Some recent results on ( P and Q )-polynomial association schemes - - PowerPoint PPT Presentation

Some recent results on (P and Q)-polynomial association schemes

Alexander Gavrilyuk (USTC, Hefei, China) based on joint work with Jack Koolen (USTC, Hefei, China) Shanghai Jiao-Tong University, 2017

Algebraic Combinatorics

Association Schemes (AS) represent one of the most important topics of Algebraic Combinatorics. In this area of mathematics, we apply:

◮ methods of abstract algebra to combinatorial problems, ◮ combinatorial techniques to problems in algebra.

It took origins mainly in:

◮ theory of finite groups, their representations and characters,

and, among other things, gave rise to:

◮ combinatorial/spherical designs, ◮ error-correcting codes, ◮ finite geometry, ◮ matroids, ◮ ...

Algebraic Combinatorics

Association Schemes (AS) represent one of the most important topics of Algebraic Combinatorics. In this area of mathematics, we apply:

◮ methods of abstract algebra to combinatorial problems, ◮ combinatorial techniques to problems in algebra.

It took origins mainly in:

◮ theory of finite groups, their representations and characters,

and, among other things, gave rise to:

◮ combinatorial/spherical designs, ◮ error-correcting codes, ◮ finite geometry, ◮ matroids, ◮ ...

Algebraic Combinatorics

Association Schemes (AS) represent one of the most important topics of Algebraic Combinatorics. In this area of mathematics, we apply:

◮ methods of abstract algebra to combinatorial problems, ◮ combinatorial techniques to problems in algebra.

It took origins mainly in:

◮ theory of finite groups, their representations and characters,

and, among other things, gave rise to:

◮ combinatorial/spherical designs, ◮ error-correcting codes, ◮ finite geometry, ◮ matroids, ◮ ...

Distance-Transitive Graphs (DTG)

Typical objects of interest in Algebraic Combinatorics are those with a lot of symmetries. A graph Γ = (V, E) is defined by:

◮ a (finite) vertex set V , ◮ an edge set E of unordered pairs of elements of V .

The distance function ∂ between vertices x, y ∈ V is the length of a shortest path x y. An automorphism: a bijection V → V that preserves the

• edges. More automorphisms = more symmetries.

Distance-Transitive Graphs (DTG) are the most symmetric

• nes: ∀ x1, x2, y1, y2 ∈ V : ∂(x1, x2) = ∂(y1, y2)

∃ γ ∈ Aut(Γ): γ(x1) = y1, γ(x2) = y2.

Distance-Transitive Graphs (DTG)

Typical objects of interest in Algebraic Combinatorics are those with a lot of symmetries. A graph Γ = (V, E) is defined by:

◮ a (finite) vertex set V , ◮ an edge set E of unordered pairs of elements of V .

The distance function ∂ between vertices x, y ∈ V is the length of a shortest path x y. An automorphism: a bijection V → V that preserves the

• edges. More automorphisms = more symmetries.

Distance-Transitive Graphs (DTG) are the most symmetric

• nes: ∀ x1, x2, y1, y2 ∈ V : ∂(x1, x2) = ∂(y1, y2)

∃ γ ∈ Aut(Γ): γ(x1) = y1, γ(x2) = y2.

Distance-Transitive Graphs (DTG)

Typical objects of interest in Algebraic Combinatorics are those with a lot of symmetries. A graph Γ = (V, E) is defined by:

◮ a (finite) vertex set V , ◮ an edge set E of unordered pairs of elements of V .

The distance function ∂ between vertices x, y ∈ V is the length of a shortest path x y. An automorphism: a bijection V → V that preserves the

• edges. More automorphisms = more symmetries.

Distance-Transitive Graphs (DTG) are the most symmetric

• nes: ∀ x1, x2, y1, y2 ∈ V : ∂(x1, x2) = ∂(y1, y2)

∃ γ ∈ Aut(Γ): γ(x1) = y1, γ(x2) = y2.

Distance-Transitive Graphs (DTG)

Typical objects of interest in Algebraic Combinatorics are those with a lot of symmetries. A graph Γ = (V, E) is defined by:

◮ a (finite) vertex set V , ◮ an edge set E of unordered pairs of elements of V .

The distance function ∂ between vertices x, y ∈ V is the length of a shortest path x y. An automorphism: a bijection V → V that preserves the

• edges. More automorphisms = more symmetries.

Distance-Transitive Graphs (DTG) are the most symmetric

• nes: ∀ x1, x2, y1, y2 ∈ V : ∂(x1, x2) = ∂(y1, y2)

∃ γ ∈ Aut(Γ): γ(x1) = y1, γ(x2) = y2.

Distance-Transitive Graphs: examples

The obvious examples of DTG come from Platonic Solids: Less trivial examples come from multiplicity-free permutation representations

• f almost simple groups or affine groups.

The classification of all DTG’s can be achieved by using CFSG. This project is almost completed, and the most of DTG’s come from classical algebraic objects like dual polar spaces and forms

• ver finite fields.

Distance-Transitive Graphs: examples

The obvious examples of DTG come from Platonic Solids: Less trivial examples come from multiplicity-free permutation representations

• f almost simple groups or affine groups.

The classification of all DTG’s can be achieved by using CFSG. This project is almost completed, and the most of DTG’s come from classical algebraic objects like dual polar spaces and forms

• ver finite fields.

Distance-Transitive Graphs: examples

The obvious examples of DTG come from Platonic Solids: Less trivial examples come from multiplicity-free permutation representations

• f almost simple groups or affine groups.

The classification of all DTG’s can be achieved by using CFSG. This project is almost completed, and the most of DTG’s come from classical algebraic objects like dual polar spaces and forms

• ver finite fields.

DTG → Association Schemes ← Finite Groups

Let Γ be a DTG. Define its distance-i matrices Ai ∈ RV ×V : (Ai)x,y :=

• 1 if ∂(x, y) = i,

0 if ∂(x, y) = i, where 0 ≤ i ≤ D (D is the diameter of Γ). Then: (1) Ai are symmetric (0,1)-matrices and

D

• i=0

Ai = J (all-1) (A0 = I, and A1 is the adjacency matrix of Γ), (2) AiAj = AjAi =

D

• k=0

pk

ijAk for some pk ij ∈ N.

We take (1) and (2) as the definition of a (symmetric) AS. There are several ways to interpret finite groups as AS’s.

DTG → Association Schemes ← Finite Groups

Let Γ be a DTG. Define its distance-i matrices Ai ∈ RV ×V : (Ai)x,y :=

• 1 if ∂(x, y) = i,

0 if ∂(x, y) = i, where 0 ≤ i ≤ D (D is the diameter of Γ). Then: (1) Ai are symmetric (0,1)-matrices and

D

• i=0

Ai = J (all-1) (A0 = I, and A1 is the adjacency matrix of Γ), (2) AiAj = AjAi =

D

• k=0

pk

ijAk for some pk ij ∈ N.

We take (1) and (2) as the definition of a (symmetric) AS. There are several ways to interpret finite groups as AS’s.

DTG → Association Schemes ← Finite Groups

Let Γ be a DTG. Define its distance-i matrices Ai ∈ RV ×V : (Ai)x,y :=

• 1 if ∂(x, y) = i,

0 if ∂(x, y) = i, where 0 ≤ i ≤ D (D is the diameter of Γ). Then: (1) Ai are symmetric (0,1)-matrices and

D

• i=0

Ai = J (all-1) (A0 = I, and A1 is the adjacency matrix of Γ), (2) AiAj = AjAi =

D

• k=0

pk

ijAk for some pk ij ∈ N.

We take (1) and (2) as the definition of a (symmetric) AS. There are several ways to interpret finite groups as AS’s.

DTG → Association Schemes ← Finite Groups

Let Γ be a DTG. Define its distance-i matrices Ai ∈ RV ×V : (Ai)x,y :=

• 1 if ∂(x, y) = i,

0 if ∂(x, y) = i, where 0 ≤ i ≤ D (D is the diameter of Γ). Then: (1) Ai are symmetric (0,1)-matrices and

D

• i=0

Ai = J (all-1) (A0 = I, and A1 is the adjacency matrix of Γ), (2) AiAj = AjAi =

D

• k=0

pk

ijAk for some pk ij ∈ N.

We take (1) and (2) as the definition of a (symmetric) AS. There are several ways to interpret finite groups as AS’s.

DTG → Association Schemes ← Finite Groups

Let Γ be a DTG. Define its distance-i matrices Ai ∈ RV ×V : (Ai)x,y :=

• 1 if ∂(x, y) = i,

0 if ∂(x, y) = i, where 0 ≤ i ≤ D (D is the diameter of Γ). Then: (1) Ai are symmetric (0,1)-matrices and

D

• i=0

Ai = J (all-1) (A0 = I, and A1 is the adjacency matrix of Γ), (2) AiAj = AjAi =

D

• k=0

pk

ijAk for some pk ij ∈ N.

We take (1) and (2) as the definition of a (symmetric) AS. There are several ways to interpret finite groups as AS’s.

Intersection numbers

Let Γ be a DTG. Define its distance-i matrices Ai ∈ RV ×V : (Ai)x,y :=

• 1 if ∂(x, y) = i,

0 if ∂(x, y) = i, Then: AiAj = AjAi =

D

• k=0

pk

ijAk for some pk ij ∈ N.

The numbers pk

ij have simple combinatorial interpretation:

∂(x, y) = k ⇒ pk

ij := |{z ∈ V : ∂(x, z) = i, ∂(y, z) = j}|.

DTG → P-polynomial Association Schemes

The equation AiAj = AjAi =

D

• k=0

pk

ijAk

shows that {A0, A1, . . . , AD} generate a matrix algebra over R (of dimension D + 1), called the Bose-Mesner algebra of Γ. Moreover, when AS comes from DTG, Ai = vi(A1), for some polynomials vi, i = 0, 1, . . . , D, of degree i. This property is called P-polynomiality. DTG P-polynomial AS

DTG → P-polynomial Association Schemes

The equation AiAj = AjAi =

D

• k=0

pk

ijAk

shows that {A0, A1, . . . , AD} generate a matrix algebra over R (of dimension D + 1), called the Bose-Mesner algebra of Γ. Moreover, when AS comes from DTG, Ai = vi(A1), for some polynomials vi, i = 0, 1, . . . , D, of degree i. This property is called P-polynomiality. DTG P-polynomial AS

DTG → P-polynomial Association Schemes

The equation AiAj = AjAi =

D

• k=0

pk

ijAk

shows that {A0, A1, . . . , AD} generate a matrix algebra over R (of dimension D + 1), called the Bose-Mesner algebra of Γ. Moreover, when AS comes from DTG, Ai = vi(A1), for some polynomials vi, i = 0, 1, . . . , D, of degree i. This property is called P-polynomiality. DTG P-polynomial AS

P-polynomial Association Schemes = DRG

DTG P-polynomial AS In a P-polynomial AS, for some ordering of A0, A1, . . . , AD: Ai = vi(A1), for some polynomials vi, i = 0, 1, . . . , D, of degree i. A1 can be considered as the adjacency matrix of a graph, while

• ther Ai’s as its distance-i matrices.

