Strengthening Wilf's Lower Bound on 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 bound for the clique number of simple graphs. Wilf's inequality is stated as follows: [EQUATION], where λ 1 is the largest eigenvalue of the adjacency matrix A(G), n is the number of vertices in G, and Ω is the clique number of G. Strengthening this bound, Elphick and Wocjan proposed a conjecture in 2018, which is stated as follows: (Formula presented), where (Formula presented) and λi are the eigenvalues of A(G). In this paper, we have settled this conjecture for some classes of graphs, such as conference graphs, strongly regular graphs with λ = μ (i.e., srg(n,d,μ,μ)) and n ≥ 2d, the line graph of Kn, the Cartesian product of strongly regular graphs, and Ramanujan graph with n ≥ 11d. © 2025 Copyright is held by the owner/author(s)