The $A_{\alpha }$ spectral radius of $k$-connected graphs with given diameter

Abstract

Let $G$ be a graph with adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$. In 2017, Nikiforov defined the matrix $A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)$ for any real $\alpha \in [0,1]$. The largest eigenvalue of $A_{\alpha }(G)$ is called the $A_{\alpha }$ spectral radius or the $A_{\alpha }$-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 $k\ge 2$ and $d\ge 2$. We generalizes the results about adjacency matrix of Theorem 3.6 in [P. Huang, W.C. Shiu, P.K. Sun, Linear Algebra Appl., 488 (2016) 350--362] and the results about signless Laplacian matrix of Theorem 3.4 in [P. Huang, J.X. Li, W.C. Shiu, Linear Algebra Appl., 617 (2021) 78--99]. Furthermore, we also obtain the upper and lower bounds of the extremal graph in ${\mathcal{G}}_{n,k}^d$.