On the first two eigenvalues of regular graphs

Let $G$ be a regular graph with $m$ edges, and let $μ_1, μ_2$ denote the two largest eigenvalues of $A_G$, the adjacency matrix of $G$. We show that, if $G$ is not complete, then $μ_1^2 + μ_2^2 \leq \frac{2(ω- 1)}ω m$ where $ω$ is the clique number of $G$. This confirms a conjecture of Bollobás and Nikiforov for regular graphs. We also show that equality holds if and only if $G$ is either a balanced Turán graph or the disjoint union of two balanced Turán graphs of the same size.

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