Edge Metric Dimension and Edge Basis of One-Heptagonal Carbon Nanocone Networks

A molecular (chemical) graph is a simple connected graph, where the vertices represent the compound’s atoms and the edges represent bonds between the atoms, and the degree (valence) of every vertex (atom) is not more than four. In this paper, we determine the edge metric basis and edge metric dimension (EMD) of the complex molecular graph of a one-heptagonal carbon nanocone (<inline-formula> <tex-math notation="LaTeX">$HCN_{7}(q)$ </tex-math></inline-formula>). We prove that only three non-adjacent vertices are the minimum requirement for the identification of all the edges in <inline-formula> <tex-math notation="LaTeX">$HCN_{7}(q)$ </tex-math></inline-formula>, uniquely.


I. INTRODUCTION
Metric graph theory is used to model the behavior of realworld distance-based systems. Chemistry, industrial chemistry, computer science, and molecular topology are all areas where it can be employed. Due to the general fascinating problems posed by the structures and symmetries involved, it attracts people from all domains. It is generally essential to identify the position of vertices in an informative network by assigning an identity corresponding to a specific set (resolving set) [16], [35]. The metric basis for a particular network is a specific set with the smallest cardinality, and this smallest cardinality of a specific set is known as the metric dimension of the graph , represented by dim v ( ). Similarly, chemical graph theory (CGT) plays a vital role in the understanding of diverse chemical networks, complex structures, and topologies in the form of a graph, that is difficult to examine in its original form by replacing the atoms with vertices and the line connecting the atoms with edges. In this paper, we determine the positions of edges in the complex molecular structure of a carbon nanocone (particularly, a single heptagon surrounded by several layers of hexagon (HCN 7 (q)) as shown in Figure 1), uniquely. A separate section has been devoted to the basics of carbon nanocones; origins, applicability, and so on, in order to comprehend the science behind the carbon nanocones.
Slater [35] and Harary and Melter [16] independently proposed the concepts of resolving set and metric dimension.
The associate editor coordinating the review of this manuscript and approving it for publication was Sun-Yuan Hsieh .
Following these two foundational publications, a plethora of studies on the theoretical properties, as well as certain applications of this invariant, have been published. As an example, Melter and Tomescu [25] studied the concept of resolvability in image processing and pattern recognition, Sebo and Tannier [30] discussed the notion of metric dimension in terms of combinatorial optimization, Cáceres et al. [8] employed the concept of metric dimension on a coin weighing problems and mastermind games, Khuller et al. [20] found an application of metric dimension in the navigation of robots, Slater [35] studied problems related to SONAR (Sound Navigation and Ranging), facility location problems and coastguard LORAN (Long-Range Navigation), etc. Several variants of the metric dimension were also investigated such as EMD [19], k-metric resolving sets [26], fault-tolerant metric dimension [18], strong resolving sets [27], k-antiresolving sets [36], mixed resolving sets [37], local resolving sets [28], etc. Among these, the EMD for a network , represented by dim e ( ), has received the greatest attention in recent years and is of great interest. This parameter has been investigated in general as well as for many structures, graphs, and networks.
Distinguishing edges rather than vertices appear to be the most natural variant of metric dimension, hence it is surprising that the EMD was just recently introduced by Kelenc et al. [19] and is properly defined in the preliminary section. When one thinks of a smart city, an intelligent transportation system (ITS) may be the first thing that comes to mind. Self-driving cars are likely to play an important role in an ITS in the near future. A self-driving car must uniquely determine its position on the streets in a city, hence each street requires a code (representation) that uniquely defines its position. If we describe the city as a planar graph , with the edges corresponding to streets, then an edge resolving set of generates unique representation for the streets.
The very first paper on the concept of resolving all the graph edges contains a large number of results [19], including proof, that obtaining the edge basis set of a graph is NP-hard, and also holds several approximation results for this graph parameter. It is also illustrated that dim e ( ) and dim v ( ) are incomparable in general. For more evidence, see [22], where it has been shown that dim e ( ) < dim v ( ) for an infinite number of graphs families. Several concerns from [19] are addressed in a recent study [41], including a categorization of the graphs of order m for which dim e ( ) = m − 1 holds. These graphs were also studied in [42], where a polynomial method was constructed to recognize them. The EMD of a graph join, the corona product of a graph, and the lexicographic product of a graph are all stated in [29]. Finally, a new study [13] provides a number of interesting results on the EMD, demonstrating, among other things, that the d-dimensional grid has at most d as its EMD. For more details, refer [5], [17], [19], [24], [31]- [33].
Moreover, EMD has already been investigated for many significant molecular graphs, like: Ahmad et al. [1] studied the EMD of benzenoid tripod structure, Sharma et al. [34] investigated the EMD of 1-pentagonal carbon nanocone structure, Azeem and Nadeem [4] studied EMD for the complex structure of polycyclic aromatic hydrocarbons, Koam et al. [23] discussed the EMD of hollow coronoid structure, etc. In this paper; we consider the complex molecular graph of HCN 7 (q) with vertices (atoms) and edges (bonds) as depicted in Figure 1, and scrutinized the recently introduced variant of MD for HCN 7 (q). To the best of the author's little knowledge, no results regarding edge metric basis and EMD of HCN 7 (q) have been published to date, thus obtaining deeper insights into the complex molecular structure of HCN 7 (q).
This manuscript is structured as follows: section 2 covers the basic overview of the CGT and carbon nanocones, section 3 focuses on the definitions of the main concept of metric dimension and EMD, section 4 emphasizes the main finding of the manuscript, and finally, the conclusion is done.

