On the distance $\alpha$-spectral radius of a connected graph

Abstract

For a connected graph $G$ and $\alpha \in [0,1)$, the distance $\alpha$-spectral radius of $G$ is 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 diagonal matrix of vertex transmissions of $G$ and $D(G)$ is the distance matrix of $G$. We give bounds for the distance $\alpha$-spectral radius, especially for graphs that are not transmission regular, propose some graft transformations that decrease or increase the distance $\alpha$-spectral radius, and determine the unique graphs with minimum and maximum distance $\alpha$-spectral radius among some classes of graphs.