Recent progress in distance-regular...

Distance-regular graphs Open Problems Characterizations About our proof

Recent progress in distance-regular graphs

J. Koolen∗

∗Department of MathematicsPOSTECH

(Based on joint work in progress with Alexander Gavrilyuk)

SCAC, August 21, 2012


Outline

1 Distance-regular graphsDefinitionsExamples

2 Open ProblemsOpen Problems

3 CharacterizationsJohnson and Hamming graphsGrassmann graphsForms Graphs

4 About our proofDefinitionsThe Terwilliger polynomialThe folded halved cubesThe Grassmann graphs


Definition

Graph: Γ = (V ,E) where V vertex set, E ⊆(V

2

)edge set.

All graphs in this talk are simple.x ∼ y if xy ∈ E .x 6∼ y if xy 6∈ E .d(x , y): length of a shortest path connecting x and y .D(Γ) diameter (max distance in Γ)


Distance-regular graphs

DefinitionΓi(x) := {y | d(x , y) = i}

DefinitionA connected graph Γ is called distance-regular(abbreviated DRG) if there are numbers ai ,bi , ci(0 ≤ i ≤ D = D(Γ)) s.t. if d(x , y) = j then

#Γ1(y) ∩ Γj−1(x) = cj#Γ1(y) ∩ Γj (x) = aj#Γ1(y) ∩ Γj+1(x) = bj

Some examples.


Hamming graphs

Definitionq ≥ 2, n ≥ 1 integers.Q = {1, . . . ,q}Hamming graph H(n,q) has vertex set Qn

x ∼ y if they differ in exactly one position.Diameter equals n.

H(n,2) = n-cube.DRG with ci = i .Gives an algebraic frame work to study codes, especiallybounds on codes.For example the Delsarte linear programming bound andthe Schrijver bound.


Hamming graphs


x ∼ y if they differ in exactly one position.Diameter equals n.H(n,2) = n-cube.DRG with ci = i .

Gives an algebraic frame work to study codes, especiallybounds on codes.For example the Delsarte linear programming bound andthe Schrijver bound.


Hamming graphs


x ∼ y if they differ in exactly one position.Diameter equals n.H(n,2) = n-cube.DRG with ci = i .Gives an algebraic frame work to study codes, especiallybounds on codes.For example the Delsarte linear programming bound andthe Schrijver bound.


Johnson graphs

Definition1 ≤ t ≤ n integers.N = {1, . . . ,n}Johnson graph J(n, t) has vertex set

(Nt

)A ∼ B if #A ∩ B = t − 1.J(n, t) ∼= J(n,n − t), diameter min(t ,n − t).DRG with ci = i2.Gives an algebraic frame work to study designs.


Distance-transitive graphs

Distance-regular graphs were introduced by Biggs in the late1960’s as a combinatorial generalization of distance-transitivegraphs, which we wil introduce now.

Let Γ = (V ,E) be a graph.A bijection φ : V → V is called an automorphism of Γ ifxy ∈ E if and only if φ(x)φ(y) ∈ E .A connected graph is called distance-transitive if for anyfour vertices x , y ,u, v such that d(x , y) = d(u, v), thereexists an automorphism φ of Γ with u = φ(x) and v = φ(y).Note that a distance-transitive graph is alwaysdistance-regular (and hence regular).




Let Γ = (V ,E) be a graph.A bijection φ : V → V is called an automorphism of Γ ifxy ∈ E if and only if φ(x)φ(y) ∈ E .

A connected graph is called distance-transitive if for anyfour vertices x , y ,u, v such that d(x , y) = d(u, v), thereexists an automorphism φ of Γ with u = φ(x) and v = φ(y).Note that a distance-transitive graph is alwaysdistance-regular (and hence regular).




Let Γ = (V ,E) be a graph.A bijection φ : V → V is called an automorphism of Γ ifxy ∈ E if and only if φ(x)φ(y) ∈ E .A connected graph is called distance-transitive if for anyfour vertices x , y ,u, v such that d(x , y) = d(u, v), thereexists an automorphism φ of Γ with u = φ(x) and v = φ(y).

