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: $\frac{n}{n-{\lambda }_1}\le \omega$, where ${\lambda }_1$ is the largest eigenvalue of the adjacency matrix $A(G)$, $n$ is the number of vertices in $G$, and $\omega$ is the clique number of $G$. Strengthening this bound, Elphick and Wocjan proposed a conjecture in 2018, which is stated as follows: $\frac{n}{n-\sqrt{s^{+}}}\le \omega$, where $s^{+}={\sum }_{{\lambda }_i>0}{\lambda }_i^2$ and ${\lambda }_i$ 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 $\lambda =\mu$ (i.e., $srg(n,d,\mu ,\mu )$) and $n\ge 2d$, the line graph of $K_n$, the Cartesian product of strongly regular graphs, and Ramanujan graph with $n\ge 11d$.