Relationship Between the Hosoya Polynomial and the Edge-Hosoya   Polynomial of Trees
  We prove the relationship between the Hosoya polynomial and the edge-Hosoya polynomial of trees. The connection between the edge-hyper-Wiener index and the edge-Hosoya polynomial is established. With these results we also prove formulas for the computation of the edge-Wiener index and the edge-hyper-Wiener index of trees using the Wiener index and the hyper-Wiener index. Moreover, the closed formulas are derived for a family of chemical trees called regular dendrimers. 
Niko Tratnik, Petra \v{Z}igert Pleter\v{s}ek
math.CO