The $A_α$ spectral radius of $k$-connected graphs with given diameter

Xichan Liu, Ligong Wang

Published 2022-10-29Version 1

The $A_\alpha$ matrix of a graph $G$ is defined as the convex combination of its adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$, that is, $A_\alpha(G)=\alpha A(G)+(1-\alpha)D(G)$, where $\alpha\in[0,1]$, whose largest eigenvalue is called the $A_\alpha$ spectral radius or $A_\alpha$-index of $G$. Let $\mathcal{G}_{n,k}^d$ be the set of $k$-connected graphs of order $n$ with diameter $d$. In this paper, we determine the graphs with maximum $A_\alpha$ spectral radius among all graphs in $\mathcal{G}_{n,k}^d$ for any $\alpha\in[0,1)$, where $k\geq2$ and $d\geq2$, which generalizes the results for adjacency matrix in [P. Huang, W.C. Shiu, P.K. Sun, Linear Algebra Appl., 2016, Theorem 3.6] and signless Laplacian matrix in [P. Huang, J.X. Li, W.C. Shiu, Linear Algebra Appl., 2021, Theorem 3.4]. Furthermore, we also obtaine some bounds of the set of graphs $\mathcal{G}_{n,k}^d$.