Skew-symmetric association schemes with two classes and strongly regular graphs of type L 2n-1(4n-1)

Abstract

A construction of a pair of strongly regular graphs Γn and Γ'n of type L 2n-1 (4n-1) from a pair of skew-symmetric association schemes W, W' of order 4n-1 is presented. Examples of graphs with the same parameters as Γn and Γ'n, i.e., of type L 2n-1 (4n-1), were known only if 4n-1 = p^S, where p is a prime. The first new graph appearing in the series has parameters (v, k, λ) = (225, 98, 45). A 4-vertex condition for relations of a skew-symmetric association scheme (very similar to one for the strongly regular graphs) is introduced and is proved to hold in any case. This has allowed us to check the 4-vertex condition for Γn and Γ'n, thus to prove that Γn and Γ'n are not rank three graphs if n > 2.