Approximately strongly regular graphs

Abstract

We give variants of the Krein bound, the absolute bound and the inertia bound for graphs with a spectrum similar to that of a strongly regular graph. In particular, we investigate what we call approximately strongly regular graphs.We apply our results to extremal problems. Among other things, we show the following:(1) Caps in PG(n,q) for which the number of secants on exterior points does not vary too much, have size O(q34n) (as q→∞ or as n→∞).(2) Optimally pseudorandom Km-free graphs of order v and degree k for which the induced subgraph on the common neighborhood of a clique of size i≤m−3 is similar to a strongly regular graph, have k=O(v1−13m−2i−5).