  It is well known that $n/(n - \mu)$, where $\mu$ is the spectral radius of a
graph with $n$ vertices, is a lower bound for the clique number. We conjecture
that $\mu$ can be replaced in this bound with $\sqrt{s^+}$, where $s^+$ is the
sum of the squares of the positive eigenvalues. We prove this conjecture for
various classes of graphs, including triangle-free graphs, and for almost all
graphs.
