ApplicationofResolvabilityTechnique to Investigate theDifferent Polyphenyl Structures for Polymer Industry

Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan Department of Aerospace Engineering, Faculty of Engineering, University Putra Malaysia, Seri Kembangan, Malaysia Sustainable Energy Technologies (SET) Center, College of Engineering, King Saud University, P O. Box 800, Riyadh 11421, Saudi Arabia Department of Chemistry, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia Department of Industrial Engineering, College of Engineering, King Saud University, P.O. Box 800, Riyadh-11421, Saudi Arabia Department of Electrical Engineering, College of Engineering, King Saud University, P.O. Box 800, Riyadh, Saudi Arabia


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. e chemical structure can be defined in the form of vertices, which mentions the atom and edges indicate the bonds types, respectively.
us, 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. e following are some mathematical definitions to indicate these concepts.
In 1975, 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 graph G(V, 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 a 1 , a 2 ∈ V is the length of geodesic between them and denoted by d(a 1 , a 2 ). Let ϕ � ϕ 1 , ϕ 2 , . . . , ϕ l be an order subset of vertices belonging to a graph G and a be a vertex. e representation r(a|ϕ) of a corresponding to ϕ is the l-tuple (d(a, ϕ 1 ), d(a, ϕ 2 ), d(a, ϕ 3 ), . . . , d(a, ϕ l )), 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. e minimum cardinality of the resolving set ϕ is called the metric dimension of G, and it is denoted by dim(G) [1]. For a given ordered set of vertices ϕ � ϕ 1 , ϕ 2 , . . . , ϕ l ⊂ V, the cth location of r(a|ϕ) � 0 if and only if a � ϕ c . us, to verify that ϕ is a resolving set, it is enough to show that r(a 1 |ϕ) ≠ r(a 2 |ϕ) for every possible distinct pair of vertices a 1 , a 2 ∈ V(G)\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 and VC 5 C 7 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 2 D 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][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28].

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, using η � 6 with arbitrary h in eorems 1-5, we can produce the para-, meta-, and ortho-polyphenyl chain of hexagons and retrieve its corresponding metric dimension as well.

Metric Dimension of O(η, h).
Let O(η, h) be a connected graph of ortho-polyphenyl network of cycle graph C η , and h are the copies of cycle graph with order λ � ηh and size h(η + 1) − 1. For the following theorems, Figure 1 shows the resolving set in dark black vertices.
Proof. To prove that dim(O(η, 2)) ≤ 2, for this assume, a resolving set ϕ � v 1 , v η+1 . We construct the following cases on vertex set of O(η, 2): Second vector representations are as follows: Hence, it follows from the above arguments in the form of representation that dim(O(η, 2)) ≤ 2 because all the vertices of O(η, 2) have unique representations with respect to resolving set ϕ.
For reverse inequality that dim(O(η, 2)) ≥ 2, by contradiction, our assertion becomes dim(O(η, 2)) < 2, implying that dim(O(η, 2)) � 1, and it is not possible because only the path graph exists having the metric dimension 1. All discussion concludes that when η ≥ 3 and h � 2, dim(O(η, 2)) � 2. (3) Proof. To show that dim (O(η, h)) � h, we will apply the induction method on h the number of copies of base graph. e base case for h � 2 is proved in eorem 1; now, assume that the assertion is true for h � m: We will show that it is true for h � m + 1. Suppose Using equations (3) and (4) in equation (5), we will get Hence, the result is true for all positive integers h ≥ 3.  M(η, h) be a connected graph of meta-polyphenyl network of cycle graph C η , and h are the copies of cycle graph with order λ � ηh and size h(η + 1) − 1. For the following theorems, Figure 2 shows the resolving set in dark black vertices.
Proof. To show that dim (M(η, h)) � h, we will apply the induction method on h showing the copies of base graph. e base case for h � 2 is proved in eorem 3; now, assume that the assertion is true for h � m: We will show that it is true for h � m + 1. Suppose dim(M(η, m + 1)) � dim(M(η, m)) + dim(M(η, 2)) − 1. (10) Using equations (8) and (9) in equation (10), we have Hence, the result is true for all positive integers h ≥ 3.  L(η, h) be a connected graph of para-polyphenyl network of cycle graph C η , and h are the copies of cycle graph with order λ � ηh and size h(η + 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. Theorem 5. If η ≥ 5 and h ≥ 2, then dim(L(η, h)) is 2.
Proof. Firstly, we prove that dim(L(η, h)) ≤ 2; for this construction, a resolving set ϕ � v 1 , v λ−η+1 from the vertex set of L(η, h). We assume the following cases on vertex set of G and on the copies of cycle graph, i.e., h:
Hence, it follows from the above discussion that dim(L S (η, h)) ≤ 2 because all the vertices of L S (η, h) have unique representations with respect to resolving set ϕ. For reverse inequality that dim(L S (η, h)) ≥ 2, by contradiction, our assertion becomes dim(L S (η, h)) < 2, implying dim(L S (η, 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 (odd) and h ≥ 2, dim L S (η, h) � 2.

Conclusion
We found the metric dimension of some chemical networks ortho-, meta-, and para-polyphenyl chains constructed with base graph C η and sun graph S η , and these networks have metric dimension dim(O(η, h)) � dim(M(η, h)) � h and dim(L(η, h)) � 2.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest.

Authors' Contributions
All the authors contributed equally to prepare this article.