An ordering theorem on the Q-spectral radius of graphs with given size and its applications

Abstract

The spectral extremal problem is a classic problem in spectral graph theory. For a simple graph $G$, let $q\left(G\right)$ denote the $Q$-spectral radius. We first characterize the graphs with maximal $Q$-spectral radius among all graphs of size $m$ with maximum degree $\Delta \ge \frac{m+1}{2}$. For two graphs $G_1$ and $G_2$ of size $m\ge 11$, employing this result, we prove that $q\left(G_1\right)>q\left(G_2\right)$ if $\Delta \left(G_1\right)>\Delta \left(G_2\right)$ and $\Delta \left(G_1\right)\ge \frac{m}{2}+3$, which improves the main result of [Bull. Malays. Math. Sci. Soc. 45(2022)2165-2174]. Let $r\ge \frac{m}{2}+3$ be an integer. Employing the above results, we completely characterize the graphs with maximal $Q$-spectral radius among all connected graphs of size $m$ with maximum degree at most $r$, with covering number $\beta$ and with independence number $\alpha \ge \frac{m}{3}$, respectively.