199011198941 1199011198942 119901

119894119895 ) where 119901

119894119895denotes

the number of paths in 119866 of length 119895 having V119894as the initial

vertex The ordered set of all such sequences arranged inlexicographic order is called the path degree sequence (PDS)of 119866 Clearly for any graph 119889

119894119895le 119901119894119895

The distance distribution of 119866 is the sequence (1198891 1198892

119889119895 ) where 119889

119895is the number of pairs of vertices in 119866 that

are at distance 119895 apartThe status 119904(V) of a vertex V in119866 is the sumof the distances

from V to all other vertices in 119866 This concept was introducedbyHarary [11]Thus using the distance degree sequence of V

119894

we have 119904(V119894) = sum

diam(119866)119895=1

119895119889119894119895

Themedian119872(119866) of119866 is the set of vertices withminimumstatus A graph is called a self-median graph if all of its verticeshave the same status

The status of a vertex is also called the distance at thatvertexThemean distance at the vertex V

119894 denoted byMD(V

119894)

is defined as MD(V119894) = 119904(V

119894)(119901 minus 1)

The mean distance for the graph denoted by MD(119866) isdefined as the mean of the distances between all pairs ofvertices of 119866 that is MD(119866) = sum

119906V 119889(119906 V) (119901

2) where 119906

and V are vertices of 119866 In terms of status we get MD(119866) =sum119901

119894=1(MD(V

119894)119901)

As an illustration consider the graph 119866 as shown inFigure 1

Figure 1

DDS(119866) = ((1 1 1 2) (1 2 2) (1 3 1) (1 2 1 1)2) and

PDS(119866) = ((1 1 1 2 2) (1 2 2 2) (1 2 3 2 1)2 (1 3 3))If we consider the 3-dimensional cube 119876

3as an example

we will get DDS(1198763) = ((1 3 3 1)

8) and PDS(119876

3) = ((1 3 6

12 12 12)6)

In [12] Kennedy and Quintas have examined how thedistance degree sequences relate to embedding trees inlattice-graphs and other spaces as well as the consequentialrelations to physical theories Chemists have also proposedand discussed other sequences for this purpose Their objec-tive and hope are to develop a ldquochemical indicatorrdquo whichwould be useful both for the graph isomorphism problemand to predict various properties of the molecule at hand Inthis connection Randic [9] proposed the structural similaritydepending on the graph property as follows

Conjecture 1 (see [9]) Between pairs of atoms in acyclicstructures there is a unique path so that the number of pathsof a given length corresponds to the number of neighbors at agiven distance

This can be rephrased in graph theoretic terms as follows

Conjecture 2 (see [9]) For a connected graph 119866 119863119863119878(119866) =119875119863119878(119866) if and only if 119866 is a tree

Anonassertive proof was given byQuintas and Slater [13]Hence this kind of study leads to the problem of finding

the smallest order for which there exists a pair of noniso-morphic graphs which are equivalent relative to some set ofgraphical invariants So the following conjecture wasmade inline with the above conjecture by Randic [9] that sought forgraph isomorphism by comparing their sequences

Conjecture 3 (see [9]) Two graphs1198661and119866

2are isomorphic

if and only if 119875119863119878(1198661) = 119875119863119878(119866

2)

This conjuncture was also shown to be invalid by citinginfinite class of pairs of nonisomorphic trees having the prop-erty that the two members of each pair have the same pathdegree sequence by Quintas and Slater [13] To extend thisnotion to nontree graphs the same examples were consideredto construct pairs of nonisomorphic graphs having a varietyof properties and invalidate the conjecture

As noted above a tree is not in general characterizedby its path (or distance) degree sequence The least order forwhich there exists a pair of nonisomorphic trees with thesame path degree sequence has been shown to be more than14 for acyclic alkanes that is trees with no vertex of degree

Journal of Discrete Mathematics 3

Figure 2 A pair of nonisomorphic trees with the same path degreesequence

Figure 3 Smallest order graphs with the same distance degreesequence

more than 4 and less than or equal to 18 for trees withoutvertex degree restrictions

On similar lines one can search for pairs of noniso-morphic graphs having the same distance degree sequenceSlater [14] constructed a pair of nonisomorphic 18 vertextrees having the same distance degree sequence as shownin Figure 2 He had subsequently conjectured that no suchsmaller order pair exists Later Gargano and Quintas [15]considered the problem with respect to the number ofindependent cycles in a graph having the same distancedegree sequence We can recall that 120573 = 119902 minus 119901 + 119896 where120573 is the number of independent cycles 119896 is the number ofcomponents in the graph and 119901 119902 are the order and sizerespectively For 120573 = 0 Slaterrsquos constructions hold goodFor 120573 = 1 Gargano and Quintas [15] constructed two pairsof graphs on 14 vertices For 120573 = 2 the smallest orderpair of nonisomorphic graphs with the same distance degreesequence is given as in Figure 3

And this is the only such pair as noted by Quintas andSlater in [13]This class of graphs was generalized by Garganoand Quintas [15] for 120573 ge 2 as follows

Theorem 4 (see [15]) If 120573 ge 2 then there exists a pairof nonisomorphic connected graphs having the same distancedegree sequence and 120573 independent cycles The smallest order119901 for such a pair is given by 119901 = lceil(3+radic33 + 8120573)2rceil where lceil119904rceildenotes the least integer greater than or equal to 119904

Two problems were asked in the same paper

Problem 5 (see [15]) What is the smallest order 119901 for whichthere exists a pair of nonisomorphic connected graphs havingthe same distance degree sequence and 120573 = 0 1 independentcycles

Problem 6 (see [15]) What is the smallest order 119901 for whichthere exists a pair of nonisomorphic connected graphs havingthe same distance degree sequence and 120573 ge 0 (120573 = 2 3 4 5and 7) independent cycles and such that the graphs have novertex with degree greater than four

With reference to Problem 6 consider the following

If 120573 = 0 then 119901 is at least 15 (see Slater [14] p17)

If 120573 = 1 Gargano and Quintas [15] conjectured 119901 =14

If 120573 = 2 the solution is 119901 = 5 (see Figure 3)

For several values of 120573 ge 3 the solution is the same asthat for Problem 5 Specifically for 120573 = 3 4 5 and 7 each ofthe graphs constructed in the proof of the theorem in [15] hasno vertex of degree greater than fourThus the open cases are120573 = 0 1 6 and 120573 ge 8

If one asks for the smallest order for which the dis-tance degree sequence fails to distinguish between 119896-regulargraphs there exists the following result due to Quintas andSlater [13]

Proposition 7 (see [13]) Let 119877(119896) denote the smallest orderfor which there exist at least two nonisomorphic connected 119896-regular graphs on 119877(119896) vertices such that each has the samedistance degree sequence Then 119877(119896) = 119896 + 3 if 119896 = 3 4 5 and 119877(119896) does not exist if 119896 = 0 1 2

As noted above there does not exist a pair of nonisomor-phic graphs with the same path degree sequence on less than12 vertices It also follows from the above result that the leastorder for which such a 119896-regular pair of graphs can exist is atleast 119896 + 3 for 119896 ge 3 and that none exist if 119896 = 0 1 2 Thus theinvestigation to produce at least one pair of nonisomorphic119896-regular graphs having the same path degree sequence hasresulted in an open problem

Problem 8 (see [13]) For 119896 ge 3 does there exist a pair ofnonisomorphic connected 119896-regular graphs having the samepath degree sequence If the answer is yes what is the smallestorder 119877(119896) realizable by such a pair


In fact a paper by Slater [14] discusses the counterexam-ples to Randicrsquos conjectures on distance degree sequences ontrees

In [16] Bussemaker et al have investigated all connectedcubic graphs on 119901 le 14 vertices and for each such graph thefollowing properties were determined and tabulated

(i) its description by means of its edges and for 119901 le 12by a drawing

(ii) the coefficients of its characteristic polynomial(iii) the eigenvalues of its (0 1)-adjacency matrix(iv) the number of its cycles of length 3 4 119901(v) its diameter connectivity and planarity(vi) the order of its automorphism group

From their table the girth the circumference and thechromatic number are easily determined The graphs wereclassified according to their order and within each suchclass the graphs were ordered lexicographically accordingto their eigenvalues which for each graph were listed innonincreasing order In addition a number of observationsconcerning the spectral properties of these graphsweremadeThe study was motivated by the importance of cubic graphsand by the search for cospectral cubic graphsThe cubic graphgeneration is looked into by Brinkmann [17] In [18] Halber-stam and Quintas have extended the results of Bussemakeret al [16] mainly considering the sequences associated withthe graphs In this report they have determined the followinginformation for each cubic graph on less than or equal to 14vertices

