On Some W-Hosoya polynomials for Several Special Connected Graphs

Lee Xu, Taher^1,*, Ahmed Jubbori²

¹University of Chinese Academy of Sciences, CAS, Mathematics Department, Beijing, China

²Computer Techniques Engineering Department, Al-Mustaqbal University, Babil, Iraq

Emails: Leexu1244@yahoo.com; taherajubbori@mustaqbal-college.edu.iq

Abstract

Let u and v be any two distinct vertices in a connected graph G. A container C(u,v) is a set of internally disjoint u - v paths. The width of C(u,v) is denoted by w or w(C(u,v)), it is equal to , and the length of  is the length of the longest u – v path in C(u,v) . Then, for a given positive integer w, the width distance between any two distinct vertices u and v in a connected graph G is define by:

, where the minimum is taken over all containers C(u, v) of width w.

In this paper, we find the Hosoya polynomials and Wiener indices of the join of two special graphs such as bipartite complete graphs, paths, cycles, star graphs and wheel graphs with respect to the width distance.

Keywords: Connected graph; Path, Width; Hosoya polynomial