On generalized distance energy of graphs

For the distance matrix D(G) and diagonal matrix of the vertex transmissions Tr(G) of a simple connected graph G, the generalized distance matrix Dα(G) is the convex combinations of Tr(G) and D(G), and is defined as Dα(G)=αTr(G)+(1−α)D(G), for 0≤α≤1. If ∂1≥∂2≥…≥∂n are the eigenvalues of Dα(G), we define the generalized distance energy of the graph G as EDα(G)=∑i=1n|∂i−2αW(G)n|, where W(G) is the Wiener index of G. This is analogous to the energies associated with the distance Laplacian and distance signless Laplacian matrices of G. We obtain upper and lower bounds for the generalized distance energy of graphs, in terms of various parameters associated with the structure of the graph G. We show that for α∈[12,1), the complete bipartite graph has the minimum generalized distance energy among all connected bipartite graphs, and for α∈(0,2n3n−2), the star graph has the minimum generalized distance energy among all trees.