Such a graph is called Distance-Regular (DRG). ∂(x, y) = k ⇒ pk

ij := |{z ∈ V : ∂(x, z) = i, ∂(y, z) = j}|.

P-polynomial Association Schemes = DRG

DTG P-polynomial AS In a P-polynomial AS, for some ordering of A0, A1, . . . , AD: Ai = vi(A1), for some polynomials vi, i = 0, 1, . . . , D, of degree i. A1 can be considered as the adjacency matrix of a graph, while

• ther Ai’s as its distance-i matrices.

Such a graph is called Distance-Regular (DRG). ∂(x, y) = k ⇒ pk

ij := |{z ∈ V : ∂(x, z) = i, ∂(y, z) = j}|.

DTG P-polynomial AS = DRG

The smallest DRG that is not DTG: The Shrikhande graph.

◮ Its distance-i matrices form P-polynomial AS; ◮ However, its group of automorphisms is not

Distance-Transitive.

Dual operation on the Bose-Mesner algebra

Let {A0, A1, . . . , AD} be a symmetric association scheme. All Ai’s can be simultaneously diagonalised, so that the matrix algebra generated by Ai’s has the second basis: E0, E1, . . . , ED, where Ej is the orthogonal projection onto a maximal common eigenspace of all {Ai}D

i=0.

Ai =

D

• k=0

θikEk, Ei =

D

• k=0

θ∗

ikAk,

AiAj =

D

• k=0

pk

ijAk,

EiEj = δijEi, Ai ◦ Aj = δijAi, Ei ◦ Ej =

D

• k=0

qk

ijEk,

pk

ij

qk

ij (the Krein parameters)

Dual operation on the Bose-Mesner algebra

Let {A0, A1, . . . , AD} be a symmetric association scheme. All Ai’s can be simultaneously diagonalised, so that the matrix algebra generated by Ai’s has the second basis: E0, E1, . . . , ED, where Ej is the orthogonal projection onto a maximal common eigenspace of all {Ai}D

i=0.

Ai =

D

• k=0

θikEk, Ei =

D

• k=0

θ∗

ikAk,

AiAj =

D

• k=0

pk

ijAk,

EiEj = δijEi, Ai ◦ Aj = δijAi, Ei ◦ Ej =

D

• k=0

qk

ijEk,

pk

ij

qk

ij (the Krein parameters)

Dual operation on the Bose-Mesner algebra

Let {A0, A1, . . . , AD} be a symmetric association scheme. All Ai’s can be simultaneously diagonalised, so that the matrix algebra generated by Ai’s has the second basis: E0, E1, . . . , ED, where Ej is the orthogonal projection onto a maximal common eigenspace of all {Ai}D

i=0.

Ai =

D

• k=0

θikEk, Ei =

D

• k=0

θ∗

ikAk,

AiAj =

D

• k=0

pk

ijAk,

EiEj = δijEi, Ai ◦ Aj = δijAi, Ei ◦ Ej =

D

• k=0

qk

ijEk,

pk

ij

qk

ij (the Krein parameters)

Dual operation on the Bose-Mesner algebra

Let {A0, A1, . . . , AD} be a symmetric association scheme. All Ai’s can be simultaneously diagonalised, so that the matrix algebra generated by Ai’s has the second basis: E0, E1, . . . , ED, where Ej is the orthogonal projection onto a maximal common eigenspace of all {Ai}D

i=0.

Ai =

D

• k=0

θikEk, Ei =

D

• k=0

θ∗

ikAk,

AiAj =

D

• k=0

pk

ijAk,

EiEj = δijEi, Ai ◦ Aj = δijAi, Ei ◦ Ej =

D

• k=0

qk

ijEk,

pk

ij

qk

ij (the Krein parameters)

Dual operation on the Bose-Mesner algebra

Let {A0, A1, . . . , AD} be a symmetric association scheme. All Ai’s can be simultaneously diagonalised, so that the matrix algebra generated by Ai’s has the second basis: E0, E1, . . . , ED, where Ej is the orthogonal projection onto a maximal common eigenspace of all {Ai}D

i=0.

Ai =

D

• k=0

θikEk, Ei =

D

• k=0

θ∗

ikAk,

AiAj =

D

• k=0

pk

ijAk,

EiEj = δijEi, Ai ◦ Aj = δijAi, Ei ◦ Ej =

D

• k=0

qk

ijEk,

pk

ij

qk

ij (the Krein parameters)

Duality and Q-polynomial Association Schemes

Two bases of the Bose-Mesner algebra: {A0, A1, . . . , AD} {E0, E1, . . . , ED} For a P-polynomial association scheme: Ai = vi(A1) Suppose now that, for some ordering of Ej’s: Ej = v∗

j (E1),

where v∗

j is a ◦-polynomial of degree j.

This property is called Q-polynomiality, and it was introduced by P. Delsarte (1973) in his PhD Thesis ’An Algebraic Approach to the Association Schemes of Coding Theory’.

Duality and Q-polynomial Association Schemes

Two bases of the Bose-Mesner algebra: {A0, A1, . . . , AD} {E0, E1, . . . , ED} For a P-polynomial association scheme: Ai = vi(A1) Suppose now that, for some ordering of Ej’s: Ej = v∗

j (E1),

where v∗

j is a ◦-polynomial of degree j.

This property is called Q-polynomiality, and it was introduced by P. Delsarte (1973) in his PhD Thesis ’An Algebraic Approach to the Association Schemes of Coding Theory’.

Duality and Q-polynomial Association Schemes

Two bases of the Bose-Mesner algebra: {A0, A1, . . . , AD} {E0, E1, . . . , ED} For a P-polynomial association scheme: Ai = vi(A1) Suppose now that, for some ordering of Ej’s: Ej = v∗

j (E1),

where v∗

j is a ◦-polynomial of degree j.

This property is called Q-polynomiality, and it was introduced by P. Delsarte (1973) in his PhD Thesis ’An Algebraic Approach to the Association Schemes of Coding Theory’.

Duality and Q-polynomial Association Schemes

Two bases of the Bose-Mesner algebra: {A0, A1, . . . , AD} {E0, E1, . . . , ED} For a P-polynomial association scheme: Ai = vi(A1) Suppose now that, for some ordering of Ej’s: Ej = v∗

j (E1),

where v∗

j is a ◦-polynomial of degree j.

This property is called Q-polynomiality, and it was introduced by P. Delsarte (1973) in his PhD Thesis ’An Algebraic Approach to the Association Schemes of Coding Theory’.

Error-Correcting Codes and Hamming Scheme

Let X be an alphabet of m symbols, Xn = X × X × . . . X (n times) is the set of words of length n over X. The main idea of ECC is to increase the Hamming distance between words by mapping them from Xk to Xn, n > k. The Hamming graph H(n, m):

◮ the vertex set V = Xn, |X| = m, ◮ the distance ∂(x, y) is the Hamming distance, i.e., the

number of different coordinates between words x and y.

◮ distance-transitive and Q-polynomial ⇒ (P and Q), ◮ a code is a subset of V (with large min distance).

Error-Correcting Codes and Hamming Scheme

Let X be an alphabet of m symbols, Xn = X × X × . . . X (n times) is the set of words of length n over X. The main idea of ECC is to increase the Hamming distance between words by mapping them from Xk to Xn, n > k. The Hamming graph H(n, m):

◮ the vertex set V = Xn, |X| = m, ◮ the distance ∂(x, y) is the Hamming distance, i.e., the

number of different coordinates between words x and y.

◮ distance-transitive and Q-polynomial ⇒ (P and Q), ◮ a code is a subset of V (with large min distance).

Error-Correcting Codes and Hamming Scheme

Let X be an alphabet of m symbols, Xn = X × X × . . . X (n times) is the set of words of length n over X. The main idea of ECC is to increase the Hamming distance between words by mapping them from Xk to Xn, n > k. The Hamming graph H(n, m):

◮ the vertex set V = Xn, |X| = m, ◮ the distance ∂(x, y) is the Hamming distance, i.e., the

number of different coordinates between words x and y.

◮ distance-transitive and Q-polynomial ⇒ (P and Q), ◮ a code is a subset of V (with large min distance).

Error-Correcting Codes and Hamming Scheme

Let X be an alphabet of m symbols, Xn = X × X × . . . X (n times) is the set of words of length n over X. The main idea of ECC is to increase the Hamming distance between words by mapping them from Xk to Xn, n > k. The Hamming graph H(n, m):

◮ the vertex set V = Xn, |X| = m, ◮ the distance ∂(x, y) is the Hamming distance, i.e., the

number of different coordinates between words x and y.

◮ distance-transitive and Q-polynomial ⇒ (P and Q), ◮ a code is a subset of V (with large min distance).

Designs and Johnson Scheme

Let X be a set of v elements, X

k

• = {all k-element subsets of X}.

A design D of strength t is a subset of X

k

• such that: every

element of X

t

• appears in the same number of elements of D.

The Johnson graph J(v, k):

◮ the vertex set V =

X

k

• ,

◮ the distance ∂(x, y) is the Johnson metric, i.e., k − |x ∩ y|. ◮ distance-transitive and Q-polynomial ⇒ (P and Q), ◮ a design is a subset of V .

Designs and Johnson Scheme

Let X be a set of v elements, X

k

• = {all k-element subsets of X}.

A design D of strength t is a subset of X

k

• such that: every

element of X

t

• appears in the same number of elements of D.

