Dual Seidel switching and Deza graphs Leonid Shalaginov Krasovskii - - PowerPoint PPT Presentation

Dual Seidel switching and Deza graphs

Leonid Shalaginov

Krasovskii Institute of Mathematics and Mechanics, Chelyabinsk State University based on joint work with S. Goryainov and V. Kabanov

Shanghai Jiao Tong University May 9, 2018

Outline

1 Basic definitions

strongly regular graphs Deza graphs strictly Deza graphs dual Seidel switching and ∆-automorphisms

2 Strictly Deza graphs (SDG)

survey of constructions SDG dual Seidel switching as a construction of SDG from SRG and SDG eigenvalues of a graph obtained by dual Seidel switching

3 A survey of results on ∆-automorphisms of Deza graphs

lattice graphs L(n) triangular graphs T(n) complements to L(n) and T(n) 2-clique extensions of SRG (4 × n)-lattice with even n Grassmann graphs Jq(n, k) using of a ∆-automorphism twice

References

[1] Haemers W. H., Dual Seidel switching. Eindhoven: Technical University Eindhoven. 1984. P. 183–191. [2] Erickson M., Fernando S., Haemers W. H., Hardy D. and Hemmeter J., Deza graphs: a generalization of strongly regular

• graphs. J. Comb. Designs. 1999. V. 7. P. 359–405.

[3] Kabanov V. V., Shalaginov L. V., On Deza graphs with parameters of lattice graphs. Trudy Inst. Mat. Mekh. UrO

• RAN. 2010. V. 3. P. 117–120. (in Russian)

[4] Shalaginov L. V., On Deza graphs with parameters of triangular graphs. Trudy Inst. Mat. Mekh. UrO RAN. 2011.

• V. 1. P. 294–298. (in Russian)

[5] Goryainov S. V., Shalaginov L. V., On Deza graphs with triangular and lattice graph complements as parameters. J.

• Appl. Industr. Math. 2013. V. 3. P. 355–362.

Strongly regular graphs (SRG)

Definition A k-regular graph G on v vertices is called strongly regular if there are also integers λ and µ such that every two adjacent vertices have λ common neighbours; every two non-adjacent vertices have µ common neighbours. We will say that it is (v, k, λ, µ)-SRG. The complement to (v, k, λ, µ)-SRG is also strongly regular. It is (v, v − k − 1, v − 2k + µ − 2, v − 2k + λ)-SRG.

Strictly Deza graphs

Definition A k-regular graph G on v vertices is a Deza graph (DG) with parameters (v, k, b, a), where v > k ≥ b ≥ a ≥ 0, if the number

• f common neighbours of two distinct vertices takes on one of

two values a or b, not necessarily depending on the adjacency of the two vertices. Definition A Deza graph is called a strictly Deza graph if it has diameter 2 and is not strongly regular.

[2] Erickson M., Fernando S., Haemers W. H., Hardy D. and Hemmeter J., Deza graphs: a generalization of strongly regular graphs.

Cayley graphs

Definition Suppose S is a subset of group Γ, such that

1 g ∈ S iff g−1 ∈ S, 2 identity of Γ is not in S.

Let G be the graph with vertex set Γ, and vertices g ∼ h iff h−1g ∈ S. Then G is called a Cayley graph of group Γ with generating set S, and denoted by Cay(Γ, S). Theorem([2], Proposition 2.1) Cay(Γ, S) where |Γ| = v, and |S| = k is a Deza graph with parameters (v, k, b, a) iff DD = aA + bB + k{e}, where A, B and {e} is a partition of group Γ.

[2] Erickson M., Fernando S., Haemers W. H., Hardy D. and Hemmeter J., Deza graphs: a generalization of strongly regular graphs.

Example SDG which is a Cayley graph

SDG with parameters (8, 4, 2, 1) is Cayley graph Cay(C8, {1, 2, 6, 7}).

Extension

Definition Let G1 = (V1, E1) and G2 = (V2, E2) be the graphs. The extension G1[G2] of G1 by G2 is a graph with vertex set V1 × V2, and (u1, u2) adjacent to (v1, v2) if and only if u1 adjacent to v1 or (u1 = v1 and u2 adjacent to v2).

The extension K3[K4]

Extension of K3 by K4 is a complete multipartite graph K3[K4] = K3×4.

SDG as an extension of SRG

Let G1 be an (v, k, λ, µ)-SRG and G2 be an (v′, k′, b, a)-DG. Then G1[G2] is a (k′ + k′v)-regular graph on vv′ vertices, and the number of common neighbours of two vertices in G1[G2] belongs to the set {a + kv′, b + kv′, µv′, λv′ + 2k′}. Theorem([2], Proposition 2.3) The graph G1[G2] is a Deza graph iff the equality |{a + kv′, b + kv′, µv′, λv′ + 2k′}| ≤ 2 holds.

