Colloquium:
Speaker: Prof. Bart De Bruyn (Ghent University, Belgium)
Title: Divisible design graphs from symplectic graphs
Time, day and date: 4:00:00 PM, Wednesday, January 29
Venue: Ramanujan Hall
Abstract:
Colloquium:
This talk is based on joint work with Sergey Goryainov, Willem Haemers and Leonid Shalaginov. A regular graph of degree k on v vertices is called a divisible design graph with parameters (v, k, λ1, λ2,m, n) if its vertex-set can be partitioned into m classes of size n such that any two vertices from the same class have λ1 common neighbours and any two vertices from different classes have λ2 common neighbours. Divisible design graphs were introduced because of their connection with divisible designs: the adjacency matrix of any such graph is an incidence matrix of such a design. In the talk, new families of divisible design graphs are constructed that are related to the symplectic graphs Sp(2e, q), e ≥ 2. Starting from a 2e dimensional vector space V over the finite field Fq that is endowed with a nondegenerate alternating bilinear form b(·, ·), the vertices of Sp(2e, q) are the one-dimensional subspaces
of V , where two distinct one-dimensional subspaces ⟨v1⟩ and ⟨v2⟩ are adjacent whenever b(v1, v2) = 0. We define and discuss a family of divisible design graphs based on a partition of Sp(4,q), q odd, in subgraphs isomorphic to K_{q+1,q+1}, and show that there is an example in this family for every odd prime power q. We have classified by computer all examples in this family for q ∈ {3, 5, 7} and we discuss the computational challenges that we faced during this process. The divisible design graphs in this family also give rise to additional examples of divisible design graphs. Finally, we also describe some families of divisible design graphs based on so-called symplectic spreads of Sp(2e, q).