Bounds on the $alpha$-distance spectrum of graphs

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 $alphain [0,1]$, $D(G)$ 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 $G$.