Localized Tur\'{a}n-type inequalities for $Q$-index

For a connected graph $G$, let $q(G)$ denote the $Q$-index of $G$, i.e., the largest eigenvalue of its signless Laplacian matrix. Abreu and Nikiforov (2013) showed that $$q(G) \leq 2n\left(1-\frac{1}{\omega(G)}\right),$$ where $\omega(G)$ denotes the clique number of $G$. We first give a short algebraic proof of this result. For a vertex $v\in V(G)$, let $c(v)$ denote the order of the largest clique of $G$ containing $v$. Our main result is the following vertex localized bound that refines the result of Abreu and Nikiforov: $$q(G) \leq 2\sum_{v\in V(G)}\left(1-\frac{1}{c(v)}\right).$$ Equality holds precisely for complete bipartite graphs when $\omega(G)=2$, and for regular complete $\omega(G)$-partite graphs when $\omega(G)\geq 3$. As a consequence, we also obtain an analogous localized inequality for the $A_\alpha$-matrix of $G$. Finally, we generalize the above localized inequality to vertex-weighted signed graphs. This contributes to the localization program for spectral Tur\'{a}n-type results.