On the first two eigenvalues of regular graphs

TL;DR

This paper proves a conjecture relating the two largest eigenvalues of regular graphs to their clique number, characterizing when equality occurs and extending spectral graph theory insights.

Contribution

It confirms a conjecture by Bollobás and Nikiforov for regular graphs and characterizes the extremal cases where equality holds.

Findings

Established an upper bound for the sum of squares of the two largest eigenvalues.

Confirmed the conjecture for regular graphs.

Characterized extremal graphs where equality holds.

Abstract

Let $G$ be a regular graph with $m$ edges, and let $\mu_1, \mu_2$ denote the two largest eigenvalues of $A_G$, the adjacency matrix of $G$. We show that, if $G$ is not complete, then $$\mu_1^2 + \mu_2^2 \leq \frac{2(\omega - 1)}{\omega} m$$ where $\omega$ is the clique number of $G$. This confirms a conjecture of Bollob\'{a}s and Nikiforov for regular graphs. We also show that equality holds if and only if $G$ is either a balanced Tur\'{a}n graph or the disjoint union of two balanced Tur\'{a}n graphs of the same size.