Approximately Strongly Regular Graphs

TL;DR

This paper introduces the concept of approximately strongly regular graphs, providing bounds and applying these to extremal problems in combinatorics and finite geometry.

Contribution

It develops variants of classical bounds for graphs with spectra similar to strongly regular graphs and applies these to problems in projective geometry and pseudorandom graph theory.

Findings

Caps in projective spaces have size at most O(q^{3n/4})

Pseudorandom K_m-free graphs have degree bounds of O(v^{1 - 1/(3m-2i-5)})

Results extend bounds and properties of strongly regular graphs to approximate cases

Abstract

We give variants of the Krein bound and the absolute 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 $\mathrm{PG}(n, q)$ for which the number of secants on exterior points does not vary too much, have size at most $O(q^{\frac34 n})$ (as $q \rightarrow \infty$ or as $n \rightarrow \infty$). (2) Optimally pseudorandom $K_m$-free graphs of order $v$ and degree $k$ for which the induced subgraph on the common neighborhood of a clique of size $i \leq m-3$ is similar to a strongly regular graph, have $k = O(v^{1 - \frac{1}{3m-2i-5}})$.