Open Journal of Discrete Mathematics
The degree distance of a graph G is D'(G)=(1/2)∑ni=1∑nj=1(di+dj)Li ,j, where di and dj are the degrees of vertices vi, vj ∈ V (G), and Li,j is the distance between them. The Wiener index is defined as W(G)=(1/2)∑ni=1 ∑nj-1Li, j. An elegant result (Gutman; Klein, Mihalic, Plavsic and Trinajstic) is known regarding their correlation, that D'(T)=4W(T)-n(n-1)for a tree T with n vertices. In this note, we extend this study for more general graphs that have frequent appearances in the study of these indices. In particular, we develop a formula regarding their correlation, with an error term that is presented with explicit formula as well as sharp bounds for unicyclic graphs and cacti with given parameters.
Gray, Daniel, Hua Wang.
"Cycles, the Degree Distance, and the Wiener Index."
Open Journal of Discrete Mathematics, 2 (4): 156-159.