(i) its distance degree sequence(ii) its distance distribution(iii) the mean distance at each vertex(iv) the mean distance for the graph(v) its path degree sequence(vi) the number of paths of specified length(vii) the total number of paths for the graph

For each graph a number of other parameters can bedetermined from these tables The radius the diameter andthe eccentricity of every vertex can be read directly by notingthe length of the appropriate distance degree sequence Byinspection one can determine the order of the center andof the median of the graph In addition the computation ofother parameters is facilitated by making use of the givendata

2 Distance Degree Regular (DDR) Graphs

Nowwehave the stage set for the discussion on the highly reg-ular class of graphs involving the distance degree sequencesnamely the distance degree regular graphsThe concept of dis-tance degree regular (DDR) graphs was introduced by Bloomet al [19] as the graphs for which all vertices have the samedistance degree sequence that is for all vertices V of a graph119866the distance degree sequence is (119889

1198940 1198891198941 1198891198942 119889

119894119895 )

For example the three-dimensional cube1198763= 1198702times1198702times

1198702is a DDR graph having (1 3 3 1) as the distance degree

sequence of each of its vertex Likewise cycles completegraphs are all DDR graphs Lattice-graphs and infinite orderregular trees are examples of infinite order DDR graphsHowever in this paper we consider finite order graphs onlyDDR graphs have peculiar relations to many parameters ofgraphs On one extreme complete graphs are DDR and at theother end there are DDR graphs having identity automor-phism group If a graph has a vertex-transitive automorphismgroup then the graph is DDR but not conversely Andthe properties of DDR graphs being self-centered and self-median find applications even in operations research Thusthe study of DDR graphs is interesting

In [19] Bloom et al have discussed many properties ofDDR graphs It is clear that the set of DDR graphs of degreele2 consists of the following types (i) a collection of 119896 isolatedvertices (ii) a collection of 119896 disjoint edges and (iii) 119896 copiesof 119899-cycles Hence the next result is a fundamental one

Proposition 9 (see [19]) Each regular graph of diameter le2is DDR and the complement of each regular graph of diameterge3 is DDR

This result ensures that ldquoevery regular graph with diame-ter at most two is DDRrdquo We know that every DDR graph isregular but the distribution of DDR graphs in regular graphsis not clear So the following question is relevant

Problem 10 (see [19]) What proportion of 119896-regular graphsare DDR

Following constructions help in getting DDR graphs ofarbitrarily chosen diameter and degree For thesemethodswerequire the following terms

Definition 11 (see [19]) Let 119866 denote a vertex-labeled graphwith vertex set 119881 = 1 2 119901 For any permutation120572 of 119881 the 119896-cycle-120572-permutation graph of 119866 denoted by119896119862119875120572(119866) consists 119896 copies of119866 joined by the additional edges

(119894 119895) (120572(119894) 119895+1) where (119894 119895) is the 119894th vertex of the 119895th copyof 119866 and 119895 + 1 = 1 when 119895 = 119896

Note that the special case 2119862119875120572(119866) is the 120572-permutation

graph of119866 119875120572(119866) introduced by Chartrand and Harary [20]

if and only if 120572 is a permutation of order less than or equalto 2 Furthermore 119896119862119875

119894119889(119866) is the cartesian product 119862

119896times 119866

of the 119896-cycle 119862119896and the graph 119866 Thus if 119866 is a 119899-cycle it

follows that 119896119862119875119894119889(119862119899) = 119899119862119875

119894119889(119862119896)

So the next result gives a family of DDR graphs

Proposition 12 (see [19]) If 119866 is any DDR graph of diameter119898 and degree 119863

1 then (i) 119875

119894119889(119866) is a DDR graph of diameter

119898+1 and degree1198631+1 and (ii) for 119896 ge 3 119896119862119875

119894119889(119866) = 119862

119896times119866

is a DDR graph of diameter119898 + lfloor1198962rfloor and degree1198631+ 2

This result can be extended to 119873 iterated cartesianproducts as shown by the following corollary

Corollary 13 (see [19]) 119875119873119894119889(119866) = 119870

119873

2times 119866 has diameter119898+119873

and degree 1198631+ 119873 and 119862119873

119896times 119866 equiv 119862

119873minus1

119896times 119866 has diameter


Figure 4

119898 + 119873lfloor1198962rfloor and degree 1198631+ 2119873 where 119875

119873

119894119889(119866) =

119875119894119889(119875119873minus1

119894119889(119866)) = 119870

119873

2times 119866 as 119875

119894119889(119866) = 119870

119873

2times 119866 that is the

119873 iterated cartesian products

Theexistence of aDDRgraphhaving diameter119898 is shownas follows

Proposition 14 (see [19]) There exist DDR graphs havingdiameter 119898 and degree 119863

1for all positive integers 119898 and 119863

1

except for119898 gt 1198631= 1

Observe that the only connected 1-regular graph is 1198702

But characterizing DDR graphs of higher diameter ischallenging In [19] the following problem was cited

Problem 15 (see [19]) CharacterizeDDRgraphs having diam-eter ge3

Since then there is no answer to this question A stepforward was taken by Huilgol et al [21] where they havecharacterized DDR graphs of diameter three with extremaldegree regularity They have shown that there are exactly 7DDR graphs of diameter three of extreme regularity Thefollowing result gives a clear picture about diameter threeDDR graphs

Theorem16 (see [21]) Let119866 be aDDRgraph of diameter threeof regularity 119889

1 Then

(i) 1198891= 119901 minus 4 if and only if 119866 = 119862

6

(ii) 1198891= 119901 minus 5 if and only if 119866 = 119862

7 or 119866 = 119876

3= 1198702times

1198702times 1198702

(iii) 1198891= 119901minus 6 if and only if 119866 is one of the graphs given in

Figure 4(iv) 119889

1= 2 if and only if 119866 = 119862

6or 119866 = 119862

7

The proof of the above theorem is primarily based on thefollowing two results

Proposition 17 (see [3]) Every connected self-centered graphof order 119901 satisfies the inequality Δ le 119901 minus 2119903 + 2 whereΔ = Δ(119866) and 119903 = rad (119866) denote respectively the maximumdegree and the radius of the graph 119866

Proposition 18 (see [10]) In a connected graph 119866 of order 119901and radius 119903 for any two arbitrarily chosen vertices 119906 and Vthe following inequality holds deg(119906) + deg(V) le 119901 minus 2119903 + 4

As the diameter three DDR graphs are considered whichare self-centered graphs satisfying the inequality 2 le Δ(119866) le119901 minus 4 In the above theorem the DDR graphs of regularity2 are characterized at one extreme and of regularities 119901 minus 4119901 minus 5 and 119901 minus 6 at the other extreme of the above mentionedinequality Using these results we can find a bound on theorder of the graph as 119901 le 2119894 minus 4 with degree regularity 119889

1=

119901 minus 119894 Hence for the degree regularity 1198891= 119901 minus 119894 119894 isin 4 5 6

of a DDR graph of order 119901 we get 119901 = 6 8 10 respectivelyBut in general the question of characterizing DDR graphsof diameter greater than two still remains open At least theexistence is answered by Huilgol et al in [21] by proving thefollowing result

Theorem 19 (see [21]) For any integer 119896 there exists a DDRgraph of diameter three and regularity 119896


So the construction of new families ofDDRgraphs is veryinteresting In [22] Huilgol et al have constructed somemoreDDR graphs of higher diameter and considered the behaviorof DDR graphs under other graph binary operations Inthis paper the authors have given several DDR graphs ofarbitrary diameter Few of them depend on the generalizedlexicographic product of graphs

The lexicographic product is defined as followsGiven graphs 119866 and 119867 the lexicographic product 119866[119867]

has the vertex set (119892 ℎ) 119892 isin 119881(119866) ℎ isin 119881(119867) and twovertices (119892 ℎ) (1198921015840 ℎ1015840) are adjacent if and only if either 1198921198921015840 isan edge of 119866 or 119892 = 1198921015840 and ℎℎ1015840 is an edge of119867

Proposition 20 (see [22]) Let 119866 be a DDR graph and let 119866119894

1 le 119894 le 119901 be undirected graphs such that |119881(119866119894)| = 119901 for all

119894 1 le 119894 le 119901 and then 119866[1198661 1198662 1198663 119866

119901] the generalized

lexicographic product is a DDR graph if and only if all119866119894 1 le

119894 le 119901 are regular graphs with the same regularity

This characterizes the generalized lexicographic prod-ucts Next result uses the generalized lexicographic productto show the existence of a DDR graph of arbitrary diameter

Proposition 21 (see [23]) There exists a DDR graph of diam-eter 119889 ge 3 of order 119901 = (2119889 + 1)119899 and 119901 = 2119889119899 where 119899 = 12 3