Note that a distance-transitive graph is alwaysdistance-regular (and hence regular).




Let Γ = (V ,E) be a graph.A bijection φ : V → V is called an automorphism of Γ ifxy ∈ E if and only if φ(x)φ(y) ∈ E .A connected graph is called distance-transitive if for anyfour vertices x , y ,u, v such that d(x , y) = d(u, v), thereexists an automorphism φ of Γ with u = φ(x) and v = φ(y).Note that a distance-transitive graph is alwaysdistance-regular (and hence regular).


Outline






More examples

All known infinite families (with unbounded diameter) ofdistance-transitive graphs come from classical objects, forexample:

Hamming graphs,Johnson graphs,Grassmann graphs,bilinear forms graphs,sesquilinear forms graphs,dual polar graphs (The vertices are the maximal totallyisotropic subspaces on a vector space over a finite fieldwith a fixed (non-degenerate) bilinear form)

Distance-regular graphs give a way to study these classicalobjects from a combinatorial view point.


Some more examples

There are four known infinite families (with unboundeddiameter) of vertex-transitive but not distance-transitive DRG,namely

The Ustimenko graphs;The Hemmeter graphs;the Doob graphs;the quadratic forms graphs

There is only one infinite family known (with unboundeddiameter) which is not vertex-transitie, namely the twistedGrassmann graphs (discovered in 2005 by Van Dam and K.). Ithas two orbits under its full automorphism group. Jungnickeland Tonchev (2009) found an infinite family of designs whichare counterexamples for Hamada’s conjecture. The blockgraphs of these designs are isomorphic to the twistedGrassmann graphs (Munemasa and Tonchev (2011)).


Some more examples



There is only one infinite family known (with unboundeddiameter) which is not vertex-transitie, namely the twistedGrassmann graphs (discovered in 2005 by Van Dam and K.). Ithas two orbits under its full automorphism group.

Jungnickeland Tonchev (2009) found an infinite family of designs whichare counterexamples for Hamada’s conjecture. The blockgraphs of these designs are isomorphic to the twistedGrassmann graphs (Munemasa and Tonchev (2011)).


Some more examples



There is only one infinite family known (with unboundeddiameter) which is not vertex-transitie, namely the twistedGrassmann graphs (discovered in 2005 by Van Dam and K.). Ithas two orbits under its full automorphism group. Jungnickeland Tonchev (2009) found an infinite family of designs whichare counterexamples for Hamada’s conjecture. The blockgraphs of these designs are isomorphic to the twistedGrassmann graphs (Munemasa and Tonchev (2011)).


Outline






Open Problems 1

Main open problemCan we classify the distance-regular graphs with largediameter? (Large probably means something like 6.)

For small diameter there is no hope asFor diameter two one has the strongly regular graphs (andtheir classification would include the classification ofSteiner triple designs, etc.).For diameter three and bipartite: These are exactly thepoint-block incidence graphs of square 2-designs, andthese include the projective planes.


Open Problems 1


For small diameter there is no hope asFor diameter two one has the strongly regular graphs (andtheir classification would include the classification ofSteiner triple designs, etc.).

For diameter three and bipartite: These are exactly thepoint-block incidence graphs of square 2-designs, andthese include the projective planes.


Open Problems 1


For small diameter there is no hope asFor diameter two one has the strongly regular graphs (andtheir classification would include the classification ofSteiner triple designs, etc.).For diameter three and bipartite: These are exactly thepoint-block incidence graphs of square 2-designs, andthese include the projective planes.


Bannai’s Problem

Bannai observed (in the 1980’s) that all the known primitiveDRG with large diameter are Q-polynomial. (I will define thislater)

So he asked:

Bannai’s ProblemCan we classify the Q-polynomial DRG with large diameter?

Terwilliger and his group has made huge contributions to thisproblem. Now we will discuss the known characterizations ofthe classical families by their intersection numbers.


Bannai’s Problem

Bannai observed (in the 1980’s) that all the known primitiveDRG with large diameter are Q-polynomial. (I will define thislater) So he asked:




Bannai’s Problem



Terwilliger and his group has made huge contributions to thisproblem.

Now we will discuss the known characterizations ofthe classical families by their intersection numbers.


