Note on the sum of the smallest and largest eigenvalues of a triangle-free graph

TL;DR

This paper investigates the sum of the largest and smallest eigenvalues of the adjacency matrix in triangle-free graphs, establishing a universal upper bound and identifying extremal cases among strongly regular graphs.

Contribution

It extends previous results by removing the regularity condition and characterizes the maximum eigenvalue sum in triangle-free strongly regular graphs.

Findings

The sum of the largest and smallest eigenvalues is at most (3-2√2)n for any triangle-free graph.

Regularity is not necessary for the eigenvalue sum bound to hold.

The Higman-Sims graph maximizes the normalized eigenvalue sum among triangle-free strongly regular graphs.

Abstract

Let $G$ be a triangle-free graph on $n$ vertices with adjacency matrix eigenvalues $\mu_1(G)\geq \mu_2(G)\geq \dots \geq \mu_n(G)$. In this paper we study the quantity $$\mu_1(G)+\mu_n(G).$$ We prove that for any triangle-free graph $G$ we have $$\mu_1(G)+\mu_n(G)\leq (3-2\sqrt{2})n.$$ This was proved for regular graphs by Brandt, we show that the condition on regularity is not necessary. We also prove that among triangle-free strongly regular graphs the Higman-Sims graph achieves the maximum of $$\frac{\mu_1(G)+\mu_n(G)}{n}.$$