Proposition 22 (see [22]) There exists a DDR graph 119866 ofdiameter 119889 such that 1198662 is also DDR with diameter 1198892 if 119889is even and (119889 + 1)2 if 119889 is odd

Proposition 23 (see [22]) If 119866 is a DDR graph with diameter119889 and regularity 119889

1 then the prism of 119866 is also a DDR graph

on 2119901 number of vertices with diameter 119889 + 1 and regularity1198891+ 1

In [23] Huilgol et al have constructed higher order DDRgraphs by considering the simplest of the products namelythe cartesian and normal product The cartesian product oftwo graphs 119866 and119867 denoted by 119866◻119867 is a graph with vertexset 119881(119866◻119867) = 119881(119866) times 119881(119867) that is the set (119892 ℎ)119892 isin

119881(119866) ℎ isin 119881(119867) The edge set of 119866◻119867 consists of all pairs[(1198921 ℎ1) (1198922 ℎ2)] of vertices with [119892

1 1198922] isin 119864(119866) and ℎ

1= ℎ2

or 1198921= 1198922and [ℎ

1 ℎ2] isin 119864(119867)

And the normal product of two graphs119866 and119867 denotedby 119866 oplus 119867 is a graph with vertex set 119881(119866 oplus 119867) = 119881(119866) times

119881(119867) that is the set (119892 ℎ)119892 isin 119881(119866) ℎ isin 119881(119867) and anedge [(119892

1 ℎ1) (1198922 ℎ2)] exists whenever any of the following

conditions hold good (i) [1198921 1198922] isin 119864(119866) and ℎ

1= ℎ2 (ii)

1198921= 1198922and [ℎ

1 ℎ2] isin 119864(119867) and (iii) [119892

1 1198922] isin 119864(119866) and

[ℎ1 ℎ2] isin 119864(119867)

Theorem 24 (see [23]) Cartesian product of two graphs 1198661

and 1198662is a DDR graph if and only if both 119866

1and 119866

2are DDR

graphs

Lemma 25 (see [23]) Let 119878 = 119866119894119894 ge 2 If there exists 119896

such that 119866119896is self-centered and diam(119866

119896) ge diam(119866

119894) for all

119894 ge 2 then normal product of all the graphs in 119878 is a self-centeredgraph with diameter equal to diam(119866

119896)

The next result characterizes the normal product of twoDDR graphs

Theorem26 (see [23]) Normal product1198661oplus 1198662of two graphs

1198661and119866

2is DDR if andonly if both119866

1and119866

2areDDR graphs

3 Relationship with Other Properties

As mentioned earlier the relation of DDR graphs with othergraph properties is relatively an unexplored area In this sectionwe try to relate DDR graphs with some symmetry properties

Let 119866 denote a connected graph with automorphismgroup Aut(119866) Then 119866 is distance-transitive if for eachquartet of vertices 119909 119910 119906 and V in 119866 such that 119889(119909 119910) =119889(119906 V) there is some 120574 in Aut(119866) satisfying 120574(119909) = 119906 and120574(119910) = V

119866 is symmetric (1-transitive) if for each quartet of vertices119909 119910 119906 and V in119866 such that 119909 is adjacent to 119910 and 119906 is adjacentto V there is some 120574 in Aut(119866) satisfying 120574(119909) = 119906 and 120574(119910) =V

119866 is vertex-transitive if for each pair of vertices 119909 and 119910in 119866 there is some 120574 in Aut(119866) satisfying 120574(119909) = 119906

119866 is edge-transitive if for each pair of edges 119909 119910 and119906 V in 119866 there is some 120574 in Aut(119866) satisfying 120574119909 119910 equiv120574(119909) 120574(119910) = 119906 V

119866 is distance-regular with diameter 119898 if it is a k-regularconnected graph with the following property

There are natural numbers 1198870= 119896 119887

1 119887

119898minus1 1198881=

1 1198882 119888

119898such that for each pair of vertices 119909 and 119910 in 119866

satisfying 119889(119909 119910) = 119895 it follows that

(i) the number of vertices at distance 119895 minus 1 from 119910 andadjacent to 119909 is 119888

119895(1 le 119895 le 119898)

(ii) the number of vertices at distance 119895 + 1 from 119910 andadjacent to 119909 is 119887

119895(0 le 119895 le 119898 minus 1)

In [24] Biggs has listed the results to show that a sym-metric graph is both vertex-transitive and edge-transitive andthat a distance-transitive graph is edge-transitive symmetricand distance regular Based on these two results Bloom et al[19] proved the following

Proposition 27 (see [19]) A graph that is vertex-transitive ordistance-regular must be DDR

In the next result Bloom et al [19] have shown thatdistance degree regularity is weaker than other symmetryconditions and it is also indicated that other well-studiedsymmetry conditions do not imply DDR

Proposition 28 (see [19]) Distance degree regularity doesnot imply edge-transitivity vertex-transitivity nor distance-regularity Also an edge-transitive graph is not necessarilyDDR(and hence neither vertex-transitive symmetric nor distancetransitive)

Vertex-transitivity does not imply edge-transitivity sym-metry nor distance-regularity Using the following Bloom etal [19] have shown that a DDR graph is vertex-transitive butnot distance-regular


1

2

x

y v

u

43

5

10

9 8

z6

7

Figure 5

The graph shown in Figure 5 is 1198625times 1198623 which is a DDR

graph This graph is not distance-regular as there are twovertices adjacent to both 119909 and 119910 whereas only one vertexis adjacent to both 119909 and V given that 119889(119909 119910) = 119889(119909 V) = 2Also since the edge 119906 V lies on a triangle and 119906 119911 doesnot and these two edges are not equivalent it shows that thegraph is not edge-transitive Hence it is not symmetric

Distance-regularity does not imply vertex-transitivityThis is shown by an example noted by Biggs in [24 p 139]and attributed to Adelrsquoson-Velskii [25] of a distance-regulargraph hence DDR with 26 vertices which is not vertex-transitive and therefore is neither symmetric nor distance-transitive

Symmetry implies neitherdistance-transitivity nor distance-regularity The former observation is demonstrated by a DDRexample due to Frucht et al [26] They showed the impli-cation did not hold for the permutation graph 119875

120572(8-119888119910119888119897119890)

where 120572 is the permutation (2 4)(3 7)(6 8) This is the sym-metric generalized Petersen graph 119875(8 3) Bloom et al [19]demonstrated the second implication by the DDR example1198752

119894119889(6-119888119910119888119897119890) = 1198702

2times 1198626= 1198624times 1198626as shown in Figure 6 This

graph can be shown to be symmetric by viewing it on a torusBut we can show that it is not a distance-regular graph onsimilar lines as done for the graph of Figure 5 by consideringthe vertex 119909 with the vertices 119910 and 119906

Bloom et al [19] have shown thatmost of the well-studiedsymmetry conditions imply the DDR property For exampleamong the four permutation graphs of the 5-cycle only themost symmetric pair that is the119875

119894119889(5-119888119910119888119897119890) and the Petersen

graph are DDR So they posed the following question tocheck the effect of symmetry on permutation graphs

Problem 29 (see [19]) Which 119896-cycle-120572-permutation graphsof a given graph are DDR

It is interesting to know that although most symmetryconditions suffice to make a graph DDR there are DDRexamples among the least symmetric graphs One such class

1

x

y

113

2

4

12

5

13

6

14

7

15

816

917

10

18

u

19

20

21

Figure 6

is DDR graphs having identity automorphism group If 119896 lt3 the only 119896-regular graph having identity automorphismgroup is the singleton graph 119870

1 which is trivially DDR

For 119896 ge 3 there exist 119896-regular graphs having identityautomorphism group For these the least order realizable hasbeen determined by [27ndash30] Bloom et al [19] got the leastordered DDR graphs of identity graphs as the complementsof the graphs given in [30] Hence they proved the followingresult

Proposition 30 (see [19]) If 119881(119896) is the least order realizableby a 119896-regular DDR graph having identity automorphismgroup then119881(5) = 10119881(6) = 11 and for 119896 ge 7119881(119896) = 119896+4if k is even and 119881(119896) = 119896 + 5 if 119896 is odd

For graphs with regularities 3 and 4 the situation is notclear Although a least-order 3-regular identity graph has 12vertices none of the five graphs (see [30]) satisfying thiscondition is DDRThus119881(3) ge 14 It has also been shown byHaigh (in a private communication to Bloom et al [19]) thatthere are noDDR graphs among the 103 cubic identity graphson 14 vertices which are contained in the complete tabulationof cubic graphs on graphs up to 14 vertices given in [16]Thus119881(3) ge 16 But the existence of a order 10 DDR identity graphis shown by Bloom et al [19] Also they posed the followingproblem