Bannai’s Problem





Outline






For a distance-regular graph with diameter D define itsintersection array byι = {b0 = k ,b1, . . . ,bD−1; c1 = 1, c2, . . . , cD}.

We are interestedwhich distance-regular graphs are determined by theirintersection number, i.e., is there a unique DRG with theseintersection numbers (or intersection array).Let us start with theJohnson and Hamming graphs.

Theorem (Egawa (1981))Any distance-regular graph with the same intersection array asa Hamming graph H(n,q) is the Hamming graph H(n,q), or, ifq = 4, a Doob graph.


For a distance-regular graph with diameter D define itsintersection array byι = {b0 = k ,b1, . . . ,bD−1; c1 = 1, c2, . . . , cD}. We are interestedwhich distance-regular graphs are determined by theirintersection number, i.e., is there a unique DRG with theseintersection numbers (or intersection array).

Let us start with theJohnson and Hamming graphs.



For a distance-regular graph with diameter D define itsintersection array byι = {b0 = k ,b1, . . . ,bD−1; c1 = 1, c2, . . . , cD}. We are interestedwhich distance-regular graphs are determined by theirintersection number, i.e., is there a unique DRG with theseintersection numbers (or intersection array).Let us start with theJohnson and Hamming graphs.



Theorem (Terwilliger (1987))

The Johnson graphs J(n, t) n ≥ 2t is determined by itsintersection array unless (n, t) = (8,2) (and there are exactly 3other graphs for (n, t) = (8,2), namely the Chang graphs).

Terwilliger (and independently Neumaier) generalized this lastresult to also include the halved cubes. They used the rootlattices in order to obtain these results.


Graphs related to the hypercube and Johnson graphs

Terwilliger (1987) showed that a folded n-cube (you identifyantipodes in the cube) is determined by its spectrum ifn ≥ 7. For n = 6 there are exactly three such graphs.

Metsch (1997), improving on earlier work by Neumaier,showed that the folded Johnson graphs are determined bytheir intersection array.Metsch (1997) also showed the uniqueness of the halvedfolded 2n-cube for n ≥ 9.Gavrilyuk and K.: The folded halved 2n-cube is uniquelydetermined by its intersection array for n ≥ 6. Our methodis completely different from the method employed byMetsch. Our method also works for the folded Johnsongraphs. I will discuss it later in this talk.



Terwilliger (1987) showed that a folded n-cube (you identifyantipodes in the cube) is determined by its spectrum ifn ≥ 7. For n = 6 there are exactly three such graphs.Metsch (1997), improving on earlier work by Neumaier,showed that the folded Johnson graphs are determined bytheir intersection array.

Metsch (1997) also showed the uniqueness of the halvedfolded 2n-cube for n ≥ 9.Gavrilyuk and K.: The folded halved 2n-cube is uniquelydetermined by its intersection array for n ≥ 6. Our methodis completely different from the method employed byMetsch. Our method also works for the folded Johnsongraphs. I will discuss it later in this talk.



Terwilliger (1987) showed that a folded n-cube (you identifyantipodes in the cube) is determined by its spectrum ifn ≥ 7. For n = 6 there are exactly three such graphs.Metsch (1997), improving on earlier work by Neumaier,showed that the folded Johnson graphs are determined bytheir intersection array.Metsch (1997) also showed the uniqueness of the halvedfolded 2n-cube for n ≥ 9.

Gavrilyuk and K.: The folded halved 2n-cube is uniquelydetermined by its intersection array for n ≥ 6. Our methodis completely different from the method employed byMetsch. Our method also works for the folded Johnsongraphs. I will discuss it later in this talk.



Terwilliger (1987) showed that a folded n-cube (you identifyantipodes in the cube) is determined by its spectrum ifn ≥ 7. For n = 6 there are exactly three such graphs.Metsch (1997), improving on earlier work by Neumaier,showed that the folded Johnson graphs are determined bytheir intersection array.Metsch (1997) also showed the uniqueness of the halvedfolded 2n-cube for n ≥ 9.Gavrilyuk and K.: The folded halved 2n-cube is uniquelydetermined by its intersection array for n ≥ 6. Our methodis completely different from the method employed byMetsch. Our method also works for the folded Johnsongraphs. I will discuss it later in this talk.


