PIMS UC Voyageur Colloquium: Qing Xiang (University of Delaware)

A strongly regular graph srg(v,k,lambda,mu) is a graph with v vertices that is regular of valency k and that has the following properties:

(1) For any two adjacent vertices x,y, there are exactly lambda vertices adjacent to both x and y.

(2) For any two nonadjacent vertices x,y, there are exactly mu vertices adjacent to both x and y.

Classical examples of strongly regular graphs include the Paley graphs.

Let q=4t+1 be a prime power. The Paley graph P(q) is the graph with the finite field F_q as vertex set, where two vertices are adjacent when they differ by a (nonzero) square. The Paley graphs are the simplest examples of the so-called cyclotomic strongly regular graphs. In this talk we will consider cyclotomic srgs Cay(F_q, D) in a broader sense, namely, D is a union of cosets of a subgroup of the multiplicative group F_q^* of F_q , not just a single coset of a subgroup of F_q^*. Twelve new infinite families of srgs are obtained this way. We also show that these srgs give rise to some very interesting association schemes. This is a joint work with Tao Feng.