Stronger Spectral Bound for Clique Number

Abstract

Given an integer $k$, deciding whether a graph has a clique of size $k$ is an NP-complete problem. Wilf's inequality provides a spectral lower bound for the clique number of a graph. Strengthening this bound, Elphick and Wocjan proposed a conjecture in 2018 using positive adjacency eigenvalues. In this paper, we present an infinite family of graphs for which the Elphick and Wocjan conjecture is true. We find a stronger spectral lower bound for the clique number for this family of graphs.