The Asymptotics of Strongly Regular Graphs

A strongly regular graph is called trivial if it or its complement is a union of disjoint cliques. We prove that the parameters n; k; ; of nontrivial strongly regular graphs satisfy = k=n+ o (n) and = k=n+ o (n) : It follows, in particular, that every in…nite family of nontrivial strongly regular graphs is quasi-random in the sense of Chung, Graham and Wilson. 1 Introduction Our graph-theoretic notation is standard (see, e.g. [1]). Given a graph G and a set R V (G) ; we write b d (R) for the number vertices in G joined to every vertex in R and call the value b d (R) the codegree of R. A strongly regular graph (srg for short) with parameters n; k; ; is a kregular graph of order n such that b d (uv) = if uv is an edge, and b d (uv) = if uv is not an edge; we denote by SR (n; k; ; ) a srg with parameters n; k; ; . Observe that any graph rKm is an SR (mr;m 1;m 2; 0) ; we call these graphs and their complements trivial srgs. Srgs have been intensively studied; we refer the reader to, e.g. [5], [2], and [4]. Among the many problems related to srgs, probably the most intriguing one is to …nd strong necessary conditions for the parameters of a srg. Despite the numerous partial results, no exact condition of wide scope is known. If we look for asymptotic conditions, however, the problem becomes more tangible. In this note we investigate the parameters of nontrivial srgs when the order tends to in…nity. Somewhat surprisingly it turns out that the parameters and are asymptotically equal. More precisely, the following theorem holds. Theorem 1 The parameters n; k; ; of nontrivial strongly regular graphs satisfy = k=n+ o (n) and = k=n+ o (n) : (1)