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.
Categories: math.CO