Automorphisms of strongly regular graphs with applications to partial difference sets

Abstract

In this article we generalize a theorem of Benson (J Algebra 15:443---454, 1970) for generalized quadrangles to strongly regular graphs, deriving numerical restrictions on the number of fixed vertices and the number of vertices mapped to adjacent vertices under an automorphism. We then use this result to develop a few new techniques to study regular partial difference sets (PDS) in Abelian groups. Ma (Des Codes Cryptogr 4:221---261, 1994) provided a list of parameter sets of regular PDS with $$k\le 100$$k≤100 in Abelian groups for which existence was known or had not been excluded. In particular there were 32 parameter sets for which existence was not known. Ma (J Stat Plan Inference 62:47---56, 1997) excluded 13 of these parameter sets. As an application of our results we here exclude the existence of a regular partial difference set for all but two of the undecided parameter sets from Ma's list.