Bounds on the $\alpha$-distance spectrum of graphs

Abstract

For a simple, undirected and connected graph $G$, $D_{\alpha }(G)=\alpha Tr(G)+(1-\alpha )D(G)$ is called the $\alpha$-distance matrix of $G$, where $\alpha \in [0,1]$, $D(G)$ is the distance matrix of $G$, and $Tr(G)$ is the vertex transmission diagonal matrix of $G$. Recently, the $\alpha$-distance energy of $G$ was defined based on the spectra of $D_{\alpha }(G)$. In this paper, we define the $\alpha$-distance Estrada index of $G$ in terms of the eigenvalues of $D_{\alpha }(G)$. And we give some bounds on the spectral radius of $D_{\alpha }(G)$, $\alpha$-distance energy and $\alpha$-distance Estrada index of $G$.