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Polyphenyl is used in a variety of applications including high-vacuum devices, optics, and electronics, and in high-temperature and radiation-resistant fluids and greases, it has low volatility, ionizing radiation stability, and high thermal-oxidative properties. The structure of polyphenyls can be represented using a molecular graph, where atoms represent vertices and bonds between atom edges. In a chemical structure, an item/vertex

Chemists require the mathematical representation of a chemical compound to work with the chemical structure. In a chemical structure, a set of selected atoms gave mathematical representations so that it gave distinct representations to distinct atoms of the structure. The chemical structure can be defined in the form of vertices, which mentions the atom and edges indicate the bonds types, respectively. Thus, a graph-theoretic analysis of this idea yields the representations of all vertices in a structure in such a way that different vertices have distinct representations with respect to some specific atoms of that structure. The following are some mathematical definitions to indicate these concepts.

In

A connected, simple graph

Metric dimension of a graph or a structure is a resolvability parameter that has been applied in numerous applications of graph theory, for the drug discovery in pharmaceutical chemistry [

Due to numerous uses of resolvability parameters in the chemical field, many works have been done with graph perspectives, and metric dimension is also considered important to study different structures with it, such as the structure of H-naphtalenic and

In the results of this article, we discuss the metric dimension of para-, meta-, and ortho-polyphenyl chemical networks constructed by different polygons. Usually, the networks are made up with the chain of hexagons using chemical operations ortho, para, and meta; in this work, we extend this to any order of polygons. Moreover, using

Let

If

To prove that

Second vector representations are as follows:

Hence, it follows from the above arguments in the form of representation that

For reverse inequality that

If

To show that

We will show that it is true for

Hence, the result is true for all positive integers

Let

If

To prove that

Hence, it follows from the above discussion that

For converse

If

To show that

We will show that it is true for

Hence, the result is true for all positive integers

Let

If

Firstly, we prove that

If

Second vector representations are as follows:

Case 1.

Case 2.

Subcase 2.1. If

Subcase 2.2. If

Subcase 2.3. If

Subcase 2.4. If

Subcase 2.5. If

Hence, it follows from the above discussion that

For reverse inequality that

Let

If

To prove that

Case 1.

Case 2. If

Case 3. If

Case 4. If

Case 5. If

The representations of all vertices with respect to the second vertex of resolving set are as follows:

Case 1.

Case 2. If

Case 3. If

Case 4. If

Case 5. If

where

Case 6. If

Case 7. If

Hence, it follows from the above discussion that

We found the metric dimension of some chemical networks ortho-, meta-, and para-polyphenyl chains constructed with base graph

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

The authors declare that there are no conflicts of interest.

All the authors contributed equally to prepare this article.

The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through Research Group no. RG-1441-453.