Eccentric topological properties of a graph associated to a finite dimensional vector space

Jia-Bao Liu,Faisal Ali,Masood Ur Rehman,Mohammad Tariq Rahim,Imran Khalid,Muhammad Salman

doi:10.1515/mgmc-2020-0020

Abstract

Abstract A topological index is actually designed by transforming a chemical structure into a number. Topological index is a graph invariant which characterizes the topology of the graph and remains invariant under graph automorphism. Eccentricity based topological indices are of great importance and play a vital role in chemical graph theory. In this article, we consider a graph (non-zero component graph) associated to a finite dimensional vector space over a finite filed in the context of the following eleven eccentricity based topological indices: total eccentricity index; average eccentricity index; eccentric connectivity index; eccentric distance sum index; adjacent distance sum index; connective eccentricity index; geometric arithmetic index; atom bond connectivity index; and three versions of Zagreb indices. Relationship of the investigated indices and their dependency with respect to the involved parameters are also visualized by evaluating them numerically and by plotting their results.

Highlights

Let a connected graph G having the sets V(G) and E(G) as the vertex set and the edge set, respectively
Topological index is a graph invariant which characterizes the topology of the graph and remains invariant under graph automorphism
We consider a graph associated to a finite dimensional vector space over a finite filed in the context of the following eleven eccentricity based topological indices: total eccentricity index; average eccentricity index; eccentric connectivity index; eccentric distance sum index; adjacent distance sum index; connective eccentricity index; geometric arithmetic index; atom bond connectivity index; and three versions of Zagreb indices

Summary

Introduction

Let a connected graph G having the sets V(G) and E(G) as the vertex set and the edge set, respectively. Liu et al.: Eccentric topological properties of a graph associated to a finite dimensional vector space 165 number (Wiener, 1947). This index is based on the concept of distance, mathematically, defined as:. We extend the study of eccentricity topological indices, based on the eccentricity, in chemical graph theory by involving an algebraic structures called a vector space. We consider a graph associated to a finite dimensional vector space over a finite field, and compute all the indices listed in the Table 1

Graph associated to a vector space

Construction

Results

Comparisons and plots