Problem 31 (see [19]) If 119896 = 3 or 4 do there exist any 119896-regular DDR graphs having identity automorphism group Ifthe answer is yes what is the least order realizable by suchgraphs

Next we will consider the relation of DDR graphs with abinary relation defined on the graph namely the eccentricdigraph The eccentric digraph of a graph or digraph is adistance based mapping defined on a relation induced bydistances in a graph or digraph that is also representedby a graph Many distance based relations can be found in


literature in the study of antipodal graphs [31] antipodaldigraphs [31] eccentric graphs [32 33] and so forth Theeccentric digraph ED(119866) of a graph (or digraph) 119866 is thedigraph that has the same vertex as 119866 and an arc from 119906 toV exists in ED(119866) if and only if V is an eccentric vertex of 119906in 119866 Eccentric digraphs were defined by Buckley in [34] In[22] Huilgol et al have considered the relations between thedistance degree regular (DDR) graphs and eccentric digraphsThe DDR graphs being self-centered yield symmetric eccen-tric digraphs The study of eccentric digraphs involves thebehavior of sequences of iterated eccentric digraphs and theyare studied by Gimbert et al [35] Given a positive integer119896 ge 2 the 119896th iterated eccentric digraph of 119866 is written asED119896(119866) = ED(ED119896minus1(119866)) where ED0(119866) = 119866 and ED1(119866) =ED(119866) For every digraph119866 there exist smallest integers 119904 gt 0and 119905 ge 0 such that ED119905(119866) = ED119904+119905(119866) In case of labeledgraphs 119904 is called the period of 119866 and 119905 the tail of 119866 whereasfor unlabeled graphs these quantities are referred to as iso-period denoted by 1199041015840(119866) and iso-tail 1199051015840(119866) respectively

In [22] Huilgol et al have showed that the uniqueeccentric vertex DDR graphs have all their iterated eccentricdigraphs as DDR graphs In the same paper many resultsare proved showing the relation between DDR graphs andeccentric digraphs

Proposition 32 (see [22]) There exists a DDR graph 119866 suchthat ED(119866) cong 119866

Proposition 33 (see [22]) There exists a DDR graph 119866 suchthat ED(119866) is a disconnected graph each of whose componentsis complete bipartite graph

Proposition 34 (see [22]) For a unique eccentric vertex DDRgraph if 119901 gt 4 then tail = 1 period = 2 and if 119901 = 4 tail = 0period = 2

Proposition 35 (see [22]) For a given diameter 119889(ge 3) thereexist 2119889 minus 6 DDR graphs with tail = 1 and period = 1

Remark 36 Theunique eccentric vertex DDR graphs have alltheir iterated eccentric digraphs as DDR graphs

For nonunique eccentric vertex DDR graphs the problemremains still open

Problem 37 (see [22]) When do nonunique eccentric vertexDDR graphs have all their iterated digraphs as DDR graphs

4 Distance Degree Injective (DDI) Graphs

In contrast to distance degree regular (DDR) graphs theDistance Degree Injective (DDI) graphs are the graphs withno two vertices of 119866 having the same distance degreesequence (dds) These were also defined by Bloom et al [36]So these graphs are highly irregular compared to the DDRgraphs But the other extreme of theDDR graphs as we notedabove to have identity automorphism group becomes aninherent property of DDI graphs which is evident by the nextresult

Theorem 38 (see [36]) If 119866 is a DDI graph then 119866 has anidentity automorphism group but not conversely

Figure 7

1

2

Figure 8

The converse is obvious as there are DDR graphs withidentity automorphism group

As DDR graphs the characterizations elude DDI graphsBut there are many existential results and examples of DDIgraphs

Theorem39 (see [36]) A smallest order nontrivial DDI graphhas order 7 and there exist DDI graphs having order 119901 for all119901 ge 7

The smallest order nontrivial identity graph has 6 verticesand there are exactly eight identity graphs on 6 vertices andthese are not DDI So the least order is 7 Actually the identitytree as shown in Figure 7 is the smallest DDI graph

The following result shows the existence of a DDI graphof arbitrary diameter

Theorem 40 (see [36]) A smallest diameter nontrivial DDIgraph has diameter 3 and there exist DDI graphs havingdiameter119898 for all119898 ge 3

The graphs shown in Figure 8 are DDI and have diameter3 and 4 respectively The graph of Figure 7 is a DDI graphwith diameter 5 For DDI graphs of diameter 6 7 we canconsider the same graph as shown in Figure 7 and go onadding vertices on the left of the path P

6

Theorem 41 (see [36]) If 119866 is a connected DDI graph and119866 = 119870

1 then the diameter of 119866 is bounded by 3 and sharply

bounded by |119866| minus 2


1

2

Figure 9

The following theorem is useful in showing that a graphis not DDI

Theorem 42 (see [36]) If 119866 contains two vertices V1 V2with

the same degree and such that no vertex of 119866 is further thandistance 2 from both V

1and V2 then 119866 is not DDI

Note that the complement of the diameter three graphshown in Figure 8 is the diameter three identity graph shownin Figure 9 We can see that the above theorem is applicablefor this graph and hence it is not DDI

But the picture is not clear when we wish to have a graphand its complement to be DDI

Problem 43 (see [36]) Does there exist a graph 119866 = 1198701such

that 119866 and its complement are both DDI

So the next question was on 119896-regular DDI graphs asposed by Bloom et al [36]

Problem 44 (see [36]) Does there exist a nontrivial 119896-regularDDI graph

This problem got resolved by the existence of a cubic DDIgraph on 24 vertices and having diameter 10 as found in [37]The general existence problem was resolved by Bollobas in[38] where he showed the following

Theorem 45 (see [38]) Let 119896 ge 3 and 120598 gt 0 be fixed Set119896 = lfloor(12 + 120598)(log119901) log(119896 minus 1)rfloor Then as 119901 goes to infinitythe probability tends to one that every vertex V

119894in a 119896-regular

labeled graph of order 119901 is uniquely determined by the initialsegments 119889

1198940 1198891198941 1198891198942 119889

119894119895of its distance degree sequence

Since the distance degree sequence of a graph is indepen-dent of a labeling this result shows that almost all 119896-regulargraphs of order119901 areDDIprovided that119901 is large enoughTheproblem that remains unresolved is that of finding minimalDDI regular graphs

In this direction there was one more problem defined byHalberstam and Quintas [37] as follows

Problem 46 (see [37]) For 119896 ge 3 what is the smallestorder andor diameter for which there exists a 119896-regular DDIgraph

a b

c d

Figure 10

a

b

c

d

middot middot middot

y1 y2

x1 x2

yk

xk

ykminus1

xkminus1

Figure 11

Figure 12

Martinez and Quintas [39] found a cubic DDI graphhaving diameter 8 and order 22 as in Figure 10 They alsoshowed that if in the graph of Figure 10 the edges ab and cdare replaced by the graph shown in Figure 11 one can obtaina cubic DDI graph with 22 + 2119898 vertices and diameter 8 +119898This was further reduced to order 18 and diameter 7 cubicgraph by Volf [40] by constructing the graph of Figure 12

From [18 36 37 39 40] it follows that

(i) if there is a cubic DDI graph having less than 18vertices then its order must be 16

(ii) if there is a cubicDDI graph having diameter less than7 then its diameter must be 4 5 or 6

So all these cases were considered by Huilgol and Rajesh-wari [41] and proved the nonexistence of such a cubic DDIgraph

Theorem47 (see [41]) There does not exist a cubic DDI graphof order 16 with diameters 4 5 and 6

So the graph of order 18 as in [40] shown in Figure 12 isthe smallest order cubic DDI graph


On the other extreme construction of new DDI graphsfrom the smaller orderdiameter is also interesting and islooked into by Huilgol et al [23] In this paper many resultsare discussed for different products and the behavior of DDIgraphs with respect to these products

Theorem 48 (see [23]) If the cartesian product of two graphs1198661and 119866

2is DDI then both 119866

1and 119866

2are DDI graphs

But the converse is not true as the cartesian product ofthe DDI graph represented by Figure 7 and diameter 4 DDIgraph of Figure 8 is not a DDI graph

Lemma 49 (see [23]) Let 1198661and 119866

2be two DDI graphs Let

119860 = 119889119889119904(119906119894)119906119894isin 119881(119866

1) and 119861 = 119889119889119904(V

119894)V119894isin 119881(119866

2) If

|119860 cap 119861| ge 2 then 1198661◻1198662is not DDI

Theorem 50 (see [23]) Let 1198661and 119866

2be any two graphs Let

119906 be a vertex in 1198661and let 119878 be a subset of 119881(119866

2) such that no

two vertices of 119878 have the same distance degree sequence thenno two vertices of 119906times119878 have the same distance degree sequencein 1198661◻1198662

