Conjectured lower bound for the clique number of a graph

Clive Elphick; Pawel Wocjan

Conjectured lower bound for the clique number of a graph

Combinatorics 2018-08-10 v2

Authors: Clive Elphick , Pawel Wocjan

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.

Keywords

clique number graph spectrum spectral radius positive eigenvalues triangle-free graphs lower bound conjecture

Cite

@article{arxiv.1804.03752,
  title  = {Conjectured lower bound for the clique number of a graph},
  author = {Clive Elphick and Pawel Wocjan},
  journal= {arXiv preprint arXiv:1804.03752},
  year   = {2018}
}

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