Strongly Regular Graphs with No Triangles

TL;DR

This paper develops a simplified theory for triangle-free strongly regular graphs, providing new proofs of key conditions, characterizations, and methods for listing feasible parameters, advancing understanding of their structure.

Contribution

It introduces a simplified framework for triangle-free strongly regular graphs, including direct proofs of Krein conditions and characterizations of specific subclasses.

Findings

Provides direct proofs of Krein conditions

Characterizes strongly regular graphs with no triangles and strongly regular second subconstituent

Offers an effective method for listing feasible parameters

Abstract

A simplified version of the theory of strongly regular graphs is developed for the case in which the graphs have no triangles. This leads to (i) direct proofs of the Krein conditions, and (ii) the characterization of strongly regular graphs with no triangles such that the second subconstituent is also strongly regular. The method also provides an effective means of listing feasible parameters for such graphs.