Let 1198781and 1198782be two subsets of119881(119866

1) For each pair (119909 119910)

with 119909 isin 1198781and 119910 isin 119878

2we can see that dds(119909) = dds(119910)

by applying the Theorem 50 Let 119911 isin 119881(1198662) be any vertex

in 1198662 and then in 119866

1◻1198662 the subsets (119909 119911)119909 isin 119878

1 and

(119910 119911)119910 isin 1198782 are such that every pair ((119909 119911) (119910 119911))119909 isin

1198781 119910 isin 119878

2satisfies dds(119909 119911) = dds(119910 119911)

Similar results are proved for normal product of DDIgraphs

Proposition 51 (see [23]) If the normal product 1198661oplus 1198662of

two graphs is DDI then both 1198661and 119866

2are DDI

As in the case of cartesian product the normal productof two DDI graphs need not be DDI and the same exampleserves the purpose

Lemma 52 (see [23]) Let 1198661and 119866


119860 = 119889119889119904(119906119894)119906119894isin 119881(119866

1) and 119861 = 119889119889119904(V

119894)V119894isin 119881(119866

2) If

|119860 cap 119861| ge 2 then 1198661oplus 1198662is not DDI

Proposition 53 (see [23]) Let 1198661and 119866

2be any two graphs

Let 119906 be a vertex in 1198661and let 119878 be a subset of 119881(119866

2) such that

no two vertices of 119878 have the same distance degree sequencethen no two vertices of 119906 times 119878 have the same distance degreesequence in 119866

1oplus 1198662

Remark 54 Let there be two sets 1198781and 1198782 subsets of 119881(119866

1)

such that every pair (119909 119910) with 119909 isin 1198781and 119910 isin 119878

2 satisfies

dds(119909) = dds(119910) Let 119911 isin 119881(1198662) be any vertex in119866

2 and then

in 1198661oplus 1198662 the subsets (119909 119911)119909 isin 119878

1 and (119910 119911)119910 isin 119878

2

are such that every pair ((119909 119911) (119910 119911))119909 isin 1198781 119910 isin 119878

2satisfies

dds(119909 119911) = dds(119910 119911)

5 Embeddings

Kennedy and Quintas [12] considered the extremality of 119891-trees when they are constrained by embedding either in

lattice-graphs or in Euclidean spaces In chemistry suchconstraints appear in the form of postulates concerning theminimal energywhich can be attained by certain connectivitypatterns of atoms in space This approach offers a frameworkensuring a physical theory to remain self-consistent withrespect to its embedding space requirements So given thedistance degree sequence of a graph it is interesting to seewhether we can get some information about the embeddingproperties of the graph Hence the next natural question willbe on embedding a graph in DDRDDI graph if not possiblethen try to find the forbidden class of subgraphs In thiscontext two problems were posed by Bloom et al [36] whichare listed below

Problem 55 (see [36]) Does there exist a pair of noniso-morphic graphs having the same distance degree sequenceand such that one of them is nonplanar with a subgraphhomeomorphic to 119870

5and the other graph is planar

Problem 56 (see [36]) Does there exist a pair of nonisomor-phic graphs having the same path degree sequence and suchthat only one of the graphs is planar

At the other extreme Huilgol et al [42] have consideredthe embedding of DDR and DDI graphsThe following resultrules out the forbidden class of subgraphs for DDR graphs byembedding any graph into a DDR graph

Theorem 57 (see [42]) Any graph can be embedded in a DDRgraph

DDR graphs exhibit high regularity in terms of thevertices and their distance distribution If we relax one vertexto have different distance degree sequence then we call sucha graph an almost distance degree regular graph or in shortan ADDR graph Similarly we define ADDI graphs

Definition 58 (see [42]) A graph 119866 of order 119901 is said tobe almost DDR if 119901 minus 1 vertices have the same distancedegree sequence and one vertexwith different distance degreesequence

Definition 59 (see [42]) A graph 119866 of order 119901 is said to bealmost DDI if 119901 minus 2 vertices have different distance degreesequences and two vertices with the same distance degreesequence

Embedding in DDI graphs seems not so easy But Huilgolet al [42] could embed only cycles paths in DDI graphs asfollows

Theorem 60 (see [42]) Any path can be embedded in analmost DDI graph

Proposition 61 (see [42]) Every cycle can be embedded in aDDI graph

So the following problems were posed in [42]

Problem 62 (see [42]) Can any graph be embedded in a DDIgraph


Problem 63 (see [42]) Does there exist a 119896-regularDDI graphfor 119896 ge 4

One more problem can be posed at this juncture asfollows

Problem 64 Does there exist a self-centered 119896-regular DDIgraph

We conclude this articlewith a comment that even thoughthere are numerous examples and results on DDR and DDIgraphs available in literature the characterizations eludeHence many open problems in these areas still persist andkeep researchers working

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] P Erdos andTGallai ldquoGraphswith prescribed degrees of verticesrdquoMatematikai Lapok vol 11 pp 264ndash274 1960 (Hungarian)

[2] V Havel ldquoA remark on the existence of finite graphsrdquo CasopisPro Pestovanı Matematiky vol 80 pp 477ndash480 1955 (Czech)

[3] SLHakimi ldquoOn realizability of a set of integers as degrees of thevertices of a linear graph Irdquo Journal of the Society for Industrialand Applied Mathematics vol 10 pp 496ndash506 1962

[4] LM Lesniak-Foster ldquoEccentric sequences in graphsrdquo PeriodicaMathematica Hungarica vol 6 no 4 pp 287ndash293 1975

[5] PAOstrandldquoGraphswithspecifiedradius and diameterrdquoDiscreteMathematics vol 4 pp 71ndash75 1973

[6] M Behzad and J E Simpson ldquoEccentric sequences and eccen-tric sets in graphsrdquo Discrete Mathematics vol 16 no 3 pp 187ndash193 1976

[7] R Nandakumar On some eccentric properties of graphs [PhDThesis] Indian Institute of Technology New Delhi India 1986

[8] J Gimbert and N Lopez ldquoEccentric sequences and eccentricitysets in digraphsrdquo Ars Combinatoria vol 86 pp 225ndash238 2008

[9] M Randic ldquoCharacterizations of atoms molecules and classesof molecules based on paths enumerationsrdquoMATCH vol 7 pp5ndash64 1979

[10] F Buckley and F Harary Distance in Graphs Addison-Wesley1990

[11] F Harary ldquoStatus and contrastatusrdquo Sociometry vol 22 pp 23ndash43 1959

[12] JW Kennedy and L V Quintas ldquoExtremal 119891-trees and embed-ding spaces for molecular graphsrdquo Discrete Applied Mathemat-ics vol 5 no 2 pp 191ndash209 1983

[13] L V Quintas and P J Slater ldquoPairs of nonisomorphic graphshaving the same path degree sequencerdquo Match no 12 pp 75ndash86 1981

[14] P J Slater ldquoCounterexamples to Randicrsquos conjecture on distancedegree sequences for treesrdquo Journal of Graph Theory vol 6 no1 pp 89ndash91 1982

[15] M Gargano and L V Quintas ldquoSmallest order pairs of non-isomorphic graphs having the same distance degree sequenceand specified number of cyclesrdquo in Notes from New York GraphTheory Day VI pp 13ndash16 New York Academy of Sciences 1983

[16] F C Bussemaker S Cobeljic D B Cvetkovic and J J SeidelldquoComputer investigation of cube Graphsrdquo T H Report 76-WSK-01 Technological University Eindhoven Eindhoven TheNetherlands 1976

[17] G Brinkmann ldquoFast generation of cubic graphsrdquo Journal ofGraph Theory vol 23 no 2 pp 139ndash149 1996

[18] F Y Halberstam and L V Quintas ANote on Tables of Distanceand Path Degree Sequences for Cubic Graphs MathematicsDepartment Pace University New York NY USA 1982

[19] G S Bloom L V Quintas and J W Kennedy ldquoDistance degreeregular graphsrdquo in The Theory and Applications of Graphs 4thInternational ConferenceWesternMichiganUniversity Kalama-zoo MI May 1980 pp 95ndash108 John Wiley amp Sons New YorkNY USA 1981

[20] G Chartrand and F Harary ldquoPlanar permutation graphsrdquoAnnales de lrsquoInstitut Henri Poincare B vol 3 pp 433ndash438 1967

[21] M I Huilgol H B Walikar and B D Acharya ldquoOn diameterthree distance degree regular graphsrdquo Advances and Applica-tions in Discrete Mathematics vol 7 no 1 pp 39ndash61 2011