The Johnson graph J(v, k):

◮ the vertex set V =

X

k

• ,

◮ the distance ∂(x, y) is the Johnson metric, i.e., k − |x ∩ y|. ◮ distance-transitive and Q-polynomial ⇒ (P and Q), ◮ a design is a subset of V .

Designs and Johnson Scheme

Let X be a set of v elements, X

k

• = {all k-element subsets of X}.

A design D of strength t is a subset of X

k

• such that: every

element of X

t

• appears in the same number of elements of D.

The Johnson graph J(v, k):

◮ the vertex set V =

X

k

• ,

◮ the distance ∂(x, y) is the Johnson metric, i.e., k − |x ∩ y|. ◮ distance-transitive and Q-polynomial ⇒ (P and Q), ◮ a design is a subset of V .

Designs and Johnson Scheme

Let X be a set of v elements, X

k

• = {all k-element subsets of X}.

A design D of strength t is a subset of X

k

• such that: every

element of X

t

• appears in the same number of elements of D.

The Johnson graph J(v, k):

◮ the vertex set V =

X

k

• ,

◮ the distance ∂(x, y) is the Johnson metric, i.e., k − |x ∩ y|. ◮ distance-transitive and Q-polynomial ⇒ (P and Q), ◮ a design is a subset of V .

Delsarte’s Theory

The main problem of coding theory (sphere-packing problem):

◮ maximize |C| subject to fixed min distance, ◮ maximize min distance subject to |C|.

There are some fundamental bounds on |C| in terms of min distance and some other parameters. There are very similar bounds on |D| for a design D in terms of its strength t and some other parameters. Delsarte reproved all these bounds in a uniform way. To do so, he realized that codes and designs should be considered as subsets of vertices of certain (P and Q)-polynomial scheme. Roughly speaking, deriving those bounds in terms of:

◮ ’distance’ requires P-polynomiality (we need a metric), ◮ ’strength’ requires Q-polynomiality (∗-metric).

Delsarte’s Theory

The main problem of coding theory (sphere-packing problem):

◮ maximize |C| subject to fixed min distance, ◮ maximize min distance subject to |C|.

There are some fundamental bounds on |C| in terms of min distance and some other parameters. There are very similar bounds on |D| for a design D in terms of its strength t and some other parameters. Delsarte reproved all these bounds in a uniform way. To do so, he realized that codes and designs should be considered as subsets of vertices of certain (P and Q)-polynomial scheme. Roughly speaking, deriving those bounds in terms of:

◮ ’distance’ requires P-polynomiality (we need a metric), ◮ ’strength’ requires Q-polynomiality (∗-metric).

Delsarte’s Theory

The main problem of coding theory (sphere-packing problem):

◮ maximize |C| subject to fixed min distance, ◮ maximize min distance subject to |C|.

There are some fundamental bounds on |C| in terms of min distance and some other parameters. There are very similar bounds on |D| for a design D in terms of its strength t and some other parameters. Delsarte reproved all these bounds in a uniform way. To do so, he realized that codes and designs should be considered as subsets of vertices of certain (P and Q)-polynomial scheme. Roughly speaking, deriving those bounds in terms of:

◮ ’distance’ requires P-polynomiality (we need a metric), ◮ ’strength’ requires Q-polynomiality (∗-metric).

Bannai’s Observation

A spherical t-design: points x1, x2, . . . , xN ∈ Sd such that

• Sd f(x)dµd(x) = 1

N

N

• i=1

f(xi) for all polynomials f in d + 1 variables, of total degree ≤ t. It turned out that some bounds for t-designs, whose proofs required Q-polynomiality, can be shown for spherical designs in the same manner. Moreover, Bannai observed that it also works for compact symmetric spaces of rank 1. Q-polynomial AS P-polynomial AS ↓ ↓ Compact symmetric Compact 2-point space of rank 1 homogeneous space

• = by H.C. Wang (1952)

Classified by E. Cartan (1926)

Bannai’s Observation

A spherical t-design: points x1, x2, . . . , xN ∈ Sd such that

• Sd f(x)dµd(x) = 1

N

N

• i=1

f(xi) for all polynomials f in d + 1 variables, of total degree ≤ t. It turned out that some bounds for t-designs, whose proofs required Q-polynomiality, can be shown for spherical designs in the same manner. Moreover, Bannai observed that it also works for compact symmetric spaces of rank 1. Q-polynomial AS P-polynomial AS ↓ ↓ Compact symmetric Compact 2-point space of rank 1 homogeneous space

• = by H.C. Wang (1952)

Classified by E. Cartan (1926)

Bannai’s Observation

A spherical t-design: points x1, x2, . . . , xN ∈ Sd such that

• Sd f(x)dµd(x) = 1

N

N

• i=1

f(xi) for all polynomials f in d + 1 variables, of total degree ≤ t. It turned out that some bounds for t-designs, whose proofs required Q-polynomiality, can be shown for spherical designs in the same manner. Moreover, Bannai observed that it also works for compact symmetric spaces of rank 1. Q-polynomial AS P-polynomial AS ↓ ↓ Compact symmetric Compact 2-point space of rank 1 homogeneous space

• = by H.C. Wang (1952)

Classified by E. Cartan (1926)

Bannai’s Observation

A spherical t-design: points x1, x2, . . . , xN ∈ Sd such that

• Sd f(x)dµd(x) = 1

N

N

• i=1

f(xi) for all polynomials f in d + 1 variables, of total degree ≤ t. It turned out that some bounds for t-designs, whose proofs required Q-polynomiality, can be shown for spherical designs in the same manner. Moreover, Bannai observed that it also works for compact symmetric spaces of rank 1. Q-polynomial AS P-polynomial AS ↓ ↓ Compact symmetric Compact 2-point space of rank 1 homogeneous space

• = by H.C. Wang (1952)

Classified by E. Cartan (1926)

Bannai’s Observation

A spherical t-design: points x1, x2, . . . , xN ∈ Sd such that

• Sd f(x)dµd(x) = 1

N

N

• i=1

f(xi) for all polynomials f in d + 1 variables, of total degree ≤ t. It turned out that some bounds for t-designs, whose proofs required Q-polynomiality, can be shown for spherical designs in the same manner. Moreover, Bannai observed that it also works for compact symmetric spaces of rank 1. Q-polynomial AS

=?

P-polynomial AS ↓ ↓ Compact symmetric Compact 2-point space of rank 1 homogeneous space

• = by H.C. Wang (1952)

Classified by E. Cartan (1926)

Bannai’s Conjecture (early 1980s)

(1) If diameter D is large enough, then primitive P-polynomial association scheme is Q-polynomial, and vice versa. Seems to be beyond the scope of our reach at this moment. (2) All (P and Q)-polynomial association schemes are known. Bannai made a list of (P and Q)-polynomial association schemes as a finite-analogue of Cartan’s classification.

Bannai’s Conjecture (early 1980s)

(1) If diameter D is large enough, then primitive P-polynomial association scheme is Q-polynomial, and vice versa. Seems to be beyond the scope of our reach at this moment. (2) All (P and Q)-polynomial association schemes are known. Bannai made a list of (P and Q)-polynomial association schemes as a finite-analogue of Cartan’s classification.

Bannai’s Conjecture (early 1980s)

(1) If diameter D is large enough, then primitive P-polynomial association scheme is Q-polynomial, and vice versa. Seems to be beyond the scope of our reach at this moment. (2) All (P and Q)-polynomial association schemes are known. Bannai made a list of (P and Q)-polynomial association schemes as a finite-analogue of Cartan’s classification.

(known) (P and Q)-Polynomial Association Schemes

Main families (except of schemes of small diameter):

◮ Schemes of dual polar spaces

◮ Vertices: maximal isotropic subspaces of a vector space

equipped with a form (symplectic/quadratic/Hermitian).

◮ Distance ∂: codimension of their intersection.

◮ Schemes of sesquilinear/quadratic forms

◮ Vertices: a collection of matrices over a finite field. ◮ Distance ∂: rank of their difference.

◮ Johnson scheme (and its derivatives) ◮ Hamming scheme (and its derivatives) ◮ Grassmann scheme ◮ This list had not been changed, until Van Dam and Koolen

(2005) discovered a new family, the so-called twisted Grassmann scheme. Every known finite simple group (except of Lie type groups of small rank and some sporadic groups) is a group of automorphisms of a (P and Q)-polynomial AS from the above.

(known) (P and Q)-Polynomial Association Schemes

Main families (except of schemes of small diameter):

◮ Schemes of dual polar spaces

◮ Vertices: maximal isotropic subspaces of a vector space

equipped with a form (symplectic/quadratic/Hermitian).

◮ Distance ∂: codimension of their intersection.

◮ Schemes of sesquilinear/quadratic forms

◮ Vertices: a collection of matrices over a finite field. ◮ Distance ∂: rank of their difference.

◮ Johnson scheme (and its derivatives) ◮ Hamming scheme (and its derivatives) ◮ Grassmann scheme ◮ This list had not been changed, until Van Dam and Koolen

(2005) discovered a new family, the so-called twisted Grassmann scheme. Every known finite simple group (except of Lie type groups of small rank and some sporadic groups) is a group of automorphisms of a (P and Q)-polynomial AS from the above.

(known) (P and Q)-Polynomial Association Schemes

Main families (except of schemes of small diameter):

◮ Schemes of dual polar spaces

◮ Vertices: maximal isotropic subspaces of a vector space

equipped with a form (symplectic/quadratic/Hermitian).

◮ Distance ∂: codimension of their intersection.

◮ Schemes of sesquilinear/quadratic forms

◮ Vertices: a collection of matrices over a finite field. ◮ Distance ∂: rank of their difference.

◮ Johnson scheme (and its derivatives) ◮ Hamming scheme (and its derivatives) ◮ Grassmann scheme ◮ This list had not been changed, until Van Dam and Koolen

(2005) discovered a new family, the so-called twisted Grassmann scheme. Every known finite simple group (except of Lie type groups of small rank and some sporadic groups) is a group of automorphisms of a (P and Q)-polynomial AS from the above.

(known) (P and Q)-Polynomial Association Schemes

Main families (except of schemes of small diameter):

◮ Schemes of dual polar spaces

◮ Vertices: maximal isotropic subspaces of a vector space

equipped with a form (symplectic/quadratic/Hermitian).

◮ Distance ∂: codimension of their intersection.

◮ Schemes of sesquilinear/quadratic forms

◮ Vertices: a collection of matrices over a finite field. ◮ Distance ∂: rank of their difference.

