Weiner Polynomials for Generalization of Distance for Some Special Graphs

Ali Ali ,Ahmed M Ali

doi:10.33899/csmj.2006.164061

Abstract

The minimum distance of a vertex v to an set of vertices of a graph G is defined as : . The n-Wiener polynomial for this distance of a graph G is defined as , where is the number of order pairs (v,S), , such that , and is the diameter for this minimum n-distance. In this paper, the n-Wiener polynomials for some types of graphs such as complete graphs, bipartite graphs, star graphs, wheel graphs, path and cycle graphs are obtained .The n-Wiener index for each of these special graphs is given. Moreover, some properties of the coefficients of are established.

Highlights

The n-Wiener polynomials for some types of graphs such as complete graphs, bipartite graphs, star graphs, wheel graphs, path and cycle graphs are obtained .The n-Wiener index for each of these special graphs is given.
‫)‪e(v) − n + 1 emin (v,n) eave (v,n) emax (v,n) = e(v‬‬ ‫استنادا إلى الا تعف المركزي‪ n 2 ، n-‬يعرف القطر‪ n-‬ونغف القطر‪ n-‬كالآتي‪:‬‬ ‫نصف القطر‪ :(n-radius) n-‬لبيا ‪ G‬هو صىر الا تعةات المركزية‪ n-‬لكل ؤوو البيا ‪، G‬‬
‫} ‪n − radM (G) = min{eM (v,n) : v V‬‬ ‫و قطر‪ :(n-diameter) n-‬لبيا ‪ G‬هو عظم الا تعةات المركزية‪ n-‬لكل ؤوو‬

Summary

Introduction

The n-Wiener polynomials for some types of graphs such as complete graphs, bipartite graphs, star graphs, wheel graphs, path and cycle graphs are obtained .The n-Wiener index for each of these special graphs is given. ‫)‪e(v) − n + 1 emin (v,n) eave (v,n) emax (v,n) = e(v‬‬ ‫استنادا إلى الا تعف المركزي‪ n 2 ، n-‬يعرف القطر‪ n-‬ونغف القطر‪ n-‬كالآتي‪:‬‬ ‫نصف القطر‪ :(n-radius) n-‬لبيا ‪ G‬هو صىر الا تعةات المركزية‪ n-‬لكل ؤوو البيا ‪، G‬‬ ‫} ‪n − radM (G) = min{eM (v,n) : v V‬‬ ‫و قطر‪ :(n-diameter) n-‬لبيا ‪ G‬هو عظم الا تعةات المركزية‪ n-‬لكل ؤوو‬ ‫البيا ‪ ، G‬ي‬ ‫} ‪n − diamM (G) = max{eM (v,n) : v V‬‬ ‫ويمكل نعرف قطر ‪ n-‬ونغف القطر‪ n-‬للبيا ‪ G‬بالاعتماد على الا تعف المركزي‬

Results

Conclusion