Strengthening Wilf's lower bound on clique number
Hareshkumar Jadav, Sreekara Madyastha, Rahul Raut, Ranveer Singh
TL;DR
This work targets strengthening Wilf's lower bound on the clique number by validating the Elphick–Wocjan conjecture $\frac{n}{n-\sqrt{s^{+}}}\leq \omega$ for several graph classes. The authors analyze eigenvalue structures of conference graphs, strongly regular graphs with $\lambda=\mu$, line graphs $L(K_n)$, Cartesian products of SRGs, and Ramanujan graphs, leveraging $s^{+}=\sum_{\lambda_i>0}\lambda_i^{2}$ to bound $\omega$. Key results include proving the conjecture for conference graphs and $srg(n,d,\lambda,\mu)$ with $\lambda=\mu$ and $n\ge 2d$, establishing the conjecture for $L(K_n)$, showing closure under Cartesian product for SRGs when $n>7$, and confirming the conjecture for Ramanujan graphs with $n\ge 11d$. These findings advance spectral bounds on clique numbers for several structurally important graph families and guide future work toward the general case.
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: $\frac{n}{n - λ_{1}} \leq ω$, 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: $\frac{n}{n - \sqrt{s^{+}}} \leq ω$, where $s^+ = \sum_{λ_{i} > 0} λ_{i}^2$ 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 $n\geq 2d$, the line graph of $K_{n}$, the Cartesian product of strongly regular graphs, and Ramanujan graph with $n\geq 11d$.