## PIMS Voyageur Colloquium

- Date: 02/04/2011

University of Calgary

Large cyclotomic strongly regular graphs and nonamorphic fusion schemes of the cyclotomic schemes.

Abstract:

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.