ORDERING GRAPHS WITH FIXED SIZE AND GIRTH BY THEIR $A_{\alpha}$-SPECTRAL RADIUS

For a graph $G$ and real number $\alpha\in [0, 1]$, the $A_\alpha$-spectral radius of $G$ is the largest eigenvalue of $A_\alpha(G):=\alpha D(G)+(1-\alpha) A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the diagonal degree matrix of $G$, respectively. Recently, for $\alpha\in [\frac{1}{2}, 1]$, Chen, Li and Huang [Discrete Appl. Math., 2023, 340, 350–362], as well as Ye, Guo and Zhang [Discrete Appl. Math., 2024, 342, 286–294] independently identified the graph with maximum $A_{\alpha}$-spectral radius among all graphs in $\mathcal{G}(m, g)$, the class of connected graphs on $m$ edges with girth $g$. In this paper, we further determine the second to the $\big(\lfloor\frac{g}{2} \rfloor+2 \big)$th largest $A_{\alpha}$-spectral radius of graphs in $\mathcal{G}(m, g)$. Moreover, for $\alpha\in [\frac{1}{2}, 1]$, we also determine the first to the $\big(\lfloor\frac{g}{2} \rfloor+3 \big)$th largest $A_{\alpha}$-spectral radius of graphs in $\mathcal{G}(m, \geq g)$, the class of connected graphs on $m$ edges with girth no less than $g$, which generalizes the recent result of Hu, Lou and Huang (2022) on $\alpha=\frac{1}{2}$.