Strengthening Wilf's Lower Bound on Clique Number

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: [EQUATION], where [EQUATION] and λ _i 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 ≥ 2 d , the line graph of K _n , the Cartesian product of strongly regular graphs, and Ramanujan graph with n ≥ 11 d.