Conjectured lower bound for the clique number of a graph

Abstract

It is well known that $n/(n - μ)$, where $μ$ is the spectral radius of a graph with $n$ vertices, is a lower bound for the clique number. We conjecture that $μ$ 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.

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