[2] Erickson M., Fernando S., Haemers W. H., Hardy D. and Hemmeter J., Deza graphs: a generalization of strongly regular graphs.

Examples of extensions of SRG

Example 1 Let Kx×y, the complete multipartite graph with x parts by y

• vertices. Then Kx×y[K2] is a

(2xy, 1 + 2y(x − 1), 2y(x − 1), 2 + 2y(x − 2))-DG.

Examples of extensions of SRG

Example 1 Let Kx×y, the complete multipartite graph with x parts by y

• vertices. Then Kx×y[K2] is a

(2xy, 1 + 2y(x − 1), 2y(x − 1), 2 + 2y(x − 2))-DG. Example 2 Let G1 be an (v, k, λ)-SRG. Let G2 be an Kv′ Then G1[G2] is an (vv′, kv′, kv′, λv′)-DG.

Examples of extensions of SRG

Example 1 Let Kx×y, the complete multipartite graph with x parts by y

• vertices. Then Kx×y[K2] is a

(2xy, 1 + 2y(x − 1), 2y(x − 1), 2 + 2y(x − 2))-DG. Example 2 Let G1 be an (v, k, λ)-SRG. Let G2 be an Kv′ Then G1[G2] is an (vv′, kv′, kv′, λv′)-DG. Example 3 Let G1 be an (v, k, λ, µ)-SRG with λ = µ − 1. Then G1[K2] is an (2v, 2k + 1, 2k, 2µ)-DG.

Dual Seidel switching

Let G be a regular graph with the adjacency matrix M. Let P be the permutation matrix that represents an automorphism π of order 2 of G. Since π is an automorphism of order 2, the matrix PM is a symmetric matrix, which can be obtained by the permutation of rows of M in pairs with respect to the automorphism π. If π interchanges only nonadjacent vertices, then the matrix PM has zeroes on the main diagonal and hence can be regarded as the adjacency matrix of a graph G′. Definition We say that the graph G′ is obtained from G by dual Seidel switching with respect to the order 2 automorphism π.

Dual Seidel switching as a construction an SRG from SRG

Dual Seidel switching was introduced in [1] as a possible way to construct strongly regular graphs with λ = µ from the existing

• nes.

Let G be a (v, k, λ)-SRG with adjacency matrix M. Let P be the permutation matrix that represents an order 2 automorphism π of G. Theorem ([1])

1 If π interchanges only nonadjacent vertices, then PM is

the adjacency matrix of an SRG with the same parameters as G.

2 If π interchanges only adjacent vertices and has no fixed

vertices, then PM − I is the adjacency matrix of an SRG with parameters (v, k − 1, λ − 2, λ).

[1] Haemers W. H., Dual Seidel switching.

Example 1

Example 1

We constructed Srikhande graph. It has parameters (16, 6, 2, 2).

Example 2

We constructed SRG with parameters (16, 5, 0, 2).

Dual Seidel switching as a construction of SDG from SRG

Let G be a (v, k, λ, µ)-SRG with adjacency matrix M, where k = µ, λ = µ, λ = 0 and µ = 0. Let P be a permutation matrix. Theorem ([2], Theorem 3.1.) The matrix PM is the adjacency matrix of an SDG iff P represents an order 2 automorphism π of G interchanging only nonadjacent vertices. Definition An automorphism π satisfying the condition of the theorem above is called a ∆-automorphism of the graph G.

[2] Erickson M., Fernando S., Haemers W. H., Hardy D. and Hemmeter J., Deza graphs: a generalization of strongly regular graphs.

Example of a ∆-automorphism

SRG with parameters (9, 4, 1, 2) and its ∆-automorphism.

Example of a ∆-automorphism

SDG with parameters (9, 4, 2, 1).

Eigenvalues of SDG constructed by dual Seidel switching

Let G be an SRG with eigenvalues {k1, rf, sg}. Let G′ be an SDG obtained from G by the dual Seidel switching. Theorem([1], Result 5) Then G′ has eigenvalues {k1, rf1, −rf2, sg1, −sg2} where f1 + f2 = f and g1 + g2 = g. In particular, the graph G′ has at most 5 eigenvalues.

[1] Haemers W. H., Dual Seidel switching.

Dual Seidel switching as a construction of SDG from SDG

Since the the dual Seidel switching is just a permutation of rows, it can be applied to an SDG as well as to an SRG.

