# Application of Resolvability Technique to Investigate the Different Polyphenyl Structures for Polymer Industry

Authors: Muhammad Faisal Nadeem; Mohsan Hassan; Muhammad Azeem; Salah Ud-Din Khan; Mohammed Rafi Shaik; Mohammed A. F. Sharaf; Abdelatty Abdelgawad; Emad Mahrous Awwad
Journal: Journal of Chemistry
 (2021)

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## Abstract

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/vertexv resolves two items v1andv2 if dv1,v≠dv2,v; similarly, the ordered subset ϕ of vertices resolves each pair of distinct vertices named as the resolving set, and its minimum cardinality is described as metric dimension. In the pharmaceutical industry, the competition to find new chemical entities for treating a disease dictates larger project teams that encompass more extensive and diverse synthetic efforts directed at increasingly complicated activity spectra. In this paper, we determine the metric dimension of para-, meta-, and ortho-polyphenyl structures, which are used for structure-activity analysis of these polyphenyl structures.


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## Body

## 1. Introduction

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.In1975, the concept of locating set was proposed by Slater [1] and called the minimum cardinality of a locating set of a graph locating number. On the same pattern, in 1976, the idea of metric dimension of a graph was individually introduced by Harary and Melter in [2], and these time metric generators were named as resolving sets. Members of metric basis set were assigned as a sonar or loran station [1].A connected, simple graphGV,E with V is the set of vertices (also can say atoms), and E is the set of edges (bond types); the distance between two vertices/bonds a1,a2∈V is the length of geodesic between them and denoted by da1,a2. Let ϕ=ϕ1,ϕ2,…,ϕl be an order subset of vertices belonging to a graph G and a be a vertex. The representation ra|ϕ of a corresponding to ϕ is the l-tuple da,ϕ1,da,ϕ2,da,ϕ3,…,da,ϕl, where ϕ is called a resolving set [2] or locating set [1], if every vertex of G is uniquely determined by its distances from the vertices of ϕ or, on the contrary, if different vertices of G have unique representations with respect to ϕ. The minimum cardinality of the resolving set ϕ is called the metric dimension of G, and it is denoted by dimG [1]. For a given ordered set of vertices ϕ=ϕ1,ϕ2,…,ϕl⊂V, the cth location of ra|ϕ=0 if and only if a=ϕc. Thus, to verify that ϕ is a resolving set, it is enough to show that ra1|ϕ≠ra2|ϕ for every possible distinct pair of vertices a1,a2∈VG\ϕ.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 [3, 4], robot navigation [5], combinatorial optimization concept studied in [6], various coin weighing problems [7, 8], and utilization of the idea in pattern recognition and processing of images, few of which also associate with the use in hierarchical data structures [1].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 andVC5C7 nanotubes discussed with metric concept [9], some upper bounds of cellulose network considering metric dimension as a point of discussion [10], resolving sets of silicate star determined in [11], metric basis of 2D lattice of alpha-boron nanotubes discussed with specific applications [12], and sharps bound on the metric dimension of honeycomb and its related network [13]; for more interesting literature work on metric dimension, metric basis, resolving set, and other resolvability parameters, refer to [13–28].


## 2. Results of Polyphenyl Chemical Networks

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, usingη=6 with arbitrary h in Theorems 1–5, we can produce the para-, meta-, and ortho-polyphenyl chain of hexagons and retrieve its corresponding metric dimension as well.