[22] M I Huilgol M Rajeshwari and S Syed Asif Ulla ldquoDistancedegree regular graphs and their eccentric digraphsrdquo Interna-tional Journal of Mathematical Sciences and Engineering Appli-cations vol 5 no 6 pp 405ndash416 2011

[23] MIHuilgolMRajeshwari andSSAUlla ldquoProducts ofdistancedegree regular and distance degree injective graphsrdquo Journal ofDiscrete Mathematical Sciences amp Cryptography vol 15 no 4-5pp 303ndash314 2012

[24] N L BiggsAlgebraic GraphTheory Cambridge Tracts inMath-ematics 67 Cambridge University Press London UK 1974

[25] GM Adelrsquoson-Velskii ldquoExample of a graph without a transitiveautomorphism grouprdquo Soviet Mathematics-Doklady vol 10 pp440ndash441 1969

[26] R Frucht J E Graver and M E Watkins ldquoThe groups ofthe generalized Petersen graphsrdquo Proceedings of the CambridgePhilosophical Society vol 70 pp 211ndash218 1971

[27] A T Balaban R O Davies F Harary A Hill and RWestwickldquoCubic identity graphs and planar graphs derived from treesrdquoJournal of the Australian Mathematical Society vol 11 pp 207ndash215 1970

[28] G Baron and W Imrich ldquoAsymmetrische regulare GraphenrdquoActa Mathematica Academiae Scientiarum Hungaricae vol 20pp 135ndash142 1969

[29] A Gewirtz A Hill and L V Quintas ldquoEl numero minimo depuntos para grafos regulares y asimetricosrdquo Scientia vol 138pp 103ndash111 1969

[30] V A Kokhov ldquoAll cubic graphs of least order with identity auto-morphism grouprdquo Programmirovanie vol 4 pp 106ndash107 1976

[31] G Johns and K Sleno ldquoAntipodal graphs and digraphsrdquo Inter-national Journal of Mathematics andMathematical Sciences vol16 no 3 pp 579ndash586 1993

[32] J Akiyama K Ando and D Avis ldquoEccentric graphsrdquo DiscreteMathematics vol 56 no 1 pp 1ndash6 1985

[33] G Chartrand W Gu M Schultz and S J Winters ldquoEccentricgraphsrdquo Networks vol 34 no 2 pp 115ndash121 1999

[34] FBuckley ldquoTheEccentric digraph of a graphrdquoCongressusNumer-antium vol 149 pp 65ndash76 2001

[35] JGimbertMMiller F Ruskey and J Ryan ldquoIterations of eccen-tric digraphsrdquo Bulletin of the Institute of Combinatorics and ItsApplications vol 45 pp 41ndash50 2005


[36] G S Bloom J W Kennedy and L V Quintas ldquoSome problemsconcerning distance and path degree sequencesrdquo in GraphTheory vol 1018 of Lecture Notes in Mathematics pp 179ndash190Springer Berlin Germany 1983

[37] F Y Halberstam and L V Quintas ldquoA note on tables of distanceand path degree sequences for cubic graphsrdquo in Proceedingsof the Silver Jubilee Conference on Combinatorics University ofWaterloo Ontario Canada JunendashJuly 1982

[38] B Bollobas ldquoDistinguishing vertices of random graphsrdquoAnnalsof Discrete Mathematics vol 13 pp 33ndash50 1982

[39] PMartinez and L VQuintas ldquoDistance degree injective regulargraphsrdquo inNotes from New York GraphTheory Day VIII pp 19ndash21 New York Academy of Sciences New York NY USA 1984

[40] J Volf ldquoA small distance degree injective cubic graphsrdquo inNotesfrom New York Graph Theory Day XIII pp 31ndash32 New YorkAcademy of Sciences 1987

[41] M I Huilgol and M Rajeshwari ldquoNon-Existence of cubic DDIgraphs of order 16 with diameter 4 5 6rdquo (Communicated)

[42] M I HuilgolM Rajeshwari and S Syed Asif Ulla ldquoEmbeddingin distance degree regular and distance degree injective graphsrdquoMalaya Journal of Mathematik vol 4 no 1 pp 134ndash141 2013

Submit your manuscripts athttpwwwhindawicom

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


of 119881 known as edges forming 119864 whose cardinality is size 119902Unless mentioned otherwise in this article by a graph wemean an undirected finite graph without multiple edges andself-loops

The distance 119889(V 119906) from a vertex V of 119866 to a vertex 119906 isthe length of shortest V to 119906 path The degree of a vertex V isthe number of vertices at distance one

The sequence of numbers of vertices having degree 0 12 3 is called the degree sequence which is the list of thedegrees of vertices of 119866 in nondecreasing order

The eccentricity 119890(V) of V is the distance to a farthest vertexfrom V

The minimum of the eccentricities is the radius rad(119866)and the maximum diameter of 119866 diam(119866)

A graph119866 is said to be self-centered if all vertices have thesame eccentricity

If 119889(V 119906) = 119890(V) (119906 = V) we say that 119906 is an eccentricvertex of V

The eccentric sequence of a connected graph 119866 is a list ofthe eccentricities of its vertices in nondecreasing order

The distance and path degree sequences of a vertexare generalizations of the degree of a vertex The distancedegree sequence (dds) of a vertex V in a graph 119866 is a listof the number of vertices at distance 1 2 119890(V) in thatorder where 119890(V) denotes the eccentricity of V in 119866 Thusthe sequence (119889

1198940 1198891198941 1198891198942 119889

119894119895 ) is the distance degree

sequence of the vertex V119894in 119866 where 119889

119894119895denotes the number

of vertices at distance 119895 from V119894 The 119901-tuple of distance

degree sequences of the vertices of 119866 with entries arrangedin lexicographic order is the distance degree sequence (DDS)of 119866 Similarly we define the path degree sequence (pds) ofV119894as the sequence (119901

1198940 1199011198941 1199011198942 119901

119894119895 ) where 119901

119894119895denotes

the number of paths in 119866 of length 119895 having V119894as the initial

vertex The ordered set of all such sequences arranged inlexicographic order is called the path degree sequence (PDS)of 119866 Clearly for any graph 119889

119894119895le 119901119894119895

The distance distribution of 119866 is the sequence (1198891 1198892

119889119895 ) where 119889

119895is the number of pairs of vertices in 119866 that

are at distance 119895 apartThe status 119904(V) of a vertex V in119866 is the sumof the distances

from V to all other vertices in 119866 This concept was introducedbyHarary [11]Thus using the distance degree sequence of V

119894

we have 119904(V119894) = sum

diam(119866)119895=1

119895119889119894119895

Themedian119872(119866) of119866 is the set of vertices withminimumstatus A graph is called a self-median graph if all of its verticeshave the same status

The status of a vertex is also called the distance at thatvertexThemean distance at the vertex V

119894 denoted byMD(V

119894)

is defined as MD(V119894) = 119904(V

119894)(119901 minus 1)

The mean distance for the graph denoted by MD(119866) isdefined as the mean of the distances between all pairs ofvertices of 119866 that is MD(119866) = sum

119906V 119889(119906 V) (119901

2) where 119906

and V are vertices of 119866 In terms of status we get MD(119866) =sum119901

119894=1(MD(V

119894)119901)

As an illustration consider the graph 119866 as shown inFigure 1

Figure 1

DDS(119866) = ((1 1 1 2) (1 2 2) (1 3 1) (1 2 1 1)2) and

PDS(119866) = ((1 1 1 2 2) (1 2 2 2) (1 2 3 2 1)2 (1 3 3))If we consider the 3-dimensional cube 119876

3as an example

we will get DDS(1198763) = ((1 3 3 1)

8) and PDS(119876

3) = ((1 3 6

12 12 12)6)

In [12] Kennedy and Quintas have examined how thedistance degree sequences relate to embedding trees inlattice-graphs and other spaces as well as the consequentialrelations to physical theories Chemists have also proposedand discussed other sequences for this purpose Their objec-tive and hope are to develop a ldquochemical indicatorrdquo whichwould be useful both for the graph isomorphism problemand to predict various properties of the molecule at hand Inthis connection Randic [9] proposed the structural similaritydepending on the graph property as follows

Conjecture 1 (see [9]) Between pairs of atoms in acyclicstructures there is a unique path so that the number of pathsof a given length corresponds to the number of neighbors at agiven distance

This can be rephrased in graph theoretic terms as follows

Conjecture 2 (see [9]) For a connected graph 119866 119863119863119878(119866) =119875119863119878(119866) if and only if 119866 is a tree

