Extremal graphs for the maximum $A_α$-spectral radius of graphs with order and size

TL;DR

This work advances the Brualdi-Solheid problem by determining the maximum $A_{\alpha}$-spectral radius for connected graphs of order $n$ and size $m$, using threshold-graph structure and a suite of graph transformations. It establishes a toolkit of transformations that do not decrease $\rho_{\alpha}$ for $\alpha\in[\tfrac12,1)$, enabling precise extremal characterizations within $\mathcal{H}_{n,m}$ under specified growth conditions $n>30r$ and $n-1\le m\le rn-\frac{r(r+1)}{2}$. The authors prove that, for $\alpha\in(\tfrac12,1)$ or $\alpha=\tfrac12$ with $m\neq (r-1)n-\frac{r(r-1)}{2}+3$, the unique extremal graph is the quasi-star $S_{n,m}$; when $\alpha=\tfrac12$ and $m$ hits the threshold, the second threshold graph $\tilde{S}_{n,m}$ also becomes extremal. These results resolve the maximal $A_{\alpha}$-spectral radius problem in the stated regimes and unify previous partial results through a threshold-graph and equitable-partition framework.

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 $A_α$-spectral radius of a simple graph of order $n$, denoted by $ρ_α(G)$, is the largest eigenvalue of the matrix $A_α(G)$ which is defined as $αD(G)+(1-α)A(G)$ for $0\le α< 1$, 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 $A_α$-spectral radius of graphs in the set $\mathcal{H}_{n,m}$. Moreover, the problem on $A_α$-spectral radius proposed by Chang and Tam [T.-C. Chang and B.-T. Tam, Graphs of fixed order and size with maximal $A_α$-index. Linear Algebra Appl. 673 (2023), 69-100] has been solved.

Figures (6)

• Figure 1: $S_{n,m}$ and $L_{n,m}$ with $n=6$, $m=10$
• Figure 2: The Ferrers matrix of a threshold graph is symmetric
• Figure 3: The Ferrers matrix of a non-threshold graph $G_{6,9}$ is asymmetrical
• Figure 4: Transformation $(7, 2; 5, 3; 2,1)$ from $L_{7, 12}$ to $S_{7, 12}$
• Figure 5: Transformation $(9,3;8,4;1,2)$ from $G_{9,23}$ to $S_{9,23}$

Theorems & Definitions (52)

• Theorem 1.3
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• Lemma 2.2
• Definition 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Corollary 2.7