Some Properties of Strongly Regular Graphs

TL;DR

This paper explores the enumeration of feasible parameters for strongly regular graphs using algebraic and structural conditions, revealing both finite possibilities and infinite families with specific properties.

Contribution

It introduces a new approach to parameter enumeration based on structural parameters and eigenvalues, and discusses algebraic reasons behind feasible parameter sets.

Findings

Finite possibilities for parameters c given a and e due to Krein bounds

Identification of algebraic conditions leading to feasible parameter sets

Existence of an infinite family of graphs with specific neighborhood properties

Abstract

An approach to the enumeration of feasible parameters for strongly regular graphs is described, based on the pair of structural parameters (a,c) and the positive eigenvalue e. The Krein bound ensures that there are only finitely many possibilities for c, given a and e, and the standard divisibility conditions can be used to reduce the possibilities further. Many sets of feasible parameters appear to be accidents of arithmetic, but in some cases the conditions are satisfied for algebraic reasons. As an example, we discuss an infinite family of feasible parameters for which the corresponding graphs necessarily have a closed neighborhood as a star complement for e.