◮ Johnson scheme (and its derivatives) ◮ Hamming scheme (and its derivatives) ◮ Grassmann scheme ◮ This list had not been changed, until Van Dam and Koolen

(2005) discovered a new family, the so-called twisted Grassmann scheme. Every known finite simple group (except of Lie type groups of small rank and some sporadic groups) is a group of automorphisms of a (P and Q)-polynomial AS from the above.

(known) (P and Q)-Polynomial Association Schemes

Main families (except of schemes of small diameter):

◮ Schemes of dual polar spaces

◮ Vertices: maximal isotropic subspaces of a vector space

equipped with a form (symplectic/quadratic/Hermitian).

◮ Distance ∂: codimension of their intersection.

◮ Schemes of sesquilinear/quadratic forms

◮ Vertices: a collection of matrices over a finite field. ◮ Distance ∂: rank of their difference.

◮ Johnson scheme (and its derivatives) ◮ Hamming scheme (and its derivatives) ◮ Grassmann scheme ◮ This list had not been changed, until Van Dam and Koolen

(2005) discovered a new family, the so-called twisted Grassmann scheme. Every known finite simple group (except of Lie type groups of small rank and some sporadic groups) is a group of automorphisms of a (P and Q)-polynomial AS from the above.

(known) (P and Q)-Polynomial Association Schemes

Main families (except of schemes of small diameter):

◮ Schemes of dual polar spaces

◮ Vertices: maximal isotropic subspaces of a vector space

equipped with a form (symplectic/quadratic/Hermitian).

◮ Distance ∂: codimension of their intersection.

◮ Schemes of sesquilinear/quadratic forms

◮ Vertices: a collection of matrices over a finite field. ◮ Distance ∂: rank of their difference.

◮ Johnson scheme (and its derivatives) ◮ Hamming scheme (and its derivatives) ◮ Grassmann scheme ◮ This list had not been changed, until Van Dam and Koolen

(2005) discovered a new family, the so-called twisted Grassmann scheme. Every known finite simple group (except of Lie type groups of small rank and some sporadic groups) is a group of automorphisms of a (P and Q)-polynomial AS from the above.

(known) (P and Q)-Polynomial Association Schemes

Main families (except of schemes of small diameter):

◮ Schemes of dual polar spaces

◮ Vertices: maximal isotropic subspaces of a vector space

equipped with a form (symplectic/quadratic/Hermitian).

◮ Distance ∂: codimension of their intersection.

◮ Schemes of sesquilinear/quadratic forms

◮ Vertices: a collection of matrices over a finite field. ◮ Distance ∂: rank of their difference.

◮ Johnson scheme (and its derivatives) ◮ Hamming scheme (and its derivatives) ◮ Grassmann scheme ◮ This list had not been changed, until Van Dam and Koolen

(2005) discovered a new family, the so-called twisted Grassmann scheme. Every known finite simple group (except of Lie type groups of small rank and some sporadic groups) is a group of automorphisms of a (P and Q)-polynomial AS from the above.

Classification Conjecture

(2) All (P and Q)-polynomial association schemes are known. The solution requires two steps:

(A) To find all feasible parameters of (P and Q)-polynomial AS. (P and Q)-polynomiality of an AS is determined by its intersection numbers {pk

ij}. We want to describe all feasible

sets {pk

ij} corresponding to (P and Q)-polynomial AS.

(In progress. Most of the work done by D. Leonard, P. Terwilliger, T. Ito and their school.) (B) To characterise (P and Q)-polynomial AS by parameters. There may exist two or more schemes with the same

• parameters. We want to find all of them up to isomorphism.

(Not much progress since 1999.)

Classification Conjecture

(2) All (P and Q)-polynomial association schemes are known. The solution requires two steps:

(A) To find all feasible parameters of (P and Q)-polynomial AS. (P and Q)-polynomiality of an AS is determined by its intersection numbers {pk

ij}. We want to describe all feasible

sets {pk

ij} corresponding to (P and Q)-polynomial AS.

(In progress. Most of the work done by D. Leonard, P. Terwilliger, T. Ito and their school.) (B) To characterise (P and Q)-polynomial AS by parameters. There may exist two or more schemes with the same

• parameters. We want to find all of them up to isomorphism.

(Not much progress since 1999.)

Classification Conjecture

(2) All (P and Q)-polynomial association schemes are known. The solution requires two steps:

(A) To find all feasible parameters of (P and Q)-polynomial AS. (P and Q)-polynomiality of an AS is determined by its intersection numbers {pk

ij}. We want to describe all feasible

sets {pk

ij} corresponding to (P and Q)-polynomial AS.

(In progress. Most of the work done by D. Leonard, P. Terwilliger, T. Ito and their school.) (B) To characterise (P and Q)-polynomial AS by parameters. There may exist two or more schemes with the same

• parameters. We want to find all of them up to isomorphism.

(Not much progress since 1999.)

The Grassmann graph Jq(n, d)

◮ Let q ≥ 2 be a prime power, n ≥ d ≥ 1 be integers. ◮ Jq(n, d) has as vertices all d-dim. subspaces U ≤ Fn q . ◮ ∂(U1, U2) = d − dim(U1 ∩ U2). ◮ All intersection numbers pk ij are expressed in q, n, d. ◮ Jq(n, d) ∼

= Jq(n, n − d), diameter equals min(d, n − d).

Theorem (K. Metsch, 1995)

The Grassmann graph Jq(n, d), d > 2, is characterized by its intersection array with the following possible exceptions:

◮ n = 2d, n = 2d ± 1, ◮ n = 2d ± 2 if q ∈ {2, 3}, ◮ n = 2d ± 3 if q = 2.

A real exception, the twisted Grassmann graph with the same parameters as Jq(2d ± 1, d) for all q, was constructed by E.R. van Dam and J.H. Koolen, Invent. Math. (2005).

The Grassmann graph Jq(n, d)

◮ Let q ≥ 2 be a prime power, n ≥ d ≥ 1 be integers. ◮ Jq(n, d) has as vertices all d-dim. subspaces U ≤ Fn q . ◮ ∂(U1, U2) = d − dim(U1 ∩ U2). ◮ All intersection numbers pk ij are expressed in q, n, d. ◮ Jq(n, d) ∼

= Jq(n, n − d), diameter equals min(d, n − d).

Theorem (K. Metsch, 1995)

The Grassmann graph Jq(n, d), d > 2, is characterized by its intersection array with the following possible exceptions:

◮ n = 2d, n = 2d ± 1, ◮ n = 2d ± 2 if q ∈ {2, 3}, ◮ n = 2d ± 3 if q = 2.

A real exception, the twisted Grassmann graph with the same parameters as Jq(2d ± 1, d) for all q, was constructed by E.R. van Dam and J.H. Koolen, Invent. Math. (2005).

The Grassmann graph Jq(n, d)

◮ Let q ≥ 2 be a prime power, n ≥ d ≥ 1 be integers. ◮ Jq(n, d) has as vertices all d-dim. subspaces U ≤ Fn q . ◮ ∂(U1, U2) = d − dim(U1 ∩ U2). ◮ All intersection numbers pk ij are expressed in q, n, d. ◮ Jq(n, d) ∼

= Jq(n, n − d), diameter equals min(d, n − d).

Theorem (K. Metsch, 1995)

The Grassmann graph Jq(n, d), d > 2, is characterized by its intersection array with the following possible exceptions:

◮ n = 2d, n = 2d ± 1, ◮ n = 2d ± 2 if q ∈ {2, 3}, ◮ n = 2d ± 3 if q = 2.

A real exception, the twisted Grassmann graph with the same parameters as Jq(2d ± 1, d) for all q, was constructed by E.R. van Dam and J.H. Koolen, Invent. Math. (2005).

The bilinear forms graphs Bilq(n × m)

◮ Let q ≥ 2 be a prime power, n ≥ m ≥ 1 be integers. ◮ Bilq(n × m) has as vertices all n × m-matrices over Fq. ◮ ∂(A, B) = rank(A − B). ◮ All intersection numbers pk ij are expressed in q, n, m. ◮ Bilq(n × m) ∼

= Bilq(m × n), diameter equals min(n, m).

Theorem (K. Metsch, 1999)

The bilinear forms graph Bilq(n × m), n ≥ m ≥ 3, is characterized by its intersection array with the following possible exceptions:

◮ q = 2 and m ∈ {n, n + 1, n + 2, n + 3}, ◮ q ≥ 3 and m ∈ {n, n + 1, n + 2}.

No actual exceptions are known.

The bilinear forms graphs Bilq(n × m)

◮ Let q ≥ 2 be a prime power, n ≥ m ≥ 1 be integers. ◮ Bilq(n × m) has as vertices all n × m-matrices over Fq. ◮ ∂(A, B) = rank(A − B). ◮ All intersection numbers pk ij are expressed in q, n, m. ◮ Bilq(n × m) ∼

= Bilq(m × n), diameter equals min(n, m).

Theorem (K. Metsch, 1999)

The bilinear forms graph Bilq(n × m), n ≥ m ≥ 3, is characterized by its intersection array with the following possible exceptions:

◮ q = 2 and m ∈ {n, n + 1, n + 2, n + 3}, ◮ q ≥ 3 and m ∈ {n, n + 1, n + 2}.

No actual exceptions are known.

q-analogues of Hamming and Johnson schemes

The Hamming and the Johnson schemes play an important roles in the classical coding and design theories, serving as underlying spaces for codes and designs. The Grassmann scheme and the bilinear forms scheme play the same roles for a new branch of coding theory: Subspace (or Network) Coding.

◮ a finite set → a vector space over Fq, ◮ a codeword over an alphabet → a subspace.

q-analogues of Hamming and Johnson schemes

The Hamming and the Johnson schemes play an important roles in the classical coding and design theories, serving as underlying spaces for codes and designs. The Grassmann scheme and the bilinear forms scheme play the same roles for a new branch of coding theory: Subspace (or Network) Coding.

◮ a finite set → a vector space over Fq, ◮ a codeword over an alphabet → a subspace.

q-analogues of Hamming and Johnson schemes

The Hamming and the Johnson schemes play an important roles in the classical coding and design theories, serving as underlying spaces for codes and designs. The Grassmann scheme and the bilinear forms scheme play the same roles for a new branch of coding theory: Subspace (or Network) Coding.