Anonassertive proof was given byQuintas and Slater [13]Hence this kind of study leads to the problem of finding

the smallest order for which there exists a pair of noniso-morphic graphs which are equivalent relative to some set ofgraphical invariants So the following conjecture wasmade inline with the above conjecture by Randic [9] that sought forgraph isomorphism by comparing their sequences

Conjecture 3 (see [9]) Two graphs1198661and119866

2are isomorphic

if and only if 119875119863119878(1198661) = 119875119863119878(119866

2)

This conjuncture was also shown to be invalid by citinginfinite class of pairs of nonisomorphic trees having the prop-erty that the two members of each pair have the same pathdegree sequence by Quintas and Slater [13] To extend thisnotion to nontree graphs the same examples were consideredto construct pairs of nonisomorphic graphs having a varietyof properties and invalidate the conjecture

As noted above a tree is not in general characterizedby its path (or distance) degree sequence The least order forwhich there exists a pair of nonisomorphic trees with thesame path degree sequence has been shown to be more than14 for acyclic alkanes that is trees with no vertex of degree






























1198940 1198891198941 1198891198942 119889

119894119895 )















119896times 119866


follows that 119896119862119875119894119889(119862119899) = 119899119862119875

119894119889(119862119896)



1 then (i) 119875



119894119889(119866) = 119862

119896times119866



Corollary 13 (see [19]) 119875119873119894119889(119866) = 119870

119873


and degree 1198631+ 119873 and 119862119873

119896times 119866 equiv 119862

119873minus1



Figure 4


119873

119894119889(119866) =

119875119894119889(119875119873minus1

119894119889(119866)) = 119870

119873

2times 119866 as 119875

119894119889(119866) = 119870

119873






1

except for119898 gt 1198631= 1






1 Then


6


7 or 119866 = 119876

3= 1198702times

1198702times 1198702


Figure 4(iv) 119889

1= 2 if and only if 119866 = 119862

6or 119866 = 119862

7





1=










119894 1 le 119894 le 119901 and then 119866[1198661 1198662 1198663 119866












1 1198922] isin 119864(119866) and ℎ

1= ℎ2

or 1198921= 1198922and [ℎ

1 ℎ2] isin 119864(119867)





1= ℎ2 (ii)

1198921= 1198922and [ℎ

1 ℎ2] isin 119864(119867) and (iii) [119892

1 1198922] isin 119864(119866) and

[ℎ1 ℎ2] isin 119864(119867)



1and 119866

2are DDR

graphs



119896) ge diam(119866

119894) for all


119896)



1198661and119866


1and119866

2areDDR graphs









1 119887

119898minus1 1198881=

1 1198882 119888




119895(1 le 119895 le 119898)


119895(0 le 119895 le 119898 minus 1)







1

2

x

y v

u

43

5

10

9 8

z6

7

Figure 5





120572(8-119888119910119888119897119890)


119894119889(6-119888119910119888119897119890) = 1198702




119894119889(5-119888119910119888119897119890) and the Petersen




1

x

y

113

2

4

12

5

13

6

14

7

15

816

917

10

18

u

19

20

21

Figure 6





















Figure 7

1

2

Figure 8








6





1

2

Figure 9















1198940 1198891198941 1198891198942 119889





a b

c d

Figure 10

a

b

c

d


y1 y2

x1 x2

yk

xk

ykminus1

xkminus1

Figure 11

Figure 12












1and 119866

2are DDI graphs


Lemma 49 (see [23]) Let 1198661and 119866


119860 = 119889119889119904(119906119894)119906119894isin 119881(119866

1) and 119861 = 119889119889119904(V

119894)V119894isin 119881(119866

2) If





2) such that no



1) For each pair (119909 119910)




in 1198662 and then in 119866

1◻1198662 the subsets (119909 119911)119909 isin 119878

1 and


1198781 119910 isin 119878





2are DDI


Lemma 52 (see [23]) Let 1198661and 119866


119860 = 119889119889119904(119906119894)119906119894isin 119881(119866

1) and 119861 = 119889119889119904(V

119894)V119894isin 119881(119866

2) If



2be any two graphs


2) such that


1oplus 1198662


1)


2 satisfies


2 and then


1 and (119910 119911)119910 isin 119878

2


2satisfies

dds(119909 119911) = dds(119910 119911)

5 Embeddings























References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































1198940 1198891198941 1198891198942 119889

119894119895 )















119896times 119866


follows that 119896119862119875119894119889(119862119899) = 119899119862119875

119894119889(119862119896)



1 then (i) 119875



119894119889(119866) = 119862

119896times119866



Corollary 13 (see [19]) 119875119873119894119889(119866) = 119870

119873


and degree 1198631+ 119873 and 119862119873

119896times 119866 equiv 119862

119873minus1



Figure 4


119873

119894119889(119866) =

119875119894119889(119875119873minus1

119894119889(119866)) = 119870

119873

2times 119866 as 119875

119894119889(119866) = 119870

119873






1

except for119898 gt 1198631= 1






1 Then


6


7 or 119866 = 119876

3= 1198702times

1198702times 1198702


Figure 4(iv) 119889

1= 2 if and only if 119866 = 119862

6or 119866 = 119862

7





1=










119894 1 le 119894 le 119901 and then 119866[1198661 1198662 1198663 119866












1 1198922] isin 119864(119866) and ℎ

1= ℎ2

or 1198921= 1198922and [ℎ

1 ℎ2] isin 119864(119867)





1= ℎ2 (ii)

1198921= 1198922and [ℎ

1 ℎ2] isin 119864(119867) and (iii) [119892

1 1198922] isin 119864(119866) and

[ℎ1 ℎ2] isin 119864(119867)



1and 119866

2are DDR

graphs



119896) ge diam(119866

119894) for all


119896)



1198661and119866


1and119866

2areDDR graphs









1 119887

119898minus1 1198881=

1 1198882 119888




119895(1 le 119895 le 119898)


119895(0 le 119895 le 119898 minus 1)







1

2

x

y v

u

43

5

10

9 8

z6

7

Figure 5





120572(8-119888119910119888119897119890)


119894119889(6-119888119910119888119897119890) = 1198702




119894119889(5-119888119910119888119897119890) and the Petersen




1

x

y

113

2

4

12

5

13

6

14

7

15

816

917

10

18

u

19

20

21

Figure 6





















Figure 7

1

2

Figure 8








6





1

2

Figure 9















1198940 1198891198941 1198891198942 119889





a b

c d

Figure 10

a

b

c

d


y1 y2

x1 x2

yk

xk

ykminus1

xkminus1

Figure 11

Figure 12












1and 119866

2are DDI graphs


Lemma 49 (see [23]) Let 1198661and 119866


119860 = 119889119889119904(119906119894)119906119894isin 119881(119866

1) and 119861 = 119889119889119904(V

119894)V119894isin 119881(119866

2) If





2) such that no



1) For each pair (119909 119910)




in 1198662 and then in 119866

1◻1198662 the subsets (119909 119911)119909 isin 119878

1 and


1198781 119910 isin 119878





2are DDI


Lemma 52 (see [23]) Let 1198661and 119866


119860 = 119889119889119904(119906119894)119906119894isin 119881(119866

1) and 119861 = 119889119889119904(V

119894)V119894isin 119881(119866

2) If



2be any two graphs


2) such that


1oplus 1198662


1)


2 satisfies


2 and then


1 and (119910 119911)119910 isin 119878

2


2satisfies

dds(119909 119911) = dds(119910 119911)

5 Embeddings























References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Figure 4


119873

119894119889(119866) =

119875119894119889(119875119873minus1

119894119889(119866)) = 119870

119873

2times 119866 as 119875

119894119889(119866) = 119870

119873






1

except for119898 gt 1198631= 1






1 Then


6


7 or 119866 = 119876

3= 1198702times

1198702times 1198702


Figure 4(iv) 119889

1= 2 if and only if 119866 = 119862

6or 119866 = 119862

7





1=










119894 1 le 119894 le 119901 and then 119866[1198661 1198662 1198663 119866












1 1198922] isin 119864(119866) and ℎ

1= ℎ2

or 1198921= 1198922and [ℎ

1 ℎ2] isin 119864(119867)





1= ℎ2 (ii)

1198921= 1198922and [ℎ

1 ℎ2] isin 119864(119867) and (iii) [119892

1 1198922] isin 119864(119866) and

[ℎ1 ℎ2] isin 119864(119867)



1and 119866

2are DDR

graphs



119896) ge diam(119866

119894) for all


119896)



1198661and119866


1and119866

2areDDR graphs









1 119887

119898minus1 1198881=

1 1198882 119888




119895(1 le 119895 le 119898)


119895(0 le 119895 le 119898 minus 1)







1

2

x

y v

u

43

5

10

9 8

z6

7

Figure 5





120572(8-119888119910119888119897119890)


