Cycles, the Degree Distance, and the Wiener Index

The degree distance of a graph G is     , 1 1 1 2 n n i j i i j D G d d L        j , where and i d j d are the degrees of vertices , and is the distance between them. The Wiener index is defined as   , i j v v V G  , i j L   , 1 1 1 2 n n i j i j W G L      . An elegant result (Gutman; Klein, Mihalić,, Plavšić and Trinajstić) is known regarding their correlation, that       4 D T W 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.