Regular weights on strongly regular graphs.

Abstract

The concept of a regular weight on a coherent configuration (cc) was introduced by D. G. Higman in connection with centralizer algebras of monomial representations, and turns out to be a natural generalization of the work of D. E. Taylor, J. J. Seidel and others on regular 2-graphs. Usually, we will think of a weight $\omega$ as a certain Hermitian matrix with values in $U\sb{t} \cup \{$0$\}$, where $U\sb{t}$ is the group of complex $t\sp{\rm th}$ roots of unity. The rank of $\omega$ is the number of classes on which $\omega$ does not vanish. The weights with values in $U\sb2$ are equivalent to 2-graphs. The regular weights on a rank 2 cc with values in $U\sb2$ are equivalent to regular 2-graphs. In this work, we restrict our attention for the most part to regular weights with values in $U\sb2$ on cc's which are association schemes with 3 relations. A cc of this type is equivalent to a strongly regular graph (srg). The goal in studying these is to gain some understanding of when they occur and why. The occurrence of feasible regular weight parameters is relatively rare, so it is reasonable to suggest that one might obtain classification theorems for regular weights on the various families and types of srg's. This work is intended as a start to that process. We examine numerous examples, discuss non-existence results, classify the feasible parameters for triangular graphs, and give a complete account of the regular weights on lattice graphs.