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

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 $n-1\le m\le rn-\frac{r(r+1)}{2}$, 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.