### 2.1. Metric Dimension ofOη,h

LetOη,h be a connected graph of ortho-polyphenyl network of cycle graph Cη, and h are the copies of cycle graph with order λ=ηh and size hη+1−1. For the following theorems, Figure 1 shows the resolving set in dark black vertices.Figure 1
Oη,h.Theorem 1.
Ifη≥3 and h=2, then dimOη,2 is 2.Proof.
To prove thatdimOη,2≤2, for this assume, a resolving set ϕ=v1,vη+1. We construct the following cases on vertex set of Oη,2:(1)r1vζ|ϕ=ζ−1;ifζ=1,2,3,…,η2+1;η−ζ+1;ifζ=η2+2,…,η;ζ−η+2;ifζ=η+1,η+2,…,η+η2;2η−ζ+2;ifζ=η+η2+1,…,2η.
Second vector representations are as follows:(2)r2vζ|ϕ=ζ+2;ifζ=1,2,3,…,η−12,η−ζ+2;ifζ=η−12+1,…,η,ζ−η;ifζ=η+1,η+2,…,η+η+22,2η−ζ+1;ifζ=η+1+η+22,η+2+η+22,…,2η.
Hence, it follows from the above arguments in the form of representation thatdimOη,2≤2 because all the vertices of Oη,2 have unique representations with respect to resolving set ϕ.
For reverse inequality thatdimOη,2≥2, by contradiction, our assertion becomes dimOη,2<2, implying that dimOη,2=1, and it is not possible because only the path graph exists having the metric dimension1. All discussion concludes that when η≥3 and h=2,(3)dimOη,2=2.Theorem 2.
Ifη,h≥3, then dimOη,h is h.Proof.
To show thatdimOη,h=h, we will apply the induction method on h the number of copies of base graph. The base case for h=2 is proved in Theorem 1; now, assume that the assertion is true for h=m:(4)dimOη,m=m.
We will show that it is true forh=m+1. Suppose(5)dimOη,m+1=dimOη,m+dimOη,2−1.Using equations (3) and (4) in equation (5), we will get(6)dimOη,m+1=m+2−1=m+1.
Hence, the result is true for all positive integersh≥3.


### 2.2. Metric Dimension ofMη,h

LetMη,h be a connected graph of meta-polyphenyl network of cycle graph Cη, and h are the copies of cycle graph with order λ=ηh and size hη+1−1. For the following theorems, Figure 2 shows the resolving set in dark black vertices.Figure 2
Mη,h.Theorem 3.
Ifη≥4 (even) and h=2, then dimMη,2 is 2.Proof.
To prove thatdimMη,2≤2, we construct a resolving set ϕ=v1,vη+2 from the vertex set of Mη,2. We assume the following cases on vertex set of Mη,2:(7)rvζ|ϕ=ζ+12,η+42−ζ2,ζ=1,3,5,…,η−1,rvζ|ϕ=ζ−12,η+22−ζ−12,ζ=2,4,6,…,η,rvζ|ϕ=ζ−12,ζ−η+12,ζ=η+1,η+3,η+5,…,2η−1,rvζ|ϕ=ζ2,ζ−η−22,ζ=η+2,η+4,η+6,…,2η.
Hence, it follows from the above discussion thatdimMη,2≤2 because all the vertices of Mη,2 have unique representations with respect to resolving set ϕ.
For conversedimMη,2≥2, we use contradiction, and dimMη,2=1 is not possible because only the path graph exists having the metric dimension1. All discussion concludes that when η≥4 (even) and h=2,(8)dimMη,2=2.Theorem 4.
Ifη≥4 (even) and h≥3, then dimMη,h is h.Proof.
To show thatdimMη,h=h, we will apply the induction method on h showing the copies of base graph. The base case for h=2 is proved in Theorem 3; now, assume that the assertion is true for h=m:(9)dimMη,m=m.
We will show that it is true forh=m+1. Suppose(10)dimMη,m+1=dimMη,m+dimMη,2−1.Using equations (8) and (9) in equation (10), we have(11)dimMη,m+1=m+2−1=m+1.
Hence, the result is true for all positive integersh≥3.