Outline






The Grassmann graphs

Theorem (Metsch (1995))

The Grassmann graphs Jq(n,D) (n ≥ 2D) are characterized bytheir intersection numbers if n ≥ max(2D + 2,2D + 6− q) andD ≥ 3.

What happens for n = 2D,n = 2D + 1?Van Dam and K. (2005) found the twisted Grassmanngraphs. They have the same parameters as Jq(2D + 1,D),so the Grassmann graphs Jq(2D + 1,D) are notdetermined by their intersection array.Gavrilyuk and K.: The Grassmann graph J2(2D,D) areuniquely determined by their intersection array if D is oddand at least three. (We also have some results for generalJq(2D,D), and I will discuss them later.)





What happens for n = 2D,n = 2D + 1?

Van Dam and K. (2005) found the twisted Grassmanngraphs. They have the same parameters as Jq(2D + 1,D),so the Grassmann graphs Jq(2D + 1,D) are notdetermined by their intersection array.Gavrilyuk and K.: The Grassmann graph J2(2D,D) areuniquely determined by their intersection array if D is oddand at least three. (We also have some results for generalJq(2D,D), and I will discuss them later.)





What happens for n = 2D,n = 2D + 1?Van Dam and K. (2005) found the twisted Grassmanngraphs. They have the same parameters as Jq(2D + 1,D),so the Grassmann graphs Jq(2D + 1,D) are notdetermined by their intersection array.

Gavrilyuk and K.: The Grassmann graph J2(2D,D) areuniquely determined by their intersection array if D is oddand at least three. (We also have some results for generalJq(2D,D), and I will discuss them later.)





What happens for n = 2D,n = 2D + 1?Van Dam and K. (2005) found the twisted Grassmanngraphs. They have the same parameters as Jq(2D + 1,D),so the Grassmann graphs Jq(2D + 1,D) are notdetermined by their intersection array.Gavrilyuk and K.: The Grassmann graph J2(2D,D) areuniquely determined by their intersection array if D is oddand at least three. (We also have some results for generalJq(2D,D), and I will discuss them later.)


Outline






Bilinear forms graphs

Metsch building on earlier work of Sprague, Ray-Chaudhuri,Huang and Cuypers showed:


The bilinear forms graph Bil(D × e,q) is characterized by itsintersection array if q = 2 and e ≥ D + 4 or q ≥ 3 ande ≥ D + 3.

The method of Gavrilyuk and my will also work in this case (ife = D), but we did not work out the details yet.


Other families

The dual polar graphs are characterized by theirintersection arrays under the extra condition that they arelocally the disjoint union of cliques.

Cameron (1982) gave a characterization the dual polarspaces .Brouwer and Wilbrink (1983) and Brouwer and Cohen(1986) have classified the thick regular near polygons withc2 ≥ 3 and diameter at least 4, using he characterization ofCameron.The Hermitian forms graphs Her(D,q2) were characterizedby Ivanov and Shpectorov (1991) and Terwilliger (1995).


Other families

The dual polar graphs are characterized by theirintersection arrays under the extra condition that they arelocally the disjoint union of cliques.Cameron (1982) gave a characterization the dual polarspaces .Brouwer and Wilbrink (1983) and Brouwer and Cohen(1986) have classified the thick regular near polygons withc2 ≥ 3 and diameter at least 4, using he characterization ofCameron.

The Hermitian forms graphs Her(D,q2) were characterizedby Ivanov and Shpectorov (1991) and Terwilliger (1995).


Other families

The dual polar graphs are characterized by theirintersection arrays under the extra condition that they arelocally the disjoint union of cliques.Cameron (1982) gave a characterization the dual polarspaces .Brouwer and Wilbrink (1983) and Brouwer and Cohen(1986) have classified the thick regular near polygons withc2 ≥ 3 and diameter at least 4, using he characterization ofCameron.The Hermitian forms graphs Her(D,q2) were characterizedby Ivanov and Shpectorov (1991) and Terwilliger (1995).


