M_n – Polynomials of Some Special Graphs

Raghad Mustafa,Abdulsattar M Khidhir,Ahmed M Ali

doi:10.24996/ijs.2021.62.6.24

Raghad Mustafa, Abdulsattar M Khidhir + Show 1 more

Open Access

https://doi.org/10.24996/ijs.2021.62.6.24

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Abstract

Let be a connected graph with vertices set and edges set . The ordinary distance between any two vertices of is a mapping from into a nonnegative integer number such that is the length of a shortest path. The maximum distance between two subsets and of is the maximum distance between any two vertices and such that belong to and belong to . In this paper, we take a special case of maximum distance when consists of one vertex and consists of vertices, . This distance is defined by: where is the order of a graph . In this paper, we defined – polynomials based on the maximum distance between a vertex in and a subset that has vertices of a vertex set of and – index. Also, we find polynomials for some special graphs, such as: complete, complete bipartite, star, wheel, and fan graphs, in addition to polynomials of path, cycle, and Jahangir graphs. Then we determine the indices of these distances.

Highlights

In 1999, Dankelmann et al defined the distance between two subsets of vertices in a connected graph, as follows: The minimum distance from to is:
We find the –Polynomials for some special graphs which have a diameter equal to two, such as complete bipartite, star, wheel, and fan graphs
This paper investigated polynomials with special structures and properties based on the maximum distance between the subset of vertices of with vertices, (

Summary

Introduction

In 1999, Dankelmann et al defined the distance between two subsets of vertices in a connected graph , as follows: The minimum distance from to is:. We define the max – – distance in as the maximum distance from a singleton , to an subset. . The eccentricity of a vertex is the maximum distance between and a set of vertices. Is the sum of max – – distances of all pairs in : The -polynomial of a graph of order is denoted by and defined by :. C. By the definition of the diameter of a graph with respect to the max- n distance, we have

Main Results

The -Polynomial of Jahangir Graph

The Relation Between the - polynomial and the Hosoya polynomial

Conclusions