On the first two eigenvalues of regular graphs

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.