Outline






Eigenvalues of graphs

Let G be a graph.The eigenvalues of its adjacency matrix are called theeigenvalues of G.

We say two graphs are cospectral if their adjacencymatrices are similar to each other, i.e. they have the samespectrum.A local eigenvalue at a vertex x of G is an eigenvalue ofthe subgraph induced by the neighbours of x .



Let G be a graph.The eigenvalues of its adjacency matrix are called theeigenvalues of G.We say two graphs are cospectral if their adjacencymatrices are similar to each other, i.e. they have the samespectrum.

A local eigenvalue at a vertex x of G is an eigenvalue ofthe subgraph induced by the neighbours of x .



Let G be a graph.The eigenvalues of its adjacency matrix are called theeigenvalues of G.We say two graphs are cospectral if their adjacencymatrices are similar to each other, i.e. they have the samespectrum.A local eigenvalue at a vertex x of G is an eigenvalue ofthe subgraph induced by the neighbours of x .


Basic theory of DRG

Let Γ be a DRG say with diameter D.Let Ai be the distance-i matrix of Γ, i.e. (Ai)xy = 1 ifd(x , y) = i and 0 otherwise.A1 = A the adjacency matrix, A0 = I.

The Bose-Mesner algebraM is the matrix algebragenerated by {A}.{Ai | i = 0,1, . . . ,D} is a basis ofM.M has also a basis of primitive idempotents{Ei | i = 0,1, . . . ,D}, where E0 = J/v (v = number ofvertices).Also AEi = θiEi , for some real numbers θi andθi , i = 0,1, . . . ,D are the eigenvalues of Γ.


Basic theory of DRG

Let Γ be a DRG say with diameter D.Let Ai be the distance-i matrix of Γ, i.e. (Ai)xy = 1 ifd(x , y) = i and 0 otherwise.A1 = A the adjacency matrix, A0 = I.The Bose-Mesner algebraM is the matrix algebragenerated by {A}.{Ai | i = 0,1, . . . ,D} is a basis ofM.

M has also a basis of primitive idempotents{Ei | i = 0,1, . . . ,D}, where E0 = J/v (v = number ofvertices).Also AEi = θiEi , for some real numbers θi andθi , i = 0,1, . . . ,D are the eigenvalues of Γ.


Basic theory of DRG

Let Γ be a DRG say with diameter D.Let Ai be the distance-i matrix of Γ, i.e. (Ai)xy = 1 ifd(x , y) = i and 0 otherwise.A1 = A the adjacency matrix, A0 = I.The Bose-Mesner algebraM is the matrix algebragenerated by {A}.{Ai | i = 0,1, . . . ,D} is a basis ofM.M has also a basis of primitive idempotents{Ei | i = 0,1, . . . ,D}, where E0 = J/v (v = number ofvertices).Also AEi = θiEi , for some real numbers θi andθi , i = 0,1, . . . ,D are the eigenvalues of Γ.


Q-polynomial DRG

Γ is called Q-polynomial if there exists an ordering of theprimitive idempotents E0,E1, . . . ,ED such that Ei = p◦i (E)where pi is a polynomial of degree i . (Here ◦ meanscoordinate-wise product)If E0,E1, . . . ,ED a Q-polynomial ordering of theidempotents, then we say that the correspondingeigenvalues θ0 = k , θ1, . . . , θD is a Q-polynomial orderingof the eigenvalues.


Outline






Alexander Gavrilyuk came in late March to POSTECH forabout two weeks.We started to discuss the folded halved cubes as therewere still some cases where uniqueness was not known.I remembered a conversation with Paul Terwilliger in June2011, during the Bled conference, where he mentionedsome polynomial for the local eigenvalues.


The Terwilliger polynomial

Let Γ be a Q-polynomial DRG with diameter D ≥ 3 andθ0, θ1, . . . , θD a Q-polynomial ordering of its eigenvalues.

