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 ≤ 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.

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A partial geometry S admitting an abelian Singer group G is called of rigid type if all lines of S have a trivial stabilizer in G. In this paper, we show that if a partial geometry of rigid type has fewer than 1000000 points it must be the Van Lint-Schrijver geometry or be a hypothetical geometry with 1024 or 4096 or 194481 points, which provides evidence that partial geometries of rigid type are very rare. Along the way we also exclude an infinite set of parameters that originally seemed very promising for the construction of partial geometries of rigid type (as it contains the Van Lint-Schrijver parameters as its smallest case and one of the other cases we cannot exclude as the second member of this parameter family). We end the paper with a conjecture on this type of geometries.

In this article we generalize a theorem of Benson for generalized quadrangles to strongly regular graphs and directed strongly regular graphs. The main result provides numerical restrictions on the number of fixed vertices and the number of vertices mapped to adjacent vertices under an automorphism. It is explained how these results can be used when studying partial difference sets in Abelian groups and projective two-weight sets. The underlying ideas are linear algebraic in nature.