Certain Distance-Based Topological Indices of Parikh Word Representable Graphs

Nobin Thomas,Lisa Mathew,K G Subramanian,Atulya K Nagar,Sastha Sriram,Ismail Naci Cangul

doi:10.1155/2021/5567663

Abstract

Relating graph structures with words which are finite sequences of symbols, Parikh word representable graphs (PWRGs) were introduced. On the other hand, in chemical graph theory, graphs have been associated with molecular structures. Also, several topological indices have been defined in terms of graph parameters and studied for different classes of graphs. In this study, we derive expressions for computing certain topological indices of PWRGs of binary core words, thereby enriching the study of PWRGs.

Highlights

Among various studies that involve graphs for analyzing and solving different kinds of problems, relating words that are finite sequences of symbols with graphs is an interesting area of investigation
Based on the notion of subwords of a word and the concept of a matrix called Parikh matrix of a word, introduced in [6] and intensively investigated by many researchers with entries of the Parikh matrix giving the counts of certain subwords in a word, a graph called Parikh word representable graph (PWRG) of a word, was introduced in [14] and its relationship with the corresponding word and partition was studied in [15]
There has been a great interest in various topological indices associated with graphs due to their application in the area of chemical graph theory [20], which deals with representations of organic compounds or equivalently their molecular structures as graphs, with atoms other than hydrogen often represented by vertices and covalent chemical bonds by edges

Summary

Introduction

Among various studies that involve graphs for analyzing and solving different kinds of problems, relating words that are finite sequences of symbols with graphs is an interesting area of investigation (for example, [1,2,3,4,5]). In the word w aababb over the ordered binary alphabet {a < b}, the number of a’s is. In [14], a simple graph, called Parikh word representable graph (PWRG), was defined corresponding to a word over an ordered alphabet. It is to be noted that PWRG G(w) of a binary word w over {a < b} is a bipartite graph [14] with as many vertices as the length |w| of w and as many edges as the number of occurrences of the subword ab in w. We deal with only binary core words and the corresponding PWRGs. we note that for a nonempty binary core word of the form w an ban b, . We note that for a nonempty binary core word of the form w an ban b, . . . , an|w|b b, where n1 ≥ 1 and nk is nonnegative for each k, 2 ≤ k ≤ |w|b, the number of edges in the corresponding PWRG G(w) is

Figure 1:

Distance-Based Topological Indices

An Illustration