◮ a finite set → a vector space over Fq, ◮ a codeword over an alphabet → a subspace.

Our results

We showed that the following possible exceptions from the Metsch results cannot be realized: The Grassmann graph Jq(n, d): q Possible exceptions by Metsch Ruled out (G., Koolen) ∀q n = 2d, n = 2d ± 1 q = 2, n = 2d 2, 3 n = 2d ± 2 q = 2, d is odd and ≫ 0 2 n = 2d ± 3 d ≡ 1(mod 3)

(in preparation)

The bilinear forms graphs Bilq(n × m): q Possible exceptions by Metsch Ruled out (G., Koolen) 2 m ∈ {n, n + 1, n + 2, n + 3} n = m ≥ 3 m ∈ {n, n + 1, n + 2}

(Combinatorica, to appear)

Our results

We showed that the following possible exceptions from the Metsch results cannot be realized: The Grassmann graph Jq(n, d): q Possible exceptions by Metsch Ruled out (G., Koolen) ∀q n = 2d, n = 2d ± 1 q = 2, n = 2d 2, 3 n = 2d ± 2 q = 2, d is odd and ≫ 0 2 n = 2d ± 3 d ≡ 1(mod 3)

(in preparation)

The bilinear forms graphs Bilq(n × m): q Possible exceptions by Metsch Ruled out (G., Koolen) 2 m ∈ {n, n + 1, n + 2, n + 3} n = m ≥ 3 m ∈ {n, n + 1, n + 2}

(Combinatorica, to appear)

Our results

We showed that the following possible exceptions from the Metsch results cannot be realized: The Grassmann graph Jq(n, d): q Possible exceptions by Metsch Ruled out (G., Koolen) ∀q n = 2d, n = 2d ± 1 q = 2, n = 2d 2, 3 n = 2d ± 2 q = 2, d is odd and ≫ 0 2 n = 2d ± 3 d ≡ 1(mod 3)

(in preparation)

The bilinear forms graphs Bilq(n × m): q Possible exceptions by Metsch Ruled out (G., Koolen) 2 m ∈ {n, n + 1, n + 2, n + 3} n = m ≥ 3 m ∈ {n, n + 1, n + 2}

(Combinatorica, to appear)

Metsch’s approach

Partial linear space is a set P of points and a set L of lines (subsets of P):

◮ any line contains at least two points, ◮ any two points are on at most one line.

Both Jq(n, d) and Bilq(n × m) naturally give rise to partial linear spaces (well studied in finite geometry):

◮ points = vertices, ◮ lines = maximal cliques.

Let X denote Jq(n, d) or Bilq(n × m). A graph Γ with the same parameters as X ↓ (from parameters) A partial linear space on cliques →? Known partial linear space ↓ Γ ∼ = X.

Metsch’s approach

Partial linear space is a set P of points and a set L of lines (subsets of P):

◮ any line contains at least two points, ◮ any two points are on at most one line.

Both Jq(n, d) and Bilq(n × m) naturally give rise to partial linear spaces (well studied in finite geometry):

◮ points = vertices, ◮ lines = maximal cliques.

Let X denote Jq(n, d) or Bilq(n × m). A graph Γ with the same parameters as X ↓ (from parameters) A partial linear space on cliques →? Known partial linear space ↓ Γ ∼ = X.

Metsch’s approach

Partial linear space is a set P of points and a set L of lines (subsets of P):

◮ any line contains at least two points, ◮ any two points are on at most one line.

Both Jq(n, d) and Bilq(n × m) naturally give rise to partial linear spaces (well studied in finite geometry):

◮ points = vertices, ◮ lines = maximal cliques.

Let X denote Jq(n, d) or Bilq(n × m). A graph Γ with the same parameters as X ↓ (from parameters) A partial linear space on cliques →? Known partial linear space ↓ Γ ∼ = X.

Metsch’s approach

Partial linear space is a set P of points and a set L of lines (subsets of P):

◮ any line contains at least two points, ◮ any two points are on at most one line.

Both Jq(n, d) and Bilq(n × m) naturally give rise to partial linear spaces (well studied in finite geometry):

◮ points = vertices, ◮ lines = maximal cliques.

Let X denote Jq(n, d) or Bilq(n × m). A graph Γ with the same parameters as X ↓ (from parameters) A partial linear space on cliques →? Known partial linear space ↓ Γ ∼ = X.

Metsch’s approach

Partial linear space is a set P of points and a set L of lines (subsets of P):

◮ any line contains at least two points, ◮ any two points are on at most one line.

Both Jq(n, d) and Bilq(n × m) naturally give rise to partial linear spaces (well studied in finite geometry):

◮ points = vertices, ◮ lines = maximal cliques.

Let X denote Jq(n, d) or Bilq(n × m). A graph Γ with the same parameters as X ↓ (from parameters) A partial linear space on cliques →? Known partial linear space ↓ Γ ∼ = X.

Metsch’s approach

Partial linear space is a set P of points and a set L of lines (subsets of P):

◮ any line contains at least two points, ◮ any two points are on at most one line.

Both Jq(n, d) and Bilq(n × m) naturally give rise to partial linear spaces (well studied in finite geometry):

◮ points = vertices, ◮ lines = maximal cliques.

Let X denote Jq(n, d) or Bilq(n × m). A graph Γ with the same parameters as X ↓ (from parameters) A partial linear space on cliques →? Known partial linear space ↓ Γ ∼ = X.

Metsch’s approach

Metsch developed a purely combinatorial method, which allows to show the existence of ’large’ cliques, if certain inequalities on pk

ij hold, and to construct a partial linear space.

However, in case of possible exceptions, these inequalities do not hold, so that the first crucial step fails: A graph Γ with the same parameters as X ↓ (from parameters) A partial linear space on cliques →? Known partial linear space ↓ Γ ∼ = X. And this is not an arithmetic problem: a partial linear space cannot be recovered from the twisted Grassmann graph.

Metsch’s approach

Metsch developed a purely combinatorial method, which allows to show the existence of ’large’ cliques, if certain inequalities on pk

ij hold, and to construct a partial linear space.

However, in case of possible exceptions, these inequalities do not hold, so that the first crucial step fails: A graph Γ with the same parameters as X ↓ (from parameters) A partial linear space on cliques →? Known partial linear space ↓ Γ ∼ = X. And this is not an arithmetic problem: a partial linear space cannot be recovered from the twisted Grassmann graph.

Metsch’s approach

Metsch developed a purely combinatorial method, which allows to show the existence of ’large’ cliques, if certain inequalities on pk

ij hold, and to construct a partial linear space.

However, in case of possible exceptions, these inequalities do not hold, so that the first crucial step fails: A graph Γ with the same parameters as X ↓ (from parameters) A partial linear space on cliques →? Known partial linear space ↓ Γ ∼ = X. And this is not an arithmetic problem: a partial linear space cannot be recovered from the twisted Grassmann graph.

Our approach

◮ Q-polynomiality → the local structure

The local graph at vertex x: Γ1(x) := {y ∈ V (Γ) : ∂(x, y) = 1}. The µ-graph of vertices x, y with ∂(x, y) = 2: Γ1(x, y) := {z ∈ V (Γ) : ∂(x, z) = 1 & ∂(z, y) = 1}. Using Q-polynomiality, we can restrict:

◮ possible eigenvalues of the local graphs, ◮ possible structure of the µ-graphs.

◮ Reconstructing the local structure

The problem of spectral characterization, i.e., reconstructing a graph from the spectrum of its adjacency matrix.

◮ Reconstructing a graph by its local structure

If we know Γ1(x), Γ1(x, y) for all x, y →? Γ

Our approach

◮ Q-polynomiality → the local structure

The local graph at vertex x: Γ1(x) := {y ∈ V (Γ) : ∂(x, y) = 1}. The µ-graph of vertices x, y with ∂(x, y) = 2: Γ1(x, y) := {z ∈ V (Γ) : ∂(x, z) = 1 & ∂(z, y) = 1}. Using Q-polynomiality, we can restrict:

◮ possible eigenvalues of the local graphs, ◮ possible structure of the µ-graphs.

◮ Reconstructing the local structure

The problem of spectral characterization, i.e., reconstructing a graph from the spectrum of its adjacency matrix.

◮ Reconstructing a graph by its local structure

If we know Γ1(x), Γ1(x, y) for all x, y →? Γ

Our approach

◮ Q-polynomiality → the local structure

The local graph at vertex x: Γ1(x) := {y ∈ V (Γ) : ∂(x, y) = 1}. The µ-graph of vertices x, y with ∂(x, y) = 2: Γ1(x, y) := {z ∈ V (Γ) : ∂(x, z) = 1 & ∂(z, y) = 1}. Using Q-polynomiality, we can restrict:

◮ possible eigenvalues of the local graphs, ◮ possible structure of the µ-graphs.

◮ Reconstructing the local structure

The problem of spectral characterization, i.e., reconstructing a graph from the spectrum of its adjacency matrix.

◮ Reconstructing a graph by its local structure

If we know Γ1(x), Γ1(x, y) for all x, y →? Γ

Our approach

◮ Q-polynomiality → the local structure

The local graph at vertex x: Γ1(x) := {y ∈ V (Γ) : ∂(x, y) = 1}. The µ-graph of vertices x, y with ∂(x, y) = 2: Γ1(x, y) := {z ∈ V (Γ) : ∂(x, z) = 1 & ∂(z, y) = 1}. Using Q-polynomiality, we can restrict:

◮ possible eigenvalues of the local graphs, ◮ possible structure of the µ-graphs.

◮ Reconstructing the local structure

The problem of spectral characterization, i.e., reconstructing a graph from the spectrum of its adjacency matrix.

◮ Reconstructing a graph by its local structure

If we know Γ1(x), Γ1(x, y) for all x, y →? Γ

Our approach

◮ Q-polynomiality → the local structure

The local graph at vertex x: Γ1(x) := {y ∈ V (Γ) : ∂(x, y) = 1}. The µ-graph of vertices x, y with ∂(x, y) = 2: Γ1(x, y) := {z ∈ V (Γ) : ∂(x, z) = 1 & ∂(z, y) = 1}. Using Q-polynomiality, we can restrict:

◮ possible eigenvalues of the local graphs, ◮ possible structure of the µ-graphs.

