Let G be a simple connected graph of order n. Denote by D(G) the distance matrix of G and by Tr(G) the diagonal matrix of its vertex transmissions. For \(0\le \alpha \le 1\) , the generalized distance matrix \(D_{\alpha }(G)\) of G is defined as \(D_{\alpha }(G)=\alpha Tr(G)+(1-\alpha )D(G)\) . The generalized distance energy of a graph G (energy of G with respect to the generalized distance matrix) is defined as \(E^{D_{\alpha }}(G)=\sum _{i=1}^{n}\left| \partial _i-\frac{2\alpha W(G)}{n}\right| ,\) where W(G) is the transmission (also called the Wiener index) of a graph G and \(\partial _{1}\ge \partial _{2}\ge \cdots \ge \partial _{n}\) are the eigenvalues of \(D_{\alpha }(G)\) . In this paper, we establish new upper and lower bounds for \(E^{D_{\alpha }}(G)\) in terms of various graph invariants, and we characterize the extremal graphs for which these bounds are attained.