Terwilliger (early 1990’s): There exists a polynomial pT ofdegree 4, which we call the Terwilliger polynomial, suchthat for any local eigenvalue λ (with an eigenvector vorthogonal to the all-one vector) at any vertex x satisfies:pT (λ) ≥ 0.pT only depends on the intersection numbers and theQ-polynomial ordering of its eigenvalues. (That is, if Γ hasmore then one Q-polynomial ordering, then the polynomialpT may be different for each of these orderings).One can say more when pT (λ) = 0, namely the modulegenerated by v under the Terwilliger (or subconstituent)algebra is so-called thin and if pT (λ) > 0, then this moduleis NOT thin.



Let Γ be a Q-polynomial DRG with diameter D ≥ 3 andθ0, θ1, . . . , θD a Q-polynomial ordering of its eigenvalues.Terwilliger (early 1990’s): There exists a polynomial pT ofdegree 4, which we call the Terwilliger polynomial, suchthat for any local eigenvalue λ (with an eigenvector vorthogonal to the all-one vector) at any vertex x satisfies:pT (λ) ≥ 0.

pT only depends on the intersection numbers and theQ-polynomial ordering of its eigenvalues. (That is, if Γ hasmore then one Q-polynomial ordering, then the polynomialpT may be different for each of these orderings).One can say more when pT (λ) = 0, namely the modulegenerated by v under the Terwilliger (or subconstituent)algebra is so-called thin and if pT (λ) > 0, then this moduleis NOT thin.



Let Γ be a Q-polynomial DRG with diameter D ≥ 3 andθ0, θ1, . . . , θD a Q-polynomial ordering of its eigenvalues.Terwilliger (early 1990’s): There exists a polynomial pT ofdegree 4, which we call the Terwilliger polynomial, suchthat for any local eigenvalue λ (with an eigenvector vorthogonal to the all-one vector) at any vertex x satisfies:pT (λ) ≥ 0.pT only depends on the intersection numbers and theQ-polynomial ordering of its eigenvalues. (That is, if Γ hasmore then one Q-polynomial ordering, then the polynomialpT may be different for each of these orderings).

One can say more when pT (λ) = 0, namely the modulegenerated by v under the Terwilliger (or subconstituent)algebra is so-called thin and if pT (λ) > 0, then this moduleis NOT thin.



Let Γ be a Q-polynomial DRG with diameter D ≥ 3 andθ0, θ1, . . . , θD a Q-polynomial ordering of its eigenvalues.Terwilliger (early 1990’s): There exists a polynomial pT ofdegree 4, which we call the Terwilliger polynomial, suchthat for any local eigenvalue λ (with an eigenvector vorthogonal to the all-one vector) at any vertex x satisfies:pT (λ) ≥ 0.pT only depends on the intersection numbers and theQ-polynomial ordering of its eigenvalues. (That is, if Γ hasmore then one Q-polynomial ordering, then the polynomialpT may be different for each of these orderings).One can say more when pT (λ) = 0, namely the modulegenerated by v under the Terwilliger (or subconstituent)algebra is so-called thin and if pT (λ) > 0, then this moduleis NOT thin.


Outline






The folded halved cubes

The folded halved 2n-cube 12Q̃(2n) is defined as follows:

You first take the halved cube 12Q(2n) and then identify

vertices at maximal distance.

For n ≥ 4, the local graph (i.e. the subgraph induced by theneighbours of some vertex) is a J(2n,2).The distinct eigenvalues of J(2n,2) are2(2n − 2),n − 4,−2.Every irreducible module for the Terwilliger algebra is thin.The diameter D of 1

2Q̃(2n) equals bn2c.





vertices at maximal distance.For n ≥ 4, the local graph (i.e. the subgraph induced by theneighbours of some vertex) is a J(2n,2).The distinct eigenvalues of J(2n,2) are2(2n − 2),n − 4,−2.

Every irreducible module for the Terwilliger algebra is thin.The diameter D of 1






vertices at maximal distance.For n ≥ 4, the local graph (i.e. the subgraph induced by theneighbours of some vertex) is a J(2n,2).The distinct eigenvalues of J(2n,2) are2(2n − 2),n − 4,−2.Every irreducible module for the Terwilliger algebra is thin.The diameter D of 1



The folded halved cubes II

The Terwilliger polynomial pT for the folded halved 2n-cubehas roots n− 4, −2, −1− D

D−1 and a root larger then n− 4.Its leading term is negative.

