On Steiner degree distance of trees

Let G be a connected graph, and u, v, w its vertices. By du is denoted the degree of the vertex u, by d(u, v) the (ordinary) distance of the vertices u and v, and by d(u, v, w) the Steiner distance of u, v, w. The degree distance DD of G is defined as the sum of terms [du+dv]d(u,v) over all pairs of vertices of G. As early as in the 1990s, a linear relation was discovered between DD of trees and the Wiener index. We now consider SDD, the Steiner-distance generalization of DD, defined as the sum of terms [du+dv+dw]d(u,v,w) over all triples of vertices of G. Also in this case, a linear relation between SDD and the Wiener index could be established.