Sharp bounds on the generalized distance spectral radius and generalized distance energy of strongly connected digraphs

Let $D(G)$ be the distance matrix of a strongly connected digraph $G$, $Tr(G)$ be the diagonal matrix with vertex transmissions of $G$ as diagonal entries. The generalized distance matrix $D_\beta(G)$ of the strongly connected digraph $G$ is defined as $D_\beta(G)=\beta Tr(G)+(1-\beta)D(G)$, for any real $0\leq\beta\leq 1$. The generalized distance spectral radius of $G$ is the spectral radius of $D_\beta(G)$. Let $\mu^\beta_1(G),\mu^\beta_2(G),\ldots,\mu^\beta_n(G)$ be the eigenvalues of $D_\beta(G)$, the generalized distance energy of the digraph $G$ is $E_{D_\beta}(G)=\sum\limits_{i=1}^n|\mu^\beta_i(G)-\frac{\beta W(G)}{n}|$, where $W(G)$ is the sum of distances between all ordered pairs of vertices of $G$. In this paper, we obtain some sharp upper and lower bounds for the generalized distance spectral radius of $G$ and characterize the extremal digraphs. Moreover, we also give some lower bounds on the generalized distance energy of digraphs.