• 3. Survey of results on

∆-automorphisms

Lattice graph(L(n))

Definition The lattice graph L(n) is the graph with vertex set {1, 2, . . . , n} × {1, 2, . . . , n}, and (x1, y1) ∼ (x2, y2) iff x1 = x2 or y1 = y2.

∆-automorphisms of L(n)

Theorem([3], L. Shalaginov 2010) If n is even then there are only two ∆-automorphisms of L(n).

1 The first involution fixes n pairwise nonadjacent vertices

and can be considered as the symmetry with respect to the main diagonal.

2 The second involution doesn’t have fixed vertices and can

be considered as the composition of symmetries with respect to main and secondary diagonals. Theorem([3], L. Shalaginov 2010) If n is odd then L(n) admits an ∆-automorphism of the first type only.

[3] Kabanov V. V., Shalaginov L. V., On Deza graphs with parameters of lattice graphs.

Triangular graph(T(n))

Definition The triangular graph T(n) is the graph with vertex set {{x, y} ⊂ {1, 2, . . . , n}}, and {x, y} ∼ {i, j} iff |{x, y} ∩ {i, j}| = 1.

∆-automorphisms of T(n)

Theorem([4], L. Shalaginov 2011) If n is even, then there is a unique ∆-automorphism of T(n). If n is odd, then there are no ∆-automorphisms of T(n). The ∆-automorphism of T(n) fixes n/2 pairwise nonadjacent vertices and interchanges any pair of cliques that have a common fixed vertex.

[4] Shalaginov L. V., On Deza graphs with parameters of triangular graphs.

∆-automorphisms of T(6)

∆-automorphisms of L(n)

Theorem([5], L. Shalaginov 2013) There are exactly ⌊n/2⌋ ∆-automorphisms of L(n). Each of them interchanges some pairs of rows. ∆-automorphisms of L(4)

∆-automorphisms of T(n)

Theorem([5], L. Shalaginov 2013) There is a unique ∆-automorphism of T(n). It interchanges two maximal cocliques from the second neighborhood of one given vertex and fixes all other vertices.

[5] Goryainov S. V., Shalaginov L. V., On Deza graphs with triangular and lattice graph complements as parameters.

Automorphisms of T(5)

∆-automorphisms of 2-clique extensions of SRG

Let G be a (v, k, λ, µ)-SRG with λ = µ − 1. Let π be a ∆-automorphism of G. Denote by π′ the automorphism which is the natural extension of π, to G[K2]. Theorem(S. Goryainov, V. V. Kabanov, L. Shalaginov 2016) Then G′ is a ∆-automorphism of G[K2].

2-clique extension of L(3)

2-clique extension of L(3)

(18, 9, 8, 4)-SDG.

Automorphisms of K4 × Kn

Theorem

1 If n is even then there are only one ∆-automorphism of

K4 × Kn.

2 If n is odd then there are no ∆-automorphisms of K4 × Kn.

The ∆-automorphism of K4 × Kn with even n doesn’t have fixed vertices and can be considered as the composition of symmetries with respect to main and secondary diagonals.

A Grassmann graph

Definition The vertices of Grassmann graph Jq(n, k) are k-dimensional subspaces of n-dimensional vector space over a finite field of

• rder q. Two subspaces are adjacent iff their intersection is

(k − 1)-dimensional.

∆-automorphisms of Grassmann graph

Theorem (S. Goryainov, L. Shalaginov 2018) Graph J2(4, 2) has a unique ∆-automorphism. It maps points to planes and vise versa in PG(3, 2). Theorem(S. Goryainov, L. Shalaginov 2018) Graph Jq(4, 2) has at least one ∆-automorphism if q = 2s and at least two different ∆-automorphisms if n is odd.

Using of a ∆-automorphism twice

Let H be the Hadamard matrix H =     1 −1 1 1 −1 1 1 1 1 1 1 −1 1 1 −1 1     Denote m-Kronecker of H with itself by Hm. Then A = 1

2(Hm − J) is adjacency matrix of (v, k, λ) graph.

Using of a ∆-automorphism twice

Define P = (J2 − I2) ⊗ I2m. Then PA is symmetric and has all

• ne diagonal. Therefore PA − I is adjacency matrix of

(v, k − 1, λ − 2, λ)-SRG. Now we use P a second time. Then matrix P(PA − I) = A − P represents (v, k − 1, λ, λ − 2)-SDG. This double using can be applied for all SRG with λ = µ if this SRG has a ∆-automorphism without fixed vertices.

Thank you for attention!