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 J q ( 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 − 2 k + µ − 2 , v − 2 k + 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 of 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 − 1 g ∈ 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 ( C 8 , { 1 , 2 , 6 , 7 } ).

Extension Definition Let G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) be the graphs. The extension G 1 [ G 2 ] of G 1 by G 2 is a graph with vertex set V 1 × V 2 , and ( u 1 , u 2 ) adjacent to ( v 1 , v 2 ) if and only if u 1 adjacent to v 1 or ( u 1 = v 1 and u 2 adjacent to v 2 ).

The extension K 3 [ K 4 ] Extension of K 3 by K 4 is a complete multipartite graph K 3 [ K 4 ] = K 3 × 4 .

SDG as an extension of SRG Let G 1 be an ( v, k, λ, µ )-SRG and G 2 be an ( v ′ , k ′ , b, a )-DG. Then G 1 [ G 2 ] is a ( k ′ + k ′ v )-regular graph on vv ′ vertices, and the number of common neighbours of two vertices in G 1 [ G 2 ] belongs to the set { a + kv ′ , b + kv ′ , µv ′ , λv ′ + 2 k ′ } . Theorem([2], Proposition 2.3) The graph G 1 [ G 2 ] is a Deza graph iff the equality |{ a + kv ′ , b + kv ′ , µv ′ , λv ′ + 2 k ′ }| ≤ 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 K x × y , the complete multipartite graph with x parts by y vertices. Then K x × y [ K 2 ] is a (2 xy, 1 + 2 y ( x − 1) , 2 y ( x − 1) , 2 + 2 y ( x − 2))-DG.

Examples of extensions of SRG Example 1 Let K x × y , the complete multipartite graph with x parts by y vertices. Then K x × y [ K 2 ] is a (2 xy, 1 + 2 y ( x − 1) , 2 y ( x − 1) , 2 + 2 y ( x − 2))-DG. Example 2 Let G 1 be an ( v, k, λ )-SRG. Let G 2 be an K v ′ Then G 1 [ G 2 ] is an ( vv ′ , kv ′ , kv ′ , λv ′ )-DG.

Examples of extensions of SRG Example 1 Let K x × y , the complete multipartite graph with x parts by y vertices. Then K x × y [ K 2 ] is a (2 xy, 1 + 2 y ( x − 1) , 2 y ( x − 1) , 2 + 2 y ( x − 2))-DG. Example 2 Let G 1 be an ( v, k, λ )-SRG. Let G 2 be an K v ′ Then G 1 [ G 2 ] is an ( vv ′ , kv ′ , kv ′ , λv ′ )-DG. Example 3 Let G 1 be an ( v, k, λ, µ )-SRG with λ = µ − 1. Then G 1 [ K 2 ] is an (2 v, 2 k + 1 , 2 k, 2 µ )-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 ones. 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 { k 1 , r f , s g } . Let G ′ be an SDG obtained from G by the dual Seidel switching. Theorem([1], Result 5) Then G ′ has eigenvalues { k 1 , r f 1 , − r f 2 , s g 1 , − s g 2 } where f 1 + f 2 = f and g 1 + g 2 = 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 ( x 1 , y 1 ) ∼ ( x 2 , y 2 ) iff x 1 = x 2 or y 1 = y 2 .

∆-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.