So this means that there are are no local eigenvaluesinside the open interval (−2,n − 4).One can show

∑λ6=2(2n−2)(λ+ 2)(λ− (n − 4)) = 0, where

the sum is over the local eigenvalues different from2(2n − 2), using easy properties of the spectrum of thelocal graph.This shows that the distinct local eigenvalues are2(2n − 2),n − 4,−2 and hence the local graph is stronglyregular.




D−1 and a root larger then n− 4.Its leading term is negative.So this means that there are are no local eigenvaluesinside the open interval (−2,n − 4).

One can show∑

λ6=2(2n−2)(λ+ 2)(λ− (n − 4)) = 0, wherethe sum is over the local eigenvalues different from2(2n − 2), using easy properties of the spectrum of thelocal graph.This shows that the distinct local eigenvalues are2(2n − 2),n − 4,−2 and hence the local graph is stronglyregular.




D−1 and a root larger then n− 4.Its leading term is negative.So this means that there are are no local eigenvaluesinside the open interval (−2,n − 4).One can show

∑λ 6=2(2n−2)(λ+ 2)(λ− (n − 4)) = 0, where

the sum is over the local eigenvalues different from2(2n − 2), using easy properties of the spectrum of thelocal graph.

This shows that the distinct local eigenvalues are2(2n − 2),n − 4,−2 and hence the local graph is stronglyregular.




D−1 and a root larger then n− 4.Its leading term is negative.So this means that there are are no local eigenvaluesinside the open interval (−2,n − 4).One can show

∑λ 6=2(2n−2)(λ+ 2)(λ− (n − 4)) = 0, where

the sum is over the local eigenvalues different from2(2n − 2), using easy properties of the spectrum of thelocal graph.This shows that the distinct local eigenvalues are2(2n − 2),n − 4,−2 and hence the local graph is stronglyregular.


The folded halved cubes III

By the classification of Seidel (1960’s) of strongly regulargraphs with smallest eigenvalue −2 we find that the localgraph is J(2n,2) (as n ≥ 6).

Now the uniqueness follows from work of Bussemaker andNeumaier (1990’s), as they characterized the halved folded2n-cubes locally.Essentially the same proof also works for the foldedJohnson graphs.



By the classification of Seidel (1960’s) of strongly regulargraphs with smallest eigenvalue −2 we find that the localgraph is J(2n,2) (as n ≥ 6).Now the uniqueness follows from work of Bussemaker andNeumaier (1990’s), as they characterized the halved folded2n-cubes locally.

Essentially the same proof also works for the foldedJohnson graphs.



By the classification of Seidel (1960’s) of strongly regulargraphs with smallest eigenvalue −2 we find that the localgraph is J(2n,2) (as n ≥ 6).Now the uniqueness follows from work of Bussemaker andNeumaier (1990’s), as they characterized the halved folded2n-cubes locally.Essentially the same proof also works for the foldedJohnson graphs.


Outline






The Grassmann graph I

The Grassmann graph Jq(n,D) (n ≥ 2D) has as verticesthe D-dim subspaces of a fixed n-dim vector space over afinite field with q elements.Two vertices are adjacent if they intersect in a (D − 1)-dimsubspace.

The local graph is isomorphic to the q-clique extension of aqD−1q−1 ×

qn−D−1q−1 -grid, i.e. the strong product of a q-clique

and a qD−1q−1 ×

qn−D−1q−1 -grid.

The distinct non-trivial local eigenvalues are−q − 1,−1,q2 qD−1−1

q−1 ,q2 qn−D−1−1q−1 if n > 2D and

−q − 1,−1,q2 qD−1−1q−1 if n = 2D.


The Grassmann graph I

The Grassmann graph Jq(n,D) (n ≥ 2D) has as verticesthe D-dim subspaces of a fixed n-dim vector space over afinite field with q elements.Two vertices are adjacent if they intersect in a (D − 1)-dimsubspace.The local graph is isomorphic to the q-clique extension of aqD−1q−1 ×

qn−D−1q−1 -grid, i.e. the strong product of a q-clique

and a qD−1q−1 ×

qn−D−1q−1 -grid.

