General validity of the Elphick–Wocjan spectral clique bound

Establish whether, for every finite simple graph G with adjacency matrix A(G) and eigenvalues λ1 ≥ λ2 ≥ ... ≥ λn, letting s+ = Σ_{i: λi>0} λ_i^2 denote the sum of squares of the positive eigenvalues, the inequality n / sqrt(s+) ≤ ω(G) holds, where n = |V(G)| and ω(G) is the clique number. This asks for a proof or disproof of the Elphick–Wocjan spectral bound for general graphs beyond the specific classes settled in this paper.

Background

Wilf’s inequality gives a classical spectral lower bound on the clique number: ω(G) ≥ n / (n − λ1), where λ1 is the largest eigenvalue of the adjacency matrix. Motivated by strengthening this, Elphick and Wocjan proposed a conjectural bound involving the sum of squares of positive eigenvalues, s+, asserting n / sqrt(s+) ≤ ω(G).

This paper proves the conjectured bound for several graph families, including conference graphs, certain strongly regular graphs (with λ = μ and n ≥ 2d), line graphs of Kn, Cartesian products of strongly regular graphs, and Ramanujan graphs with n ≥ 11d. However, the authors explicitly note that the conjecture is unresolved for general graphs.