### 2.3. Metric Dimension ofLη,h

LetLη,h be a connected graph of para-polyphenyl network of cycle graph Cη, and h are the copies of cycle graph with order λ=ηh and size hη+1−1. For the following theorems, vertices are labeled, as shown in Figure 3; moreover, it also shows the resolving set in dark black vertices.Figure 3
Lη,h.Theorem 5.
Ifη≥5 and h≥2, then dimLη,h is 2.Proof.
Firstly, we prove thatdimLη,h≤2; for this construction, a resolving set ϕ=v1,vλ−η+1 from the vertex set of Lη,h. We assume the following cases on vertex set of G and on the copies of cycle graph, i.e., h:(12)r1vζ|ϕ=ζ−1,ifζ=1,2,3,…,η+12,η−ζ+1,ifζ=η+12+1,η+12+2,…,η,3ζη+ζ−1−ηζη,ifζ≡1,2,3,…,η−12modη,ηζη+2−ζ,ifζ≡0,η−12+1,η−12+2,…,η−1modη.
Ifζ≡0,η−1/2+1,η−1/2+2,…,η−1modηandζ>2η,(13)r1vζ|ϕ=ηζ+η−1/2+1η−ζ+2+3ζ+η−1/2+1η−2.
Second vector representations are as follows:Case 1.h=2:(14)r2vζ|ϕ=3,ifζ=1;ζ,ifζ=2,3,…,η+22,η−ζ+4,ifζ=η+22+1,…,η.Case 2.h≥3:(15)r2vζ|ϕ=3ηh−1+1η−1+ζ,ifζ=1,3ζη−1+ζ,ifζ=2,3,…,η+22,η−ζ+4+3ηh−1+1η−1,ifζ=η+22+1,…,η.Subcase 2.1. Ifζ=ηh−1+1,ηh−1+2,…,ηh−1+η/2,(16)r2vζ|ϕ=ζ−ηh−1−1.Subcase 2.2. Ifζ=ηh−1+η/2+1,ηh−1+η/2+2,…,ηh,(17)r2vζ|ϕ=ηh−ζ+1.Subcase 2.3. Ifζ≡2,3,…,η+2/2modη, andζ≥η+1,(18)r2vζ|ϕ=3h−ζ−1η−2+2+ζ−2+ηζη.Subcase 2.4. Ifζ≡0,η+2/2+1,…,η−1modη, andζ≥η+1,(19)r2vζ|ϕ=3h−ζ−1η−2+4−ζ+ηζ+η+2/2η.Subcase 2.5. Ifζ≡1modη, andζ≥η+1,(20)r2vζ|ϕ=3h−ζη−1.
Hence, it follows from the above discussion thatdimLη,h≤2 because all the vertices of Lη,h have unique representations with respect to resolving set ϕ.
For reverse inequality thatdimLη,h≥2, by contradiction, our assertion becomes dimLη,h<2, implying that dimLη,h=1, and it is not possible because only the path graph exists having the metric dimension 1. All discussion concluding that when η≥5 and h≥2,(21)dimLη,h=2.


### 2.4. Metric Dimension ofLSη,h

LetLSη,h be a connected graph of para-polyphenyl network of sun graph Sη, where h are the copies of sun graph and η is the order of interior cycle of sun graph. The order and size of network are VLSη,h=λ=2ηh and ELSη,h=h2η+1−1, respectively. The vertices are labeled, as shown in Figure 4.Figure 4
LSη,h.Theorem 6.
Ifη≥5 (odd) and h≥2, then dimLSη,h is 2.Proof.
To prove thatdimLSη,h≤2, for this construct, a resolving set ϕ=vη+3/2,vξ, where ξ=η4h−3−1/2 from the vertex set of LSη,h. We assume the following cases on vertex set of LSη,h:Case 1.1≤ζ≤2η:(22)r1vζ|ϕ=η+32−ζ,ifζ=1,2,…,η+32,ζ−η+32,ifζ=η+52,η+52,…,η,η+12,ifζ=η+1,3η+52−ζ,ifζ=η+2,η+3,…,3η+32,ζ−3η+12,ifζ=3η+52,3η+72,…,2η.Case 2. Ifζ≡η+1/2,η+3/2,…,ηmod2η, ζ≥5η+1/2,(23)r1vζ|ϕ=η−12+3η+2ζ−13ηη−ζ+3ζ2η+2ζ2η−1.Case 3. Ifζ≡1,2,…,η−1/2mod2η, ζ≥2η+1,(24)r1vζ|ϕ=η−12+ζ+5ζ2η−η4h−3−12+η−52.Case 4. Ifζ≡η+1,η+2,…,3η−1/2mod2η, ζ≥3η+1,(25)r1vζ|ϕ=ζ−η4ζ/2η+1−3−12+3ζ2η+2ζ2η−1.Case 5. Ifζ≡0,3η+1/2,3η+3/2,…,2η−1mod2η, ζ≥7η+1/2,(26)r1vζ|ϕ=η−12+4ζ−12η+2ζ2η−1−ζ+η2ζ−12η2−ζ−η−13η+z,where z=−2 when ζ≡0mod2η and ζ≥4η; otherwise, z=0.
The representations of all vertices with respect to the second vertex of resolving set are as follows:Case 1.1≤ζ≤3η+1:(27)r2vζ|ϕ=ζ−ξ,ifζ=ξ+1,ξ+2,…,2ξ+η−12,η+12,ifζ=2ξ+η+12,ξ−ζ,ifζ=2ξ+η+12,2ξ+η+12+1,…,ξ,ζ−ξ−η+1,ifζ=ξ+η,ξ+η+1,ξ+η+2,…,2ξ+3η−12,η+32,ifζ=2ξ+3η+12,η−12,ifζ=3η+1.Case 2. Ifζ≡2,3,4,…,η+3/2mod2η,(28)r2vζ|ϕ=η−12+3h−ζ2η−1+2h−2−ζ2η+ζ−2+2ηζ2η.Case 3. Ifζ≡η+5/2,η+7/2,…,ηmod2η,(29)r2vζ|ϕ=3η+32+3h−1−ζ2η+2h−2−ζ2η−ζ+2ηζ2η.Case 4. Ifζ≡1mod2η,(30)r2vζ|ϕ=η−12+3h−ζ2η−1+2h−2−ζ2η+ζ−2ηζ2η.Case 5. Ifζ≡η+2,η+3,…,3η+3/2mod2η,(31)r2vζ|ϕ=η−32+3h−ζ2η−1+2h−2−ζ2η+ζ−2ηζ2η−η+z,wherez=−2 when ζ≡7mod2η and, otherwise, z=0.Case 6. Ifζ≡0,3η+5/2,3η+7/2,…,2η−1mod2η,(32)r2vζ|ϕ=5η+52+3h−ζ2η−1+2h−2−ζ2η−ζ+2ηζ2η.Case 7. Ifζ≡η+1mod2η,(33)r2vζ|ϕ=3η+52+3h−ζ2η−1+2h−2−ζ2η−ζ+2ηζ2η.
Hence, it follows from the above discussion thatdimLSη,h≤2 because all the vertices of LSη,h have unique representations with respect to resolving set ϕ. For reverse inequality that dimLSη,h≥2, by contradiction, our assertion becomes dimLSη,h<2, implying dimLSη,h=1, and it is not possible because only the path graph exists having the metric dimension1. All discussion concluding that when η≥5 (odd) and h≥2, (34)dimLSη,h=2.