119894119889(6-119888119910119888119897119890) = 1198702




119894119889(5-119888119910119888119897119890) and the Petersen




1

x

y

113

2

4

12

5

13

6

14

7

15

816

917

10

18

u

19

20

21

Figure 6





















Figure 7

1

2

Figure 8








6





1

2

Figure 9















1198940 1198891198941 1198891198942 119889





a b

c d

Figure 10

a

b

c

d


y1 y2

x1 x2

yk

xk

ykminus1

xkminus1

Figure 11

Figure 12












1and 119866

2are DDI graphs


Lemma 49 (see [23]) Let 1198661and 119866


119860 = 119889119889119904(119906119894)119906119894isin 119881(119866

1) and 119861 = 119889119889119904(V

119894)V119894isin 119881(119866

2) If





2) such that no



1) For each pair (119909 119910)




in 1198662 and then in 119866

1◻1198662 the subsets (119909 119911)119909 isin 119878

1 and


1198781 119910 isin 119878





2are DDI


Lemma 52 (see [23]) Let 1198661and 119866


119860 = 119889119889119904(119906119894)119906119894isin 119881(119866

1) and 119861 = 119889119889119904(V

119894)V119894isin 119881(119866

2) If



2be any two graphs


2) such that


1oplus 1198662


1)


2 satisfies


2 and then


1 and (119910 119911)119910 isin 119878

2


2satisfies

dds(119909 119911) = dds(119910 119911)

5 Embeddings























References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119894 1 le 119894 le 119901 and then 119866[1198661 1198662 1198663 119866












1 1198922] isin 119864(119866) and ℎ

1= ℎ2

or 1198921= 1198922and [ℎ

1 ℎ2] isin 119864(119867)





1= ℎ2 (ii)

1198921= 1198922and [ℎ

1 ℎ2] isin 119864(119867) and (iii) [119892

1 1198922] isin 119864(119866) and

[ℎ1 ℎ2] isin 119864(119867)



1and 119866

2are DDR

graphs



119896) ge diam(119866

119894) for all


119896)



1198661and119866


1and119866

2areDDR graphs









1 119887

119898minus1 1198881=

1 1198882 119888




119895(1 le 119895 le 119898)


119895(0 le 119895 le 119898 minus 1)







1

2

x

y v

u

43

5

10

9 8

z6

7

Figure 5





120572(8-119888119910119888119897119890)


119894119889(6-119888119910119888119897119890) = 1198702




119894119889(5-119888119910119888119897119890) and the Petersen




1

x

y

113

2

4

12

5

13

6

14

7

15

816

917

10

18

u

19

20

21

Figure 6





















Figure 7

1

2

Figure 8








6





1

2

Figure 9















1198940 1198891198941 1198891198942 119889





a b

c d

Figure 10

a

b

c

d


y1 y2

x1 x2

yk

xk

ykminus1

xkminus1

Figure 11

Figure 12












1and 119866

2are DDI graphs


Lemma 49 (see [23]) Let 1198661and 119866


119860 = 119889119889119904(119906119894)119906119894isin 119881(119866

1) and 119861 = 119889119889119904(V

119894)V119894isin 119881(119866

2) If





2) such that no



1) For each pair (119909 119910)




in 1198662 and then in 119866

1◻1198662 the subsets (119909 119911)119909 isin 119878

1 and


1198781 119910 isin 119878





2are DDI


Lemma 52 (see [23]) Let 1198661and 119866


119860 = 119889119889119904(119906119894)119906119894isin 119881(119866

1) and 119861 = 119889119889119904(V

119894)V119894isin 119881(119866

2) If



2be any two graphs


2) such that


1oplus 1198662


1)


2 satisfies


2 and then


1 and (119910 119911)119910 isin 119878

2


2satisfies

dds(119909 119911) = dds(119910 119911)

5 Embeddings























References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

2

x

y v

u

43

5

10

9 8

z6

7

Figure 5





120572(8-119888119910119888119897119890)


119894119889(6-119888119910119888119897119890) = 1198702




119894119889(5-119888119910119888119897119890) and the Petersen




1

x

y

113

2

4

12

5

13

6

14

7

15

816

917

10

18

u

19

20

21

Figure 6





















Figure 7

1

2

Figure 8








6





1

2

Figure 9















1198940 1198891198941 1198891198942 119889





a b

c d

Figure 10

a

b

c

d


y1 y2

x1 x2

yk

xk

ykminus1

xkminus1

Figure 11

Figure 12












1and 119866

2are DDI graphs


Lemma 49 (see [23]) Let 1198661and 119866


119860 = 119889119889119904(119906119894)119906119894isin 119881(119866

1) and 119861 = 119889119889119904(V

119894)V119894isin 119881(119866

2) If





2) such that no



1) For each pair (119909 119910)




in 1198662 and then in 119866

1◻1198662 the subsets (119909 119911)119909 isin 119878

1 and


1198781 119910 isin 119878





2are DDI


Lemma 52 (see [23]) Let 1198661and 119866


119860 = 119889119889119904(119906119894)119906119894isin 119881(119866

1) and 119861 = 119889119889119904(V

119894)V119894isin 119881(119866

2) If



2be any two graphs


2) such that


1oplus 1198662


1)


2 satisfies


2 and then


1 and (119910 119911)119910 isin 119878

2


2satisfies

dds(119909 119911) = dds(119910 119911)

5 Embeddings























References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















Figure 7

1

2

Figure 8








6





1

2

Figure 9















1198940 1198891198941 1198891198942 119889





a b

c d

Figure 10

a

b

c

d


y1 y2

x1 x2

yk

xk

ykminus1

xkminus1

Figure 11

Figure 12












1and 119866

2are DDI graphs


Lemma 49 (see [23]) Let 1198661and 119866


119860 = 119889119889119904(119906119894)119906119894isin 119881(119866

1) and 119861 = 119889119889119904(V

119894)V119894isin 119881(119866

2) If





2) such that no



1) For each pair (119909 119910)




in 1198662 and then in 119866

1◻1198662 the subsets (119909 119911)119909 isin 119878

1 and


1198781 119910 isin 119878





2are DDI


Lemma 52 (see [23]) Let 1198661and 119866


119860 = 119889119889119904(119906119894)119906119894isin 119881(119866

1) and 119861 = 119889119889119904(V

119894)V119894isin 119881(119866

2) If



2be any two graphs


2) such that


1oplus 1198662


1)


2 satisfies


2 and then


1 and (119910 119911)119910 isin 119878

2


2satisfies

dds(119909 119911) = dds(119910 119911)

5 Embeddings























References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

2

Figure 9















1198940 1198891198941 1198891198942 119889





a b

c d

Figure 10

a

b

c

d


y1 y2

x1 x2

yk

xk

ykminus1

xkminus1

Figure 11

Figure 12












1and 119866

2are DDI graphs


Lemma 49 (see [23]) Let 1198661and 119866


119860 = 119889119889119904(119906119894)119906119894isin 119881(119866

1) and 119861 = 119889119889119904(V

119894)V119894isin 119881(119866

2) If





2) such that no



1) For each pair (119909 119910)




in 1198662 and then in 119866

1◻1198662 the subsets (119909 119911)119909 isin 119878

1 and


1198781 119910 isin 119878





2are DDI


Lemma 52 (see [23]) Let 1198661and 119866


119860 = 119889119889119904(119906119894)119906119894isin 119881(119866

1) and 119861 = 119889119889119904(V

119894)V119894isin 119881(119866

2) If



2be any two graphs


2) such that


1oplus 1198662


1)


2 satisfies


2 and then


1 and (119910 119911)119910 isin 119878

2


2satisfies

dds(119909 119911) = dds(119910 119911)

5 Embeddings























References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1and 119866

2are DDI graphs


Lemma 49 (see [23]) Let 1198661and 119866


119860 = 119889119889119904(119906119894)119906119894isin 119881(119866

1) and 119861 = 119889119889119904(V

119894)V119894isin 119881(119866

2) If





2) such that no



1) For each pair (119909 119910)




in 1198662 and then in 119866

1◻1198662 the subsets (119909 119911)119909 isin 119878

1 and


1198781 119910 isin 119878





2are DDI


Lemma 52 (see [23]) Let 1198661and 119866


119860 = 119889119889119904(119906119894)119906119894isin 119881(119866

1) and 119861 = 119889119889119904(V

119894)V119894isin 119881(119866

2) If



2be any two graphs


2) such that


1oplus 1198662


1)


2 satisfies


2 and then


1 and (119910 119911)119910 isin 119878

2


2satisfies

dds(119909 119911) = dds(119910 119911)

5 Embeddings























References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Review Article Distance Degree Regular Graphs and Distance...

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