◮ Reconstructing the local structure

The problem of spectral characterization, i.e., reconstructing a graph from the spectrum of its adjacency matrix.

◮ Reconstructing a graph by its local structure

If we know Γ1(x), Γ1(x, y) for all x, y →? Γ

Our approach

◮ Q-polynomiality → the local structure

The local graph at vertex x: Γ1(x) := {y ∈ V (Γ) : ∂(x, y) = 1}. The µ-graph of vertices x, y with ∂(x, y) = 2: Γ1(x, y) := {z ∈ V (Γ) : ∂(x, z) = 1 & ∂(z, y) = 1}. Using Q-polynomiality, we can restrict:

◮ possible eigenvalues of the local graphs, ◮ possible structure of the µ-graphs.

◮ Reconstructing the local structure

The problem of spectral characterization, i.e., reconstructing a graph from the spectrum of its adjacency matrix.

◮ Reconstructing a graph by its local structure

If we know Γ1(x), Γ1(x, y) for all x, y →? Γ

The Bose-Mesner algebra and intersection numbers

The equation AiAj = AjAi =

D

• k=0

pk

ijAk

shows that {A0, A1, . . . , AD} generate a matrix algebra over R (of dimension D + 1), called the Bose-Mesner algebra of Γ. The numbers pk

ij have simple combinatorial interpretation:

∂(x, y) = k ⇒ pk

ij := |{z ∈ V : ∂(x, z) = i, ∂(y, z) = j}|.

Triple intersection numbers

Let Γ be a Q-polynomial DRG with diameter D ≥ 3. Fix a triple of vertices x, y, z such that ∂(x, y) = ∂(x, z) = 1. Denote a triple intersection number [ℓ, m, n]: [ℓ, m, n]x,y,z = |{w : ∂(x, w) = ℓ, ∂(y, w) = m, ∂(x, w) = n}|. For example, Terwilliger (1995) showed that for i ≥ 2 [i, i ± 1, i ± 1] = κi,δ[1, 1, 1] + τi where δ = d(y, z) ∈ {1, 2}, and κi,δ and τi are real scalars that do not depend on the choice of x, y, z. x y z

Triple intersection numbers

Let Γ be a Q-polynomial DRG with diameter D ≥ 3. Fix a triple of vertices x, y, z such that ∂(x, y) = ∂(x, z) = 1. Denote a triple intersection number [ℓ, m, n]: [ℓ, m, n]x,y,z = |{w : ∂(x, w) = ℓ, ∂(y, w) = m, ∂(x, w) = n}|. For example, Terwilliger (1995) showed that for i ≥ 2 [i, i ± 1, i ± 1] = κi,δ[1, 1, 1] + τi where δ = d(y, z) ∈ {1, 2}, and κi,δ and τi are real scalars that do not depend on the choice of x, y, z. x y z

Triple intersection numbers

Let Γ be a Q-polynomial DRG with diameter D ≥ 3. Fix a triple of vertices x, y, z such that ∂(x, y) = ∂(x, z) = 1. Denote a triple intersection number [ℓ, m, n]: [ℓ, m, n]x,y,z = |{w : ∂(x, w) = ℓ, ∂(y, w) = m, ∂(x, w) = n}|. For example, Terwilliger (1995) showed that for i ≥ 2 [i, i ± 1, i ± 1] = κi,δ[1, 1, 1] + τi where δ = d(y, z) ∈ {1, 2}, and κi,δ and τi are real scalars that do not depend on the choice of x, y, z. x y z

Triple intersection numbers → Terwilliger algebra

Fix any vertex x ∈ V (Γ). For 0 ≤ i ≤ D, denote by E∗

i := E∗ i (x) a diagonal matrix with rows and columns indexed

by V (Γ), and defined by (E∗

i )y,y :=

• 1 if ∂(x, y) = i,

0 if ∂(x, y) = i. The dual Bose-Mesner algebra (w.r.t. x) M∗ := M∗(x) = span{E∗

0, E∗ 1, . . . , E∗ D}.

The Terwilliger (or subconstituent) algebra (w.r.t. x) T := T (x) = M, M∗, where M is the Bose-Mesner algebra of Γ. The Terwilliger algebra is semi-simple, so that RV decomposes into an orthogonal direct sum of irreducible T -modules, the structure of which is closely related to the local structure of Γ.

Triple intersection numbers → Terwilliger algebra

Fix any vertex x ∈ V (Γ). For 0 ≤ i ≤ D, denote by E∗

i := E∗ i (x) a diagonal matrix with rows and columns indexed

by V (Γ), and defined by (E∗

i )y,y :=

• 1 if ∂(x, y) = i,

0 if ∂(x, y) = i. The dual Bose-Mesner algebra (w.r.t. x) M∗ := M∗(x) = span{E∗

0, E∗ 1, . . . , E∗ D}.

The Terwilliger (or subconstituent) algebra (w.r.t. x) T := T (x) = M, M∗, where M is the Bose-Mesner algebra of Γ. The Terwilliger algebra is semi-simple, so that RV decomposes into an orthogonal direct sum of irreducible T -modules, the structure of which is closely related to the local structure of Γ.

Triple intersection numbers → Terwilliger algebra

Fix any vertex x ∈ V (Γ). For 0 ≤ i ≤ D, denote by E∗

i := E∗ i (x) a diagonal matrix with rows and columns indexed

by V (Γ), and defined by (E∗

i )y,y :=

• 1 if ∂(x, y) = i,

0 if ∂(x, y) = i. The dual Bose-Mesner algebra (w.r.t. x) M∗ := M∗(x) = span{E∗

0, E∗ 1, . . . , E∗ D}.

The Terwilliger (or subconstituent) algebra (w.r.t. x) T := T (x) = M, M∗, where M is the Bose-Mesner algebra of Γ. The Terwilliger algebra is semi-simple, so that RV decomposes into an orthogonal direct sum of irreducible T -modules, the structure of which is closely related to the local structure of Γ.

Triple intersection numbers → Terwilliger algebra

Fix any vertex x ∈ V (Γ). For 0 ≤ i ≤ D, denote by E∗

i := E∗ i (x) a diagonal matrix with rows and columns indexed

by V (Γ), and defined by (E∗

i )y,y :=

• 1 if ∂(x, y) = i,

0 if ∂(x, y) = i. The dual Bose-Mesner algebra (w.r.t. x) M∗ := M∗(x) = span{E∗

0, E∗ 1, . . . , E∗ D}.

The Terwilliger (or subconstituent) algebra (w.r.t. x) T := T (x) = M, M∗, where M is the Bose-Mesner algebra of Γ. The Terwilliger algebra is semi-simple, so that RV decomposes into an orthogonal direct sum of irreducible T -modules, the structure of which is closely related to the local structure of Γ.

Terwilliger algebra

Denote A := E∗

1A1E∗ 1 and

J := E∗

1JE∗

• 1. One can see
• A =

N

• .

where N — the adjacency matrix of Γ1(x). Note [ℓ, m, n]x,y,z = (E∗

1AmE∗ ℓ AnE∗ 1)y,z and [1, 1, 1] = (

A)2

y,z

Then [i, i ± 1, i ± 1] = κi,δ[1, 1, 1] + τi, imply that E∗

1Ai−1E∗ i Ai−1E∗ 1 and E∗ 1AiE∗ i−1AiE∗ 1

are the polynomials (of degree 2) in A and J := E∗

1JE∗ 1.

Terwilliger algebra

Denote A := E∗

1A1E∗ 1 and

J := E∗

1JE∗

• 1. One can see
• A =

N

• .

where N — the adjacency matrix of Γ1(x). Note [ℓ, m, n]x,y,z = (E∗

1AmE∗ ℓ AnE∗ 1)y,z and [1, 1, 1] = (

A)2

y,z

Then [i, i ± 1, i ± 1] = κi,δ[1, 1, 1] + τi, imply that E∗

1Ai−1E∗ i Ai−1E∗ 1 and E∗ 1AiE∗ i−1AiE∗ 1

are the polynomials (of degree 2) in A and J := E∗

1JE∗ 1.

Terwilliger algebra

Denote A := E∗

1A1E∗ 1 and

J := E∗

1JE∗

• 1. One can see
• A =

N

• .

where N — the adjacency matrix of Γ1(x). Note [ℓ, m, n]x,y,z = (E∗

1AmE∗ ℓ AnE∗ 1)y,z and [1, 1, 1] = (

A)2

y,z

Then [i, i ± 1, i ± 1] = κi,δ[1, 1, 1] + τi, imply that E∗

1Ai−1E∗ i Ai−1E∗ 1 and E∗ 1AiE∗ i−1AiE∗ 1

are the polynomials (of degree 2) in A and J := E∗

1JE∗ 1.

The Terwilliger polynomial of a Q-DRG

E∗

1Ai−1E∗ i Ai−1E∗ 1 and E∗ 1AiE∗ i−1AiE∗ 1

are the polynomials (of degree 2) in A and J := E∗

1JE∗ 1. ◮ Terwilliger (early 1990’s): There exists a polynomial pT of

degree 4 such that, for any vertex x ∈ Γ, and any non-principal eigenvalue η of Γ1(x) we have pT (η) ≥ 0.

◮ pT only depends on the intersection numbers of Γ and the

Q-polynomial ordering of primitive idempotents of its Bose-Mesner algebra.

◮ We call pT the Terwilliger polynomial.

References:

• P. Terwilliger, Lecture Note on Terwilliger algebra (edited by
• H. Suzuki), 1993.
• A.L.G., J.H. Koolen, The Terwilliger polynomial of a

Q-polynomial distance-regular graph and its application to pseudo-partition graphs // Linear Algebra and Its Appl. (2015).

The Terwilliger polynomial of a Q-DRG

E∗

1Ai−1E∗ i Ai−1E∗ 1 and E∗ 1AiE∗ i−1AiE∗ 1

are the polynomials (of degree 2) in A and J := E∗

1JE∗ 1. ◮ Terwilliger (early 1990’s): There exists a polynomial pT of

degree 4 such that, for any vertex x ∈ Γ, and any non-principal eigenvalue η of Γ1(x) we have pT (η) ≥ 0.

◮ pT only depends on the intersection numbers of Γ and the

Q-polynomial ordering of primitive idempotents of its Bose-Mesner algebra.

◮ We call pT the Terwilliger polynomial.

