The Second Subconstituent of some Strongly Regular Graphs

TL;DR

This paper investigates the construction of triangle-free strongly regular graphs via second subconstituents with equitable partitions, successfully identifying candidates for infinitely many prime powers but only confirming known cases.

Contribution

It introduces a novel approach to constructing strongly regular graphs using second subconstituents with four-part equitable partitions, providing new candidate graphs for many prime powers.

Findings

Constructed plausible second subconstituents for infinitely many prime powers.

Confirmed strong regularity only when the prime power is 3, corresponding to a known graph.

Showed limitations of the approach in producing new strongly regular graphs.

Abstract

This is a report on a failed attempt to construct new graphs that are strongly regular with no triangles. The approach is based on the assumption that the second subconstituent has an equitable partition with four parts. For infinitely many odd prime powers we construct a graph that is a plausible candidate for the second subconstituent. Unfortuantely we also show that the corresponding graph is strongly regular only when the prime power is 3, in which case the graph is already known.