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

TL;DR

This paper establishes a spectral radius condition based on the $A_eta$-spectral radius that guarantees the existence of a perfect matching in graphs of even order, generalizing previous adjacency spectral radius results.

Contribution

It introduces a new spectral radius threshold involving the $A_eta$-spectral radius that ensures perfect matchings, extending prior adjacency spectral radius conditions.

Findings

Spectral radius threshold guarantees perfect matchings.

Identifies the extremal graph for the condition.

Generalizes previous adjacency spectral radius results.

Abstract

Let $\alpha\in[0,1)$, and let $G$ be a graph of even order $n$ with $n\geq f(\alpha)$, where $f(\alpha)=10$ for $0\leq \alpha\leq1/2$, $f(\alpha)=14$ for $1/2<\alpha\leq 2/3$ 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 $x^3 - ((\alpha + 1)n +\alpha-4)x^2 + (\alpha n^2 + (\alpha^2 - 2\alpha - 1)n - 2\alpha+1)x -\alpha^2n^2 + (5\alpha^2 - 3\alpha + 2)n - 10\alpha^2 + 15\alpha - 8=0$ 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.