Computing Edge Metric Dimension of One-Pentagonal Carbon Nanocone

Minimum resolving sets (edge or vertex) have become an integral part of molecular topology and combinatorial chemistry. Resolving sets for a specific network provide crucial information required for the identification of each item contained in the network, uniquely. The distance between an edge e = cz and a vertex u is defined by d(e, u) = min{d(c, u), d(z, u)}. If d(e 1, u) ≠ d(e 2, u), then we say that the vertex u resolves (distinguishes) two edges e 1 and e 2 in a connected graph G. A subset of vertices R E in G is said to be an edge resolving set for G, if for every two distinct edges e 1 and e 2 in G we have d(e 1, u) ≠ d(e 2, u) for at least one vertex u ∈ R E . An edge metric basis for G is an edge resolving set with minimum cardinality and this cardinality is called the edge metric dimension edim(G) of G. In this article, we determine the edge metric dimension of one-pentagonal carbon nanocone (1-PCNC). We also show that the edge resolving set for 1-PCNC is independent.


INTRODUCTION
Carbon nanocones (CNC) made their first appearance in 1968, or perhaps earlier, on the surface of graphite occurring naturally [25]. These chemical structures are exciting due to their conceivable uses in gas sensors, gas storage, bio-sensors, energy storage, chemical probes, and nano-electronic devices, see [2,3,8,17,29]. Nanocones are the networks of the carbon that can be represented mathematically as cubic planar infinite graphs. Iijima [19], mainly addressed graphitic carbon helical microtubules. The existence of CNC and their combinatorial properties have been discussed in [14,24]. Depending upon the positive signed curvature, Klein et al. [25] categorized CNC into eight groups. Brinkmann et al. have classified these structures [7]. Justus et al. [20] have given the expander constants and boundaries of these nanocones triangle patches. CNC has recently gained considerable scientific attention due to its peculiar properties and promising applications such as hydrogen storage and energy [7].
The chemical graph of carbon nanocones CNC p [m], as shown in Figure 1, comprises of conical structures with a cycle of size and order p at their center and (m − 1)-layers of six-sided faces (hexagons) placed at the conical surface around its center. Here we are interested for the case p 5 i.e., CNC 5 [m]. When one pentagon is inserted in the honeycomb layer, a disclination defect in the graphenic plane is generated, resulting in the formation of a conic structure with positive curvature in which the pentagon is surrounded by the first belt of five hexagons. CNC 5 [m] denotes an (m − 1)-dimensional one-pentagonal carbon nanocone (1-PCNC), where m represents (m − 1)-number of layers consisting of six-sided faces, which include the conical surface of the nanocone, and five denotes the presence of a single five-sided face on the tip known as its center. Along with it, a two-dimensional planar graph of a 1-PCNC is constructed, with carbon atoms representing vertices and bonds representing edges between them (see Figure 2).
Resolvability in graph theory aims to understand the behavior of real-world distance-based frameworks. It has been used in chemistry, molecular topology, industrial chemistry, and computer science. It attracts authors from various fields, including mathematics, because of the fascinating problems that arise from the symmetries and structures involved. It is always highly beneficial in an enigmatic network to identify uniquely the location of vertices (such as atoms) by establishing an identity with respect to a specific set. Such a specific set with minimum cardinality is called the metric basis and this cardinality is the metric dimension [16,34]. These findings have been used effectively in drug patterns to access specific atoms.
The researchers are motivated by the fact that the metric dimension has a variety of practical applications in everyday life and so it has been extensively investigated. Metric dimension is utilized in a wide range of fields of sciences, including robot navigation [23], geographical routing protocols [27], connected joints in network and chemistry [10,11], telecommunications [6], combinatorial optimization [31], network discovery and verification [6], etc. NP-hardness and computational complexity for the resolvability parameters are addressed in [15,26].
Many authors have introduced and analyzed certain variations of resolving sets, such as local resolving set, partition resolving set, fault-tolerant resolving set, resolving  dominating set, strong resolving set, independent resolving set, and so on. For further details the reader is referred to [1,6,10,21,22,32,33]. In addition to defining other variants of resolving sets in graphs, Kelenc et al. [22], introduced a parameter used to uniquely distinguish graph edges and called it the edge metric dimension. In general, a graph metric was used to describe each pair of edges based on distances to a specific set of vertices. This was based on the assumption that a minimum resolving set R of a connected graph G identifies uniquely all the vertices of G using distance-vector, but does not necessarily recognize all the edges of G. CNCs are a significant class of carbon nanomaterials that have been discovered in 1994 by Ge and Sattler [23]. This class of CNC has recently received considerable attention. Bultheel and Ori [9], analyzed topological modeling techniques used to study 1-PCNC and obtained significant findings about the chemical reactivity and desired sizes. Moreover, they also addressed the topological roundness and efficiency of CNC 5 [m] as the long-range topological potential whose local minima correspond to magic sizes of nanocones with a greater percentage of formation. In [ [4,18,28].
The flexibility and strength of carbon nanotubes make them suitable for manipulating another nanoscale structures, implying that they will play an important part in nanotechnology engineering. These 3D all-carbon architectures may be used to create the next generation of power storage, field emission transistors, photovoltaics, supercapacitors, biomedical devices & implants, and high-performance catalysis. Because of its applications, uses, and significance in numerous fields of study, we are interested in contributing more to this subject. For our purpose, in 1-PCNC bonds represent the edges and carbon atoms represent vertices. Recently, a study [17] reveals that 1-PCNC possesses the minimum metric generator of cardinality three correspondings to the atoms (vertices) therein. We now obtain some important results regarding the edges (bonds) present in 1-PCNC, as there is no such study regarding the edge metric dimension of 1-PCNC network. So, in this article, we study some basic properties of 1-PCNC along with its edge metric dimension.
The main results obtained are as follows: • The edge metric dimension of 1-PCNC is three.
The remainder of this paper is structured as: Section 2 introduces some basic concepts related to the metric dimension and the edge metric dimension. Some proven outcomes of 1-PCNC with respect to the metric dimension are also discussed. We study the edge metric dimension of 1-PCNC in Sect. 3 and discuss some of its properties. Finally, the conclusion and future work of the present study are discussed in Sect. 4.

