Extension of the Wiener index and Wiener polynomial

The Wiener index W(G)W(G) of a connected graph GG is defined as the sum of distances between all pairs of vertices. The Wiener polynomial H(G,x)H(G,x) has the property that its first derivative evaluated at x=1x=1 equals the Wiener index, i.e. H′(G,1)=W(G)H′(G,1)=W(G). The hyper-Wiener polynomial HH(G,x)HH(G,x) satisfies the condition HH′(G,1)=WW(G)HH′(G,1)=WW(G), the hyper-Wiener index of GG. In this paper we introduce a new generalization W(G,y)W(G,y) of the Wiener index and H(G,x,y)H(G,x,y) of the Wiener polynomial. One of the advantages of our definitions is that one can handle the Wiener and hyper-Wiener index (respectively polynomial) with the same formula, i.e. W(G)=W(G,1)W(G)=W(G,1), WW(G)=W(G,2)WW(G)=W(G,2), H(G,x)=H(G,x,1)H(G,x)=H(G,x,1) and HH(G,x)=H(G,x,2)HH(G,x)=H(G,x,2).