| Volume 15, Number 2, 2025, Pages - DOI：10.11948/JAAC-2024-0091 |
| Ordering graphs with fixed size and girth by their $A_{\alpha}$-spectral radius |
| Yirong Zheng,Hongzhang Chen,Long Jin |
| Keywords:$A_\alpha$-spectral radius, size, girth, order. |
| Abstract: |
| 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~[{\it Discrete Appl. Math.}, 340(2023), 350-362] 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
$\left(\lfloor\frac{g}{2} \rfloor+2 \right)$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 $\left(\lfloor\frac{g}{2} \rfloor+3 \right)$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 $\alpha=\frac{1}{2}$. |
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