References:

• P. Terwilliger, Lecture Note on Terwilliger algebra (edited by
• H. Suzuki), 1993.
• A.L.G., J.H. Koolen, The Terwilliger polynomial of a

Q-polynomial distance-regular graph and its application to pseudo-partition graphs // Linear Algebra and Its Appl. (2015).

The Terwilliger polynomial of a Q-DRG

E∗

1Ai−1E∗ i Ai−1E∗ 1 and E∗ 1AiE∗ i−1AiE∗ 1

are the polynomials (of degree 2) in A and J := E∗

1JE∗ 1. ◮ Terwilliger (early 1990’s): There exists a polynomial pT of

degree 4 such that, for any vertex x ∈ Γ, and any non-principal eigenvalue η of Γ1(x) we have pT (η) ≥ 0.

◮ pT only depends on the intersection numbers of Γ and the

Q-polynomial ordering of primitive idempotents of its Bose-Mesner algebra.

◮ We call pT the Terwilliger polynomial.

References:

• P. Terwilliger, Lecture Note on Terwilliger algebra (edited by
• H. Suzuki), 1993.
• A.L.G., J.H. Koolen, The Terwilliger polynomial of a

Q-polynomial distance-regular graph and its application to pseudo-partition graphs // Linear Algebra and Its Appl. (2015).

The Terwilliger polynomial of a Q-DRG

E∗

1Ai−1E∗ i Ai−1E∗ 1 and E∗ 1AiE∗ i−1AiE∗ 1

are the polynomials (of degree 2) in A and J := E∗

1JE∗ 1. ◮ Terwilliger (early 1990’s): There exists a polynomial pT of

degree 4 such that, for any vertex x ∈ Γ, and any non-principal eigenvalue η of Γ1(x) we have pT (η) ≥ 0.

◮ pT only depends on the intersection numbers of Γ and the

Q-polynomial ordering of primitive idempotents of its Bose-Mesner algebra.

◮ We call pT the Terwilliger polynomial.

References:

• P. Terwilliger, Lecture Note on Terwilliger algebra (edited by
• H. Suzuki), 1993.
• A.L.G., J.H. Koolen, The Terwilliger polynomial of a

Q-polynomial distance-regular graph and its application to pseudo-partition graphs // Linear Algebra and Its Appl. (2015).

Terwilliger algebra theory: Summary, 1

For a Q-DRG Γ and a base vertex x ∈ Γ:

◮ Triple intersection numbers:

[i, i − 1, i − 1] = κi,δ[1, 1, 1] + τi x y z [i, i + 1, i + 1] = σi,δ[1, 1, 1] + ρi x y z

Terwilliger algebra theory: Summary, 2

◮ The Terwilliger polynomial pT (of degree 4) such that

pT (η) ≥ 0 for any non-principal eigenvalue η of Γ1(x). This restricts possible eigenvalues of Γ1(x).

Our approach

Suppose that Γ is a DRG with the same intersection array as Bil2(n × n).

◮ Q-polynomiality → the local structure

The local graph at vertex x: Γ1(x) := {y ∈ V (Γ) : ∂(x, y) = 1}. The µ-graph of vertices x, y with ∂(x, y) = 2: Γ1(x, y) := {z ∈ V (Γ) : ∂(x, z) = 1 & ∂(z, y) = 1}. Using Q-polynomiality, we can restrict:

◮ possible eigenvalues of the local graphs, ◮ possible structure of the µ-graphs.

◮ Reconstructing the local structure

The problem of spectral characterization, i.e., reconstructing a graph from the spectrum of its adjacency matrix.

◮ Reconstructing a graph by its local structure

If we know Γ1(x), Γ1(x, y) for all x, y →? Γ

Bilq(n × n)

Suppose that Γ is a DRG with the same intersection array as Bilq(n × n).

◮ The Terwilliger polynomial pT has the three distinct roots:

−q − 1, −1, and qn − q − 1 (of multiplicity 2), while the leading coefficient of pT is negative.

◮ Hence a local non-principal eigenvalue η at any vertex

x ∈ Γ satisfies: η ∈ [−q − 1, −1] or η = qn − q − 1.

Bilq(n × n)

Suppose that Γ is a DRG with the same intersection array as Bilq(n × n).

◮ The Terwilliger polynomial pT has the three distinct roots:

−q − 1, −1, and qn − q − 1 (of multiplicity 2), while the leading coefficient of pT is negative.

◮ Hence a local non-principal eigenvalue η at any vertex

x ∈ Γ satisfies: η ∈ [−q − 1, −1] or η = qn − q − 1.

Bilq(n × n)

Suppose that Γ is a DRG with the same intersection array as Bilq(n × n).

◮ The Terwilliger polynomial pT has the three distinct roots:

−q − 1, −1, and qn − q − 1 (of multiplicity 2), while the leading coefficient of pT is negative.

◮ Hence a local non-principal eigenvalue η at any vertex

x ∈ Γ satisfies: η ∈ [−q − 1, −1] or η = qn − q − 1.

Bil2(n × n)

For q = 2, a local non-principal eigenvalue η at any vertex x ∈ Γ satisfies: η ∈ [−3, −1] or η = 2n − 3.

Claim

Γ1(x) has integral eigenvalues, i.e., −3, −2, −1, or 2n − 3. Proof:

• 1. The adjacency matrix is (0,1)-matrix, hence the eigenvalues
• f a graph are algebraic integers, and their product is integral.
• 2. Let η1, . . . , ηs be all irrational eigenvalues of Γ1(x).

Then ηi ∈ (−3, −1) and Πs

i=1ηi is an integer, so

Πs

i=1(ηi + 2)

is an integer. ηi ∈ (−3, −1) ⇒ |ηi + 2| < 1 ⇒ Πs

i=1(ηi + 2) = 0.

Bil2(n × n)

For q = 2, a local non-principal eigenvalue η at any vertex x ∈ Γ satisfies: η ∈ [−3, −1] or η = 2n − 3.

Claim

Γ1(x) has integral eigenvalues, i.e., −3, −2, −1, or 2n − 3. Proof:

• 1. The adjacency matrix is (0,1)-matrix, hence the eigenvalues
• f a graph are algebraic integers, and their product is integral.
• 2. Let η1, . . . , ηs be all irrational eigenvalues of Γ1(x).

Then ηi ∈ (−3, −1) and Πs

i=1ηi is an integer, so

Πs

i=1(ηi + 2)

is an integer. ηi ∈ (−3, −1) ⇒ |ηi + 2| < 1 ⇒ Πs

i=1(ηi + 2) = 0.

Bil2(n × n)

For q = 2, a local non-principal eigenvalue η at any vertex x ∈ Γ satisfies: η ∈ [−3, −1] or η = 2n − 3.

Claim

Γ1(x) has integral eigenvalues, i.e., −3, −2, −1, or 2n − 3. Proof:

• 1. The adjacency matrix is (0,1)-matrix, hence the eigenvalues
• f a graph are algebraic integers, and their product is integral.
• 2. Let η1, . . . , ηs be all irrational eigenvalues of Γ1(x).

Then ηi ∈ (−3, −1) and Πs

i=1ηi is an integer, so

Πs

i=1(ηi + 2)

is an integer. ηi ∈ (−3, −1) ⇒ |ηi + 2| < 1 ⇒ Πs

i=1(ηi + 2) = 0.

Bil2(n × n)

For q = 2, a local non-principal eigenvalue η at any vertex x ∈ Γ satisfies: η ∈ [−3, −1] or η = 2n − 3.

Claim

Γ1(x) has integral eigenvalues, i.e., −3, −2, −1, or 2n − 3. Proof:

• 1. The adjacency matrix is (0,1)-matrix, hence the eigenvalues
• f a graph are algebraic integers, and their product is integral.
• 2. Let η1, . . . , ηs be all irrational eigenvalues of Γ1(x).

Then ηi ∈ (−3, −1) and Πs

i=1ηi is an integer, so

Πs

i=1(ηi + 2)

is an integer. ηi ∈ (−3, −1) ⇒ |ηi + 2| < 1 ⇒ Πs

i=1(ηi + 2) = 0.

Bil2(n × n)

For q = 2, a local non-principal eigenvalue η at any vertex x ∈ Γ satisfies: η ∈ [−3, −1] or η = 2n − 3.

Claim

Γ1(x) has integral eigenvalues, i.e., −3, −2, −1, or 2n − 3. Proof:

• 1. The adjacency matrix is (0,1)-matrix, hence the eigenvalues
• f a graph are algebraic integers, and their product is integral.
• 2. Let η1, . . . , ηs be all irrational eigenvalues of Γ1(x).

Then ηi ∈ (−3, −1) and Πs

i=1ηi is an integer, so

Πs

i=1(ηi + 2)

is an integer. ηi ∈ (−3, −1) ⇒ |ηi + 2| < 1 ⇒ Πs

i=1(ηi + 2) = 0.

Bil2(n × n)

For q = 2, a local non-principal eigenvalue η at any vertex x ∈ Γ satisfies: η ∈ [−3, −1] or η = 2n − 3.

Claim

Γ1(x) has integral eigenvalues, i.e., −3, −2, −1, or 2n − 3. Proof:

• 1. The adjacency matrix is (0,1)-matrix, hence the eigenvalues
• f a graph are algebraic integers, and their product is integral.
• 2. Let η1, . . . , ηs be all irrational eigenvalues of Γ1(x).

Then ηi ∈ (−3, −1) and Πs

i=1ηi is an integer, so

Πs

i=1(ηi + 2)

is an integer. ηi ∈ (−3, −1) ⇒ |ηi + 2| < 1 ⇒ Πs

i=1(ηi + 2) = 0.

Bil2(n × n)

For q = 2, a local non-principal eigenvalue η at any vertex x ∈ Γ satisfies: η ∈ [−3, −1] or η = 2n − 3.

Claim

Γ1(x) has integral eigenvalues, i.e., −3, −2, −1, or 2n − 3. Proof:

• 1. The adjacency matrix is (0,1)-matrix, hence the eigenvalues
• f a graph are algebraic integers, and their product is integral.
• 2. Let η1, . . . , ηs be all irrational eigenvalues of Γ1(x).

Then ηi ∈ (−3, −1) and Πs

i=1ηi is an integer, so

Πs

i=1(ηi + 2)

is an integer. ηi ∈ (−3, −1) ⇒ |ηi + 2| < 1 ⇒ Πs

i=1(ηi + 2) = 0.