PRELIMINARIES
In this section, we list some basic properties of 1-PCNC, the definition of metric dimension & edge metric dimension, and recall some existing results regarding these notions.
Suppose G (V, E) is a non-trivial, simple, and connected graph, where V represents a set of vertices and E represents a set of edges. The distance between two vertices u and w in an undirected graph G, denoted by d(u, w), is the length of a shortest u − w path in G.
. The metric dimension of G, denoted by dim(G), is defined as dim(G) min{|R| : R is resoving set in G}. These notions were introduced, independently by Slater [34] and Harary and Melter [16].
Definition 4. Independent set: A set of vertices I in a graph G is said to be an independent set (also known as stable set) if no two vertices in I are adjacent [12].
Definition 5. Independent resolving set: A subset of vertices R in G is said to be an independent resolving set for G, if R E is resolving as well as independent set [12].
One can see that the metric dimension deals with the vertices of the graph by its definition, a similar concept dealing with the edges of the graph introduced by Kelenc et al. in [22], called the edge metric dimension of graph G, which uniquely identifies the edges related to graph G. Definition 6. Edge metric dimension: For an edge e cz and a vertex w the distance between them is defined as d(e, w) min {d(c, w), d(z, w)}. A subset R E is called an edge resolving set for G, if for any two distinct edges e 1 and e 2 of G are recognized by at least one vertex w of R E . For a subset of distinct (ordered) vertices R E {w 1 , w 2 , w 3 , . . ., w z }4 V(G), the edge metric co- w 3 ), . . ., d(e, w z )). The edge resolving set with minimum cardinality is termed as edge metric basis, and that cardinality is known as the edge metric dimension of graph G, denoted by edim(G) [22]. Definition 7. Independent edge resolving set (IERS): A subset R E of distinct vertices in G is said to be an IERS for G, if R E is edge resolving as well as independent set.
In [17], Hussain et al. obtained the metric dimension of CNC 5 [m]. They proved that CNC 5 [m] denotes a class of plane graph with constant and bounded metric dimension i.e., the metric dimension does not depend upon the value of m. For metric dimension, they gave the following result.
Hence, from the above cases, we conclude that there is no edge metric generator R E for 1-PCNC such that |R E | 2. However, symmetry of graphs can be used to derive alternative relations that show the same kind of contradictions. Therefore, we must have |R E |≥ 3 i.e., edim (CNC 5 [m]) ≥ 3.
Next, we prove that the upper bound for the edge metric dimension of 1-PCNC is also three. Proof. Suppose R E is an edge resolving set for CNC 5 [m]. To prove that the edge resolving set R E of 1-PCNC has cardinality less than or equal to three (i.e., |R E | 3, because of Theorem 1), we have to show that the edge codes corresponding to R E are distinct for any pair of different edges in CNC 5 [m]. Let R E {u m,1 , u m,3 , u m,2m+2 }. Then, we will show that R E is an edge resolving set for CNC 5 [m] with cardinality three. Next, we give edge codes to every edge of CNC 5 [m] with respect to R E . For our convenience, we denote the edges on the CNC 5 [m] by u i,j u i,k u i j,k and u i,j u k,l u i,k j,l . The edge metric codes for the edges of first cycle and edges joining first and second cycles are as shown in Case 3A.
The edge metric codes for the edges of second cycle and edges joining second and third cycles are as shown in Case 3B.
The edge metric codes for the edges of third cycle are as shown in Case 3C.
The edge metric codes for the edges joining the third and fourth cycle are as shown in Case 3D.
The edge metric codes for the edges of fourth cycle are as Case 3E. The edge metric codes for the edges joining the fourth and fifth cycle are as shown in Case 3F.
The edge metric codes for the edges of fifth cycle are Case 3G. Next, the edge metric codes for the edges joining the i th and (i + 1) th ; (5 ≤ i ≤ m − 2) cycles are as shown in Case 3H.
Frontiers in Physics | www.frontiersin.org November 2021 | Volume 9 | Article 749166 The edge metric codes for the edges of i th (6 ≤ i ≤ m − 1) cycle are as shown in Case 3J.
The edge metric codes for the edges of i m th cycle are as shown in Case 3K.
From these edge codes, we find that these are distinct from one and another in at least one coordinate, implying R E to be an edge resolving set with cardinality three for CNC 5 [m]. Hence, edimCNC 5 [m] ≤ 3.
By using Theorem 1 and Theorem 2, we obtain the following result

CONCLUSION
Edge metric generators for a given connected chemical graph contain crucial information required for the identification of each bond (edge) present in the graph, uniquely. In this article, for an important class of carbon nanocone, viz., one-pentagonal carbon nanocone CNC 5 [m], we prove that edim (CNC 5 [m]) 3 and it does not depend upon the value of m. We show that the minimum edge resolving set for 1-PCNC is also independent. The contributions of this research may be beneficial to those working in the fields of micro-devices built with CNC 5 [m], nanodevices, nano-biotechnology, nano-engineering, and pharmacy. Following the metric dimension and edge metric dimension of CNC 5 [m], the natural problem that arises from the text is: What should be the minimal cardinality of mixed metric resolving set (edge, as well as vertex, resolving set [35]) for CNC 5 [m]?

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.