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 D-alpha(G) is the convex combinations of Tr(G) and D(G), and is defined as D-alpha(G) = alpha Tr(G) + (1 - alpha)D(G), for 0 <= alpha <= 1. If partial derivative(1) > partial derivative(2) >= ... >= partial derivative(n) are the eigenvalues of D-alpha(G), we define the generalized distance energy of the graph G as E-D alpha(G) = Sigma(n)(i=1) vertical bar partial derivative(i) - 2 alpha W(G)/n vertical bar, 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 alpha is an element of [1/2, 1), the complete bipartite graph has the minimum generalized distance energy among all connected bipartite graphs, and for alpha is an element of (0, 2n/3n-2), the star graph has the minimum generalized distance energy among all trees. (C) 2020 Elsevier Inc. All rights reserved.