The maximum $A_{\alpha}$-spectral radius of $t$-connected graphs with bounded matching number

TL;DR

This paper determines the graphs with the maximum $A_{\alpha}$-spectral radius among $t$-connected graphs with bounded matching number, generalizing previous results for certain parameters.

Contribution

It introduces a characterization of extremal graphs maximizing the $A_{\alpha}$-spectral radius under connectivity and matching constraints for $\alpha \in [0, 1/2]$, extending prior work.

Findings

Identifies extremal graphs for maximum $A_{\alpha}$-spectral radius

Generalizes previous spectral graph theory results

Provides bounds for $t$-connected graphs with matching constraints

Abstract

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be a diagonal matrix of the degrees of $G$. In 2017, Nikiforov defined the $A_{\alpha}$-matrix of $G$ as \begin{equation*} A_{\alpha}(G)=\alpha G)+(1-\alpha)A(G), \end{equation*}d where $\alpha\in[0,1]$ is an arbitrary real number. The largest eigenvalue of $A_{\alpha}(G)$ is called the $A_{\alpha}$-spectral radius of $G$. Let $n$, $t$, $k$ be positive integers, satisfying $t\geq1$, $k\geq2$, $n\geq k+2$, and $n\equiv k$ (mod $2$). In this paper, for $\alpha\in[0,\frac{1}{2}]$, we determine the extremal graphs with the maximum $A_{\alpha}$-spectral radius among all $t$-connected graphs on $n$ vertices with matching number $\frac{n-k}{2}$ at most. This generalizes some results of O (2021) and Zhang (2022).