Bil2(n × n)

Recall a basic fact: Let θm0

0 , θm1 1 , . . . , θms s

be the spectrum of a regular (with valency k) graph on v vertices, and A be its adjacency matrix. Then:

s

• i=0

mi = v, tr(A) =

s

• i=0

miθi = 0, tr(A2) =

s

• i=0

miθ2

i = vk.

We may put θ0 = k and, moreover, m0 = 1 if the graph is connected.

Bil2(n × n)

Claim

Γ1(x) has spectrum 2(2n − 2)1, (2n − 3)2(2n−2), (−2)(2n−1)2. Proof: Apply the previous fact to our case: p0

11 = v = (2n − 1)2,

θ0 = k = p1

11 = 2(2n − 2),

θ1 = 2n − 3, θ2 = −1, θ3 = −2, θ4 = −3, m1 =?, m2 =?, m3 =?, m4 =?, and m0 = 1 (as Γ1(x) is connected). The system of (three) linear equations with respect to (four) unknowns m1, . . . , m4 has the only non-negative integral solution: m1 = 2(2n − 2), m2 = 0, m3 = (2n − 1)2, m4 = 0.

Bil2(n × n)

Claim

Γ1(x) has spectrum 2(2n − 2)1, (2n − 3)2(2n−2), (−2)(2n−1)2. Proof: Apply the previous fact to our case: p0

11 = v = (2n − 1)2,

θ0 = k = p1

11 = 2(2n − 2),

θ1 = 2n − 3, θ2 = −1, θ3 = −2, θ4 = −3, m1 =?, m2 =?, m3 =?, m4 =?, and m0 = 1 (as Γ1(x) is connected). The system of (three) linear equations with respect to (four) unknowns m1, . . . , m4 has the only non-negative integral solution: m1 = 2(2n − 2), m2 = 0, m3 = (2n − 1)2, m4 = 0.

Bil2(n × n)

Claim

Γ1(x) has spectrum 2(2n − 2)1, (2n − 3)2(2n−2), (−2)(2n−1)2. Proof: Apply the previous fact to our case: p0

11 = v = (2n − 1)2,

θ0 = k = p1

11 = 2(2n − 2),

θ1 = 2n − 3, θ2 = −1, θ3 = −2, θ4 = −3, m1 =?, m2 =?, m3 =?, m4 =?, and m0 = 1 (as Γ1(x) is connected). The system of (three) linear equations with respect to (four) unknowns m1, . . . , m4 has the only non-negative integral solution: m1 = 2(2n − 2), m2 = 0, m3 = (2n − 1)2, m4 = 0.

Our approach

Suppose that Γ is a DRG with the same intersection array as Bil2(n × n).

◮ Q-polynomiality → the local structure

The local graph at vertex x: Γ1(x) := {y ∈ V (Γ) : ∂(x, y) = 1}. The µ-graph of vertices x, y with ∂(x, y) = 2: Γ1(x, y) := {z ∈ V (Γ) : ∂(x, z) = 1 & ∂(z, y) = 1}. Using Q-polynomiality, we can restrict:

◮ possible eigenvalues of the local graphs, ◮ possible structure of the µ-graphs.

◮ Reconstructing the local structure

The problem of spectral characterization, i.e., reconstructing a graph from the spectrum of its adjacency matrix.

◮ Reconstructing a graph by its local structure

If we know Γ1(x), Γ1(x, y) for all x, y →? Γ

Bil2(n × n)

For every vertex x of Γ: Γ1(x) is a regular graph with exactly 3 distinct eigenvalues ↓ Γ1(x) is DRG with diameter 2 (strongly regular graph) ↓ Γ1(x) has the parameters of the Hamming graph H(2, 2n − 1) ↓ Γ1(x) is the Hamming graph H(2, 2n − 1) But we cannot recover a partial linear space!

Bil2(n × n)

For every vertex x of Γ: Γ1(x) is a regular graph with exactly 3 distinct eigenvalues ↓ Γ1(x) is DRG with diameter 2 (strongly regular graph) ↓ Γ1(x) has the parameters of the Hamming graph H(2, 2n − 1) ↓ Γ1(x) is the Hamming graph H(2, 2n − 1) But we cannot recover a partial linear space!

Bil2(n × n)

For every vertex x of Γ: Γ1(x) is a regular graph with exactly 3 distinct eigenvalues ↓ Γ1(x) is DRG with diameter 2 (strongly regular graph) ↓ Γ1(x) has the parameters of the Hamming graph H(2, 2n − 1) ↓ Γ1(x) is the Hamming graph H(2, 2n − 1) But we cannot recover a partial linear space!

Bil2(n × n)

For every vertex x of Γ: Γ1(x) is a regular graph with exactly 3 distinct eigenvalues ↓ Γ1(x) is DRG with diameter 2 (strongly regular graph) ↓ Γ1(x) has the parameters of the Hamming graph H(2, 2n − 1) ↓ Γ1(x) is the Hamming graph H(2, 2n − 1) But we cannot recover a partial linear space!

Our approach

Suppose that Γ is a DRG with the same intersection array as Bil2(n × n).

◮ Q-polynomiality → the local structure

The local graph at vertex x: Γ1(x) := {y ∈ V (Γ) : ∂(x, y) = 1}. The µ-graph of vertices x, y with ∂(x, y) = 2: Γ1(x, y) := {z ∈ V (Γ) : ∂(x, z) = 1 & ∂(z, y) = 1}. Using Q-polynomiality, we can restrict:

◮ possible eigenvalues of the local graphs, ◮ possible structure of the µ-graphs.

◮ Reconstructing the local structure

The problem of spectral characterization, i.e., reconstructing a graph from the spectrum of its adjacency matrix.

◮ Reconstructing a graph by its local structure

If we know Γ1(x), Γ1(x, y) for all x, y →? Γ

Bil2(n × n)

So, Γ has the same local graphs as Bil2(n × n). In other words, for a any vertex x of B := Bil2(n × n) and any vertex x ∈ Γ, we have an isomorphism: ϕ : {x} ∪ B1(x) → {x} ∪ Γ1(x). Then we are able to show this isomorphism may be extended to: ϕ′ : {x} ∪ B1(x) ∪ B2(x) → {x} ∪ Γ1(x) ∪ Γ2(x), and, moreover, for any vertex y ∈ B1(x), an isomorphism ϕ′ |y∪B1(y): y ∪ B1(y) → ϕ(y) ∪ Γ1(ϕ(y)) is also extendable.

Bil2(n × n)

So, Γ has the same local graphs as Bil2(n × n). In other words, for a any vertex x of B := Bil2(n × n) and any vertex x ∈ Γ, we have an isomorphism: ϕ : {x} ∪ B1(x) → {x} ∪ Γ1(x). Then we are able to show this isomorphism may be extended to: ϕ′ : {x} ∪ B1(x) ∪ B2(x) → {x} ∪ Γ1(x) ∪ Γ2(x), and, moreover, for any vertex y ∈ B1(x), an isomorphism ϕ′ |y∪B1(y): y ∪ B1(y) → ϕ(y) ∪ Γ1(ϕ(y)) is also extendable.

Bil2(n × n)

◮ A homomorphism from a graph

Γ to a graph Γ is a map ρ : V ( Γ) → V (Γ) that preserves the edges.

◮ A local isomorphism: a homomorphism ρ :

Γ → Γ such that, for each vertex x ∈ Γ, the induced mapping

• Γ1(ρ−1(x)) → Γ1(x) is bijective.

◮ A covering map: a surjective local isomorphism.

The theorem∗ by Munemasa and Shpectorov shows that Γ covers B ⇒ Γ ∼ = B. (In order to use this theorem, we need to check some additional assumptions, which indeed hold in our case.)

∗ A. Munemasa, S.V. Shpectorov, A local characterization of the

graphs of alternating forms // Finite geometry and combinatorics,

• 1993. P. 289-302.

Bil2(n × n)

◮ A homomorphism from a graph

Γ to a graph Γ is a map ρ : V ( Γ) → V (Γ) that preserves the edges.

◮ A local isomorphism: a homomorphism ρ :

Γ → Γ such that, for each vertex x ∈ Γ, the induced mapping

• Γ1(ρ−1(x)) → Γ1(x) is bijective.

◮ A covering map: a surjective local isomorphism.

The theorem∗ by Munemasa and Shpectorov shows that Γ covers B ⇒ Γ ∼ = B. (In order to use this theorem, we need to check some additional assumptions, which indeed hold in our case.)

∗ A. Munemasa, S.V. Shpectorov, A local characterization of the

graphs of alternating forms // Finite geometry and combinatorics,

• 1993. P. 289-302.

Bil2(n × n)

◮ A homomorphism from a graph

Γ to a graph Γ is a map ρ : V ( Γ) → V (Γ) that preserves the edges.

◮ A local isomorphism: a homomorphism ρ :

Γ → Γ such that, for each vertex x ∈ Γ, the induced mapping

• Γ1(ρ−1(x)) → Γ1(x) is bijective.

◮ A covering map: a surjective local isomorphism.

The theorem∗ by Munemasa and Shpectorov shows that Γ covers B ⇒ Γ ∼ = B. (In order to use this theorem, we need to check some additional assumptions, which indeed hold in our case.)

∗ A. Munemasa, S.V. Shpectorov, A local characterization of the

graphs of alternating forms // Finite geometry and combinatorics,

• 1993. P. 289-302.

Bil2(n × n)

◮ A homomorphism from a graph

Γ to a graph Γ is a map ρ : V ( Γ) → V (Γ) that preserves the edges.

◮ A local isomorphism: a homomorphism ρ :

Γ → Γ such that, for each vertex x ∈ Γ, the induced mapping

• Γ1(ρ−1(x)) → Γ1(x) is bijective.

◮ A covering map: a surjective local isomorphism.

The theorem∗ by Munemasa and Shpectorov shows that Γ covers B ⇒ Γ ∼ = B. (In order to use this theorem, we need to check some additional assumptions, which indeed hold in our case.)

∗ A. Munemasa, S.V. Shpectorov, A local characterization of the

graphs of alternating forms // Finite geometry and combinatorics,

• 1993. P. 289-302.

My coauthors

Jack Koolen ↓ ↑ Klaus Metsch Thank you!

My coauthors

Jack Koolen ↓ ↑ Klaus Metsch Thank you!