An improved bound for strongly regular graphs with smallest eigenvalue $-m$

TL;DR

This work improves Neumaier's bound for primitive strongly regular graphs with smallest eigenvalue $-m$ by introducing a sharper threshold $f(m,\mu)$, and shows that, except for two infinite families, either $\lambda \le f(m,\mu)$ or the graph is the collinearity graph of a partial geometry with parameters determined by $m$ and $\mu$. The authors develop and leverage the SPLS($\sigma$) framework and show that large parameter regimes force the SRG to be geometric, i.e., the point graph of a partial geometry, which then yields a precise classification. The approach provides elementary, general methods that not only strengthen previous bounds but also connect to geometric constructions, Bruck's Completion Theorem for orthogonal arrays, and consequences for graph isomorphism bounds. Overall, the paper advances the understanding of how fixed smallest eigenvalue constraints restrict SRG structure and offers tighter computable bounds useful in combinatorial design theory and related algorithmic problems.

Abstract

In 1979, Neumaier gave a bound on $λ$ in terms of $m$ and $μ$, where $-m$ is the smallest eigenvalue of a primitive strongly regular graph, unless the graph in question belongs to one of the two infinite families of strongly regular graphs. We improve this result. We also indicate how our methods can be used to give an alternate derivation of Bruck's Completion Theorem for orthogonal arrays.

Theorems & Definitions (18)

• Theorem 1.1: cf. SRG
• Theorem 1.2: cf. SRG
• Theorem 1.3
• Lemma 2.1
• Theorem 2.2: cf.metsch1999characterization, Metsch
• Theorem 2.3
• Lemma 2.4
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3