The distinct non-trivial local eigenvalues are−q − 1,−1,q2 qD−1−1

q−1 ,q2 qn−D−1−1q−1 if n > 2D and

−q − 1,−1,q2 qD−1−1q−1 if n = 2D.


The Grassmann graphs II

The roots of the Terwilliger polynomial pT are−q − 1,−1,q2 qD−1−1

q−1 ,q2 qn−D−1−1q−1 if n > 2D and

−q − 1,−1,q2 qD−1−1q−1 ,q2 qD−1−1

q−1 if n = 2D.The leading coefficient of pT is negative.


The Grassmann graphs III

Using a similar but more difficult method as above, Gavrilyukand I showed:

TheoremAny DRG with the same parameters as Jq(2D,D) D ≥ 3 haslocal graphs cospectral with the q-clique extension of aqD−1q−1 ×

qD−1q−1 -grid.

This in particular means that any irreducibleT -module with end point one is thin, and that there are exactlythree non-isomorphic ones.

For q = 2 and D ≥ 3 odd, we were able to show that the localgraph must be isomorphic to q-clique extension of aqD−1q−1 ×

qD−1q−1 -grid, and hence they are unique with these

parameters, using local charaterizations, dating back toSprague and Ray-Chaudhuri.


The Grassmann graphs III

Using a similar but more difficult method as above, Gavrilyukand I showed:

TheoremAny DRG with the same parameters as Jq(2D,D) D ≥ 3 haslocal graphs cospectral with the q-clique extension of aqD−1q−1 ×

qD−1q−1 -grid. This in particular means that any irreducible

T -module with end point one is thin, and that there are exactlythree non-isomorphic ones.

For q = 2 and D ≥ 3 odd, we were able to show that the localgraph must be isomorphic to q-clique extension of aqD−1q−1 ×

qD−1q−1 -grid, and hence they are unique with these

parameters, using local charaterizations, dating back toSprague and Ray-Chaudhuri.


The Grassmann graphs IV

Some remarks:

The q-clique extension of a t × t-grid is not determined bytheir spectrum in general as Van Dam in his thesisconstructed some examples with the same spectrum. Buton this moment we do not have graphs cospectral with aq-clique extension of a t × t-grid with large t .There should be a theory of DRG with exactly three thinmodules with end point one, like the one developed for thecase when there are exactly two thin modules with endpoint one, as developed by Curtin. We are confident thatusing such a theory would show that the Grassmanngraphs Jq(2D,D) are determined by their intersectionnumbers.



Some remarks:The q-clique extension of a t × t-grid is not determined bytheir spectrum in general as Van Dam in his thesisconstructed some examples with the same spectrum. Buton this moment we do not have graphs cospectral with aq-clique extension of a t × t-grid with large t .

There should be a theory of DRG with exactly three thinmodules with end point one, like the one developed for thecase when there are exactly two thin modules with endpoint one, as developed by Curtin. We are confident thatusing such a theory would show that the Grassmanngraphs Jq(2D,D) are determined by their intersectionnumbers.



Some remarks:The q-clique extension of a t × t-grid is not determined bytheir spectrum in general as Van Dam in his thesisconstructed some examples with the same spectrum. Buton this moment we do not have graphs cospectral with aq-clique extension of a t × t-grid with large t .There should be a theory of DRG with exactly three thinmodules with end point one, like the one developed for thecase when there are exactly two thin modules with endpoint one, as developed by Curtin. We are confident thatusing such a theory would show that the Grassmanngraphs Jq(2D,D) are determined by their intersectionnumbers.


The bilinear forms graphs

The same approach may also work for the bilinear formsgraph Bil(D × e,q), e ≥ D especially in the case whene = D.

We are able to show that any DRG with the sameparameters as Bil(D × D,2) is locally a square grid.

Thank you for your attention.


The bilinear forms graphs

The same approach may also work for the bilinear formsgraph Bil(D × e,q), e ≥ D especially in the case whene = D.We are able to show that any DRG with the sameparameters as Bil(D × D,2) is locally a square grid.

Thank you for your attention.

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