The $A_{\alpha }$-spectral radius and perfect matchings of graphs

Abstract

Let $\alpha \in [0,1)$, and let $G$ be a graph of even order $n$ with $n\ge f(\alpha )$, where $f(\alpha )=10$ for $0\le \alpha \le 1/2$, $f(\alpha )=14$ for $1/2<\alpha \le 2/3$ and $f(\alpha )=5/(1-\alpha )$ for $2/3<\alpha <1$. In this paper, it is shown that if the $A_{\alpha }$-spectral radius of $G$ is not less than the largest root of $x^3-((\alpha +1)n+\alpha -4)x^2+(\alpha n^2+({\alpha }^2-2\alpha -1)n-2\alpha +1)x-{\alpha }^2{n}^2+(5{\alpha }^2-3\alpha +2)n-10{\alpha }^2+15\alpha -8=0$ then $G$ has a perfect matching unless $G=K_1\nabla (K_{n-3}\cup 2K_1)$. This generalizes a result of S. O [Spectral radius and matchings in graphs, Linear Algebra Appl. 614 (2021) 316--324], which gives a sufficient condition for the existence of a perfect matching in a graph in terms of the adjacency spectral radius.

13 pages