BOUNDING THE Aα-SPECTRAL RADIUS OF k-CONNECTED IRREGULAR GRAPHS

Abstract

Let $G$ be a simple graph of order $n$. For $\alpha \in [0,1]$, the $A_{\alpha }$-matrix of $G$ is defined as $A_{\alpha }=\alpha D(G)+(1-\alpha )A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the diagonal matrix of vertex degrees of $G$, respectively. The largest eigenvalue of $A_{\alpha }(G)$, denoted by ${\rho }_{\alpha }(G)$, is called the $A_{\alpha }$ spectral radius of $G$. We give an upper bound on ${\rho }_{\alpha }(G)$ for $k$-connected irregular graphs. Moreover, we also derive an upper bound on ${\rho }_{\alpha }(G)$ when $G$ is a subgraph of a $k$-connected regular graph. Our results improve or extend the existing results, respectively.