On $D_{\alpha}$ spectrum of connected graphs

Abstract

Let $G$ be a connected graph with $\alpha \in [0,1]$, the $D_{\alpha}$-spectral radius of $G$ is defined to be the spectral radius of the matrix $D_{\alpha}(G)$, defined as $D_{\alpha}(G)= \alpha T(G)+(1 - \alpha)D(G)$, where $T(G)$ is a transmission diagonal matrix of $G$ and $D(G)$ denotes the distance matrix of $G$. In this paper, we give some sharp upper and lower bounds for the $D_{\alpha}$-spectral radius with respect to different graph parameters.