On the first two eigenvalues of regular graphs

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 $${\mu }_1^2+{\mu }_2^2\le \frac{2(\omega -1)}{\omega }m$$ where $\omega$ 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.