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

TL;DR

The paper addresses the inverse eigenvalue problem for graphs by focusing on the minimum number of distinct eigenvalues $q(G)$ and, in particular, the existence of a matrix with exactly two eigenvalues for triangle-free strongly regular graphs. It develops and applies Gram-matrix characterizations and plus-graph techniques to analyze structural constraints, proving $q(\mathsf{C})=2$ for the Clebsch graph and $q(\Gamma)=3$ for the Sims-Gewirtz graph, thereby resolving an open question for the latter and determining six of the seven known graphs. The Clebsch analysis reveals a $2$-eigenvalue realization with a Gram matrix in $\mathcal{S}(\mathsf{C})$, while the Sims-Gewirtz result hinges on a detailed decomposition into incidence-like and $4$-cycle structures and the connectivity of the plus-graph $\Gamma^+$. The Higman–Sims graph remains open, highlighting the role of plus-graph connectivity and cycle-sign constraints in the inverse eigenvalue problem for triangle-free SRGs and guiding future explorations in this area.

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.

Figures (12)

• Figure 1: Clebsch graph shown as the 4-cube missing eight additional edges $(v,-v)$ for $v = 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111$.
• Figure 2: The diamond graph and the banner graph.
• Figure 3: The trapezohedral graph $T_4$.
• Figure 4: The subgraph of $T_n$ whose vertical edges comprise the set $S$.
• Figure 5: The set $P\cup L$.

Theorems & Definitions (45)

• Lemma 2.1
• proof
• Corollary 2.2
• Proposition 2.3
• proof
• Corollary 2.4
• proof
• Corollary 2.5
• Lemma 3.1
• proof