  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.