## 2.1. Metric Dimension ofOη,h

LetOη,h be a connected graph of ortho-polyphenyl network of cycle graph Cη, and h are the copies of cycle graph with order λ=ηh and size hη+1−1. For the following theorems, Figure 1 shows the resolving set in dark black vertices.Figure 1
Oη,h.Theorem 1.
Ifη≥3 and h=2, then dimOη,2 is 2.Proof.
To prove thatdimOη,2≤2, for this assume, a resolving set ϕ=v1,vη+1. We construct the following cases on vertex set of Oη,2:(1)r1vζ|ϕ=ζ−1;ifζ=1,2,3,…,η2+1;η−ζ+1;ifζ=η2+2,…,η;ζ−η+2;ifζ=η+1,η+2,…,η+η2;2η−ζ+2;ifζ=η+η2+1,…,2η.
Second vector representations are as follows:(2)r2vζ|ϕ=ζ+2;ifζ=1,2,3,…,η−12,η−ζ+2;ifζ=η−12+1,…,η,ζ−η;ifζ=η+1,η+2,…,η+η+22,2η−ζ+1;ifζ=η+1+η+22,η+2+η+22,…,2η.
Hence, it follows from the above arguments in the form of representation thatdimOη,2≤2 because all the vertices of Oη,2 have unique representations with respect to resolving set ϕ.
For reverse inequality thatdimOη,2≥2, by contradiction, our assertion becomes dimOη,2<2, implying that dimOη,2=1, and it is not possible because only the path graph exists having the metric dimension1. All discussion concludes that when η≥3 and h=2,(3)dimOη,2=2.Theorem 2.
Ifη,h≥3, then dimOη,h is h.Proof.
To show thatdimOη,h=h, we will apply the induction method on h the number of copies of base graph. The base case for h=2 is proved in Theorem 1; now, assume that the assertion is true for h=m:(4)dimOη,m=m.
We will show that it is true forh=m+1. Suppose(5)dimOη,m+1=dimOη,m+dimOη,2−1.Using equations (3) and (4) in equation (5), we will get(6)dimOη,m+1=m+2−1=m+1.
Hence, the result is true for all positive integersh≥3.


