Cycles, the Degree Distance, and the Wiener Index

The degree distance of a graph G is D'(G)=(1/2)∑ ⁿ _i=1 ∑ ⁿ _{j= 1} ( d _i +d _j )L _{i ,j} , where d _i and d _j are the degrees of vertices v _i , v _j ∈ V (G) , and L _i,j is the distance between them. The Wiener index is defined as W(G) =(1/2)∑ ⁿ i ₌₁ ∑ ⁿ _j-1 L _{i, 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.