^{1}

^{2}

^{1}

^{3}

^{4}

^{4}

^{1}

^{2}

^{3}

^{4}

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.

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, [

On the other hand, there has been a great interest in various topological indices associated with graphs (for example, [

There are a number of studies (for example, [

The basic definitions and notations relating to words are as given in [

An ordered alphabet

The set of all words over an alphabet

[

Clearly, a nonempty binary word

In [

[

It is to be noted that

In this study, we deal with only binary core words and the corresponding PWRGs. Also, we note that for a nonempty binary core word of the form

Note that in the word

The

We consider only simple graphs, and for notions related to graphs, we refer to [

[

The Harary index [

The Harary index of the PWRG

We consider pairs of vertices

Both

Both

They are given as a pair of type 1, 2, 3, and 4, respectively. There are

The Harary index

The bounds are attained on

Since

[

The total eccentricity index [

The topological index, namely, eccentric connectivity index, was introduced in [

The eccentric connectivity index of the

Let

The remaining

Therefore,

Now, if

Similarly, if

It can be seen that the total eccentricity index of the

The eccentric connectivity index

The upper bound is attained on

Since

In order to maximize this expression with the constraint

On the other hand, for

The total eccentricity index

If

Yet another topological index, called eccentricity connectivity coindex [

[

The eccentric connectivity coindex

It has been shown [

Therefore,

Now, if

Therefore,

Now, if

If

Bipartite graphs have been used in investigating structural features in the areas of molecular biology and chemistry (for example, [

First, we observe the following facts relating to

For

In the graph

The degree of each vertex

We now illustrate the computation of the topological indices considered in the earlier sections, both directly from the definition and from the formulae in Theorems

The Harary index of

The same value is obtained from the formula for the Harary index, namely,

The eccentric connectivity index of

The same value is obtained from the formula for the eccentric connectivity index, namely,

The total eccentricity index of

The same value is obtained from the formula for the total eccentricity index, namely,

The eccentric connectivity coindex of

The same value is obtained from the formula for the eccentric connectivity coindex, namely,

The distance-based topological indices considered in this study have been extensively investigated by researchers for different classes of graphs, and so we were motivated to study these indices for a recently introduced special class of graphs, called PWRGs. An advantage of this study is that this provides a link between two different areas of research, namely, word combinatorics and graph theory. Specifically, we have obtained expressions for evaluating certain distance-based topological indices for PWRGs [

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

The authors would like to thank the reviewers for their very useful comments which helped to revise the study and improve the presentation of the study.