## 2.2. Metric Dimension ofMη,h

LetMη,h be a connected graph of meta-polyphenyl network of cycle graph Cη, and h are the copies of cycle graph with order λ=ηh and size hη+1−1. For the following theorems, Figure 2 shows the resolving set in dark black vertices.Figure 2
Mη,h.Theorem 3.
Ifη≥4 (even) and h=2, then dimMη,2 is 2.Proof.
To prove thatdimMη,2≤2, we construct a resolving set ϕ=v1,vη+2 from the vertex set of Mη,2. We assume the following cases on vertex set of Mη,2:(7)rvζ|ϕ=ζ+12,η+42−ζ2,ζ=1,3,5,…,η−1,rvζ|ϕ=ζ−12,η+22−ζ−12,ζ=2,4,6,…,η,rvζ|ϕ=ζ−12,ζ−η+12,ζ=η+1,η+3,η+5,…,2η−1,rvζ|ϕ=ζ2,ζ−η−22,ζ=η+2,η+4,η+6,…,2η.
Hence, it follows from the above discussion thatdimMη,2≤2 because all the vertices of Mη,2 have unique representations with respect to resolving set ϕ.
For conversedimMη,2≥2, we use contradiction, and dimMη,2=1 is not possible because only the path graph exists having the metric dimension1. All discussion concludes that when η≥4 (even) and h=2,(8)dimMη,2=2.Theorem 4.
Ifη≥4 (even) and h≥3, then dimMη,h is h.Proof.
To show thatdimMη,h=h, we will apply the induction method on h showing the copies of base graph. The base case for h=2 is proved in Theorem 3; now, assume that the assertion is true for h=m:(9)dimMη,m=m.
We will show that it is true forh=m+1. Suppose(10)dimMη,m+1=dimMη,m+dimMη,2−1.Using equations (8) and (9) in equation (10), we have(11)dimMη,m+1=m+2−1=m+1.
Hence, the result is true for all positive integersh≥3.


## 2.3. Metric Dimension ofLη,h

LetLη,h be a connected graph of para-polyphenyl network of cycle graph Cη, and h are the copies of cycle graph with order λ=ηh and size hη+1−1. For the following theorems, vertices are labeled, as shown in Figure 3; moreover, it also shows the resolving set in dark black vertices.Figure 3
Lη,h.Theorem 5.
Ifη≥5 and h≥2, then dimLη,h is 2.Proof.
Firstly, we prove thatdimLη,h≤2; for this construction, a resolving set ϕ=v1,vλ−η+1 from the vertex set of Lη,h. We assume the following cases on vertex set of G and on the copies of cycle graph, i.e., h:(12)r1vζ|ϕ=ζ−1,ifζ=1,2,3,…,η+12,η−ζ+1,ifζ=η+12+1,η+12+2,…,η,3ζη+ζ−1−ηζη,ifζ≡1,2,3,…,η−12modη,ηζη+2−ζ,ifζ≡0,η−12+1,η−12+2,…,η−1modη.
Ifζ≡0,η−1/2+1,η−1/2+2,…,η−1modηandζ>2η,(13)r1vζ|ϕ=ηζ+η−1/2+1η−ζ+2+3ζ+η−1/2+1η−2.
Second vector representations are as follows:Case 1.h=2:(14)r2vζ|ϕ=3,ifζ=1;ζ,ifζ=2,3,…,η+22,η−ζ+4,ifζ=η+22+1,…,η.Case 2.h≥3:(15)r2vζ|ϕ=3ηh−1+1η−1+ζ,ifζ=1,3ζη−1+ζ,ifζ=2,3,…,η+22,η−ζ+4+3ηh−1+1η−1,ifζ=η+22+1,…,η.Subcase 2.1. Ifζ=ηh−1+1,ηh−1+2,…,ηh−1+η/2,(16)r2vζ|ϕ=ζ−ηh−1−1.Subcase 2.2. Ifζ=ηh−1+η/2+1,ηh−1+η/2+2,…,ηh,(17)r2vζ|ϕ=ηh−ζ+1.Subcase 2.3. Ifζ≡2,3,…,η+2/2modη, andζ≥η+1,(18)r2vζ|ϕ=3h−ζ−1η−2+2+ζ−2+ηζη.Subcase 2.4. Ifζ≡0,η+2/2+1,…,η−1modη, andζ≥η+1,(19)r2vζ|ϕ=3h−ζ−1η−2+4−ζ+ηζ+η+2/2η.Subcase 2.5. Ifζ≡1modη, andζ≥η+1,(20)r2vζ|ϕ=3h−ζη−1.
Hence, it follows from the above discussion thatdimLη,h≤2 because all the vertices of Lη,h have unique representations with respect to resolving set ϕ.
For reverse inequality thatdimLη,h≥2, by contradiction, our assertion becomes dimLη,h<2, implying that dimLη,h=1, and it is not possible because only the path graph exists having the metric dimension 1. All discussion concluding that when η≥5 and h≥2,(21)dimLη,h=2.


