Energy of graphs with respect to generalized distance matrix: Extremal results and bounds

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≤α≤1, the generalized distance matrix Dα(G) of G is defined as Dα(G)=αTr(G)+(1-α)D(G). The generalized distance energy of a graph G (energy of G with respect to the generalized distance matrix) is defined as EDα(G)=∑i=1n∂i-2αW(G)n, where W(G) is the transmission (also called the Wiener index) of a graph G and ∂1≥∂2≥⋯≥∂n are the eigenvalues of Dα(G). In this paper, we establish new upper and lower bounds for EDα(G) in terms of various graph invariants, and we characterize the extremal graphs for which these bounds are attained.