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Conjectured lower bound for the clique number of a graph
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abstract
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.
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Cited by 1 Pith paper
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A positive square-energy strengthening of Tur\'an's theorem
Every n-vertex graph with clique number ω has √s⁺(G) ≤ (1−1/ω)n, where s⁺(G) is the sum of squared positive adjacency eigenvalues.
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