Strongly regular Cayley graphs with λ − μ = −1

AbstractThe classification problem for strongly regular graphs for which the parameters are related by the equation λ − μ = −1 is still completely open. Restricting attention to those examples which are simultaneously Cayley graphs based on an abelian group (which are equivalent to abelian partial difference sets with λ − μ = −1), we obtain the following classification result: any such graph is—up to complementation—either of Paley type (i.e., it has parameters (ν, (ν − 1)/2, (ν − 5)/4, (ν − 1)/4)) or it has parameters (243, 22, 1, 2). The proof of this theorem combines recent results on the structure of partial difference sets with some results concerning diophantine equations. Our theorem has interesting applications to the theory of divisible difference sets, since it allows us to improve previous classification results concerning abalian DDS's satisfying k − λ1 = 1 and reversible abelian DDS's