Equiseparability on Terminal Wiener Index

Wiener index as one of the oldest chemical index has been well studied. It has been extensive used in Computational Biology, Preliminary screening of drugs and Complex Network. Based on variable Wiener index, I.Gutman et al [6] introduced the concept of equiseparable pairs of trees and chemical trees, meanwhile they gave a rule on how to construct such equiseparable pairs. D.Vukičević and I.Gutman [8] proved almost all trees and chemical trees have equiseparable mates, which is a disadvantageous property of many molecular-structure graph-based descriptors. Recently, I.Gutman et al [9] proposed the concept of Terminal Wiener Index, which equals to the summation of distance between all pairs of pendent vertices of trees. Following this line, we explore the properties of terminal Wiener index, and show the fact that there still exist pairs of trees and chemical trees which can not be distinguished by it, therefore we give some general methods to construct equiseparable pairs and compare the methods in the case of Wiener index. More specifically, we show that terminal Wiener index is degenerative to some extent.