On the minimum number of distinct eigenvalues of triangle-free strongly regular graphs

TL;DR

This paper investigates the minimum number of distinct eigenvalues in triangle-free strongly regular graphs, proving that most have at least three, with only the Clebsch graph having exactly two, thus addressing an open question.

Contribution

It establishes the minimum number of distinct eigenvalues for several triangle-free strongly regular graphs, including resolving an open question for the Sims-Gewirtz graph.

Findings

The Clebsch graph has exactly two distinct eigenvalues.

Five of the known graphs have more than two eigenvalues.

The minimum eigenvalue count for the Sims-Gewirtz graph is three.

Abstract

Among the seven known (non-degenerate) triangle-free strongly regular graphs, we prove that the Clebsch graph describes a matrix with exactly two distinct eigenvalues while five of the graphs do not. In showing that the minimum number of distinct eigenvalues of the Sims-Gewirtz graph is three, we answer a recently stated open question.