Extremal graphs for the maximum $A_{\alpha}$-spectral radius of graphs with order and size

TL;DR

This paper characterizes extremal graphs with maximum $A_{\alpha}$-spectral radius among graphs with fixed order and size, solving a longstanding problem and extending previous work on spectral graph optimization.

Contribution

It provides a complete characterization of extremal graphs for the maximum $A_{\alpha}$-spectral radius under certain conditions, advancing spectral graph theory.

Findings

Identifies extremal graphs with maximum $A_{\alpha}$-spectral radius for specified parameters.

Solves an open problem on $A_{\alpha}$-spectral radius maximization.

Extends previous results to broader graph classes.

Abstract

In 1986, Brualdi and Solheid firstly proposed the problem of determining the maximum spectral radius of graphs in the set $\mathcal{H}_{n,m}$ consisting of all simple connected graphs with $n$ vertices and $m$ edges, which is a very tough problem and far from resolved. The $A_{\alpha}$-spectral radius of a simple graph of order $n$, denoted by $\rho_\alpha(G)$, is the largest eigenvalue of the matrix $A_{\alpha}(G)$ which is defined as $\alpha D(G)+(1-\alpha)A(G)$ for $0\le \alpha< 1$, where $D(G)$ and $A(G)$ are the degree diagonal and adjacency matrices of $G$, respectively. In this paper, if $r$ is a positive integer, $n>30r$ and $n-1\leq m \le rn-\frac{r(r+1)}{2}$, we characterize all extremal graphs which have the maximum $A_{\alpha}$-spectral radius of graphs in the set $\mathcal{H}_{n,m}$. Moreover, the problem on $A_{\alpha}$-spectral radius proposed by Chang and Tam [T.-C. Chang and B.-T. Tam, Graphs of fixed order and size with maximal $A_{\alpha}$-index. Linear Algebra Appl. 673 (2023), 69-100] has been solved.