## 2.4. Metric Dimension ofLSη,h

LetLSη,h be a connected graph of para-polyphenyl network of sun graph Sη, where h are the copies of sun graph and η is the order of interior cycle of sun graph. The order and size of network are VLSη,h=λ=2ηh and ELSη,h=h2η+1−1, respectively. The vertices are labeled, as shown in Figure 4.Figure 4
LSη,h.Theorem 6.
Ifη≥5 (odd) and h≥2, then dimLSη,h is 2.Proof.
To prove thatdimLSη,h≤2, for this construct, a resolving set ϕ=vη+3/2,vξ, where ξ=η4h−3−1/2 from the vertex set of LSη,h. We assume the following cases on vertex set of LSη,h:Case 1.1≤ζ≤2η:(22)r1vζ|ϕ=η+32−ζ,ifζ=1,2,…,η+32,ζ−η+32,ifζ=η+52,η+52,…,η,η+12,ifζ=η+1,3η+52−ζ,ifζ=η+2,η+3,…,3η+32,ζ−3η+12,ifζ=3η+52,3η+72,…,2η.Case 2. Ifζ≡η+1/2,η+3/2,…,ηmod2η, ζ≥5η+1/2,(23)r1vζ|ϕ=η−12+3η+2ζ−13ηη−ζ+3ζ2η+2ζ2η−1.Case 3. Ifζ≡1,2,…,η−1/2mod2η, ζ≥2η+1,(24)r1vζ|ϕ=η−12+ζ+5ζ2η−η4h−3−12+η−52.Case 4. Ifζ≡η+1,η+2,…,3η−1/2mod2η, ζ≥3η+1,(25)r1vζ|ϕ=ζ−η4ζ/2η+1−3−12+3ζ2η+2ζ2η−1.Case 5. Ifζ≡0,3η+1/2,3η+3/2,…,2η−1mod2η, ζ≥7η+1/2,(26)r1vζ|ϕ=η−12+4ζ−12η+2ζ2η−1−ζ+η2ζ−12η2−ζ−η−13η+z,where z=−2 when ζ≡0mod2η and ζ≥4η; otherwise, z=0.
The representations of all vertices with respect to the second vertex of resolving set are as follows:Case 1.1≤ζ≤3η+1:(27)r2vζ|ϕ=ζ−ξ,ifζ=ξ+1,ξ+2,…,2ξ+η−12,η+12,ifζ=2ξ+η+12,ξ−ζ,ifζ=2ξ+η+12,2ξ+η+12+1,…,ξ,ζ−ξ−η+1,ifζ=ξ+η,ξ+η+1,ξ+η+2,…,2ξ+3η−12,η+32,ifζ=2ξ+3η+12,η−12,ifζ=3η+1.Case 2. Ifζ≡2,3,4,…,η+3/2mod2η,(28)r2vζ|ϕ=η−12+3h−ζ2η−1+2h−2−ζ2η+ζ−2+2ηζ2η.Case 3. Ifζ≡η+5/2,η+7/2,…,ηmod2η,(29)r2vζ|ϕ=3η+32+3h−1−ζ2η+2h−2−ζ2η−ζ+2ηζ2η.Case 4. Ifζ≡1mod2η,(30)r2vζ|ϕ=η−12+3h−ζ2η−1+2h−2−ζ2η+ζ−2ηζ2η.Case 5. Ifζ≡η+2,η+3,…,3η+3/2mod2η,(31)r2vζ|ϕ=η−32+3h−ζ2η−1+2h−2−ζ2η+ζ−2ηζ2η−η+z,wherez=−2 when ζ≡7mod2η and, otherwise, z=0.Case 6. Ifζ≡0,3η+5/2,3η+7/2,…,2η−1mod2η,(32)r2vζ|ϕ=5η+52+3h−ζ2η−1+2h−2−ζ2η−ζ+2ηζ2η.Case 7. Ifζ≡η+1mod2η,(33)r2vζ|ϕ=3η+52+3h−ζ2η−1+2h−2−ζ2η−ζ+2ηζ2η.
Hence, it follows from the above discussion thatdimLSη,h≤2 because all the vertices of LSη,h have unique representations with respect to resolving set ϕ. For reverse inequality that dimLSη,h≥2, by contradiction, our assertion becomes dimLSη,h<2, implying dimLSη,h=1, and it is not possible because only the path graph exists having the metric dimension1. All discussion concluding that when η≥5 (odd) and h≥2, (34)dimLSη,h=2.


## 3. Conclusion

We found the metric dimension of some chemical networks ortho-, meta-, and para-polyphenyl chains constructed with base graphCη and sun graph Sη, and these networks have metric dimension dimOη,h=dimMη,h=h and dimLη,h=2.



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*Source: 6633227-2021-04-20.xml*