II. CHEMICAL GRAPH THEORY AND CARBON NANOCONES
CGT applies graph theory to chemistry and focuses on the concept of a molecular graph, also known as a chemical graph, structural graph, or constitutional graph, which all refer to a graph with vertices representing the compound's atoms and the edges representing the chemical bonds between the atoms. It is associated with the graph-based analysis of diverse networks, chemical structures, and topologies. Chemical structures that are incredibly large and complex in their original form are hard to study, and then CGT is employed to make these structures approachable. After converting a molecular structure to a graph, detailed structural analysis can easily be performed. Topological indices (molecular structure descriptors) are also the significant graph parameters that are being used in the studies of QSPR and QSAR (Quantitative Structure-Property Relationships and Quantitative Structure-Activity Relationships). The structure-dependent chemical behaviors of molecules are the subject of QSPR and QSAR research. Metric dimension and its recently introduced variants are considered as the most implicative and studied parameters of graph theory. So, continuing in the same direction, we study EMD for the molecular structure of HCN 7 (q) in this work. Next, we give a brief overview of carbon nanocones.
Carbon nanocones (CNCs) with a single wall are the graphene sheets, coiled up into a cone topologically. CNCs exhibits unique electronic, chemical, thermal, mechanical, and structural properties of sp 2 related carbons, but they have the potential to be even harder than carbon nanotubes [39]. There are many types of CNCs theoretically, each having a different apex angle defined by the number of disclinations formed during the nucleation process [14]. CNCs can possess bases with micrometer-sizes, tip radii varying from less then one nanometer (nm) to several tens of nms, and are composed of a single layer or even several layers. CNCs were observed for the first time in 1994 and are introduced as mechanically stable tips for scanning probe microscopy [12]. They were then synthesized subsequently with distinct chemical structures. Annealed cones have the less structural disorder, larger crystallite size, and higher electrochemical properties than non-annealed cones [40]. Additionally, pentagons and other polygons at the tip apex produce magnetism due to the intense resonant peaks in the LDOS (local density of states) and the presence of unpaired electrons [14].
CNCs were employed to completely cover the ultra-fine gold needles. Because of their great electrical conductivity and chemical stability, such needles are commonly applied in scanning probe microscopy, although their tips are prone to mechanical wear due to gold's high plasticity. Adding a thin carbon cap at the tip mechanically stabilizes it without compromising its other qualities [6]. Because of their strength and flexibility, CNCs are perfect for manipulating other nanoscale structures, indicating that they will play a significant role in the domain of engineering related to nanotechnology. These 3D all-carbon designs could help with the next generation of field power storage, emission transistors, biomedical devices & implants, supercapacitors, high-performance catalysis, photovoltaics, etc. So, we are interested in adding more to this subject because of its uses, applications, and relevance in a variety of disciplines of study.
CNCs are an extremely interesting family of carbon nanomaterials identified firstly in 1994 by Ge and Sattler [12]. These are conical formations comprised primarily of carbon and having at least one dimension of the order of one micrometer or less. Nanocones have the same height as their diameter which separates them from tipped nanowires, which are significantly longer than their diameter. These CNCs are VOLUME 10, 2022 constructed by extracting a 60-degree wedge from a graphene sheet and connecting the edges produces a cone with a single pentagonal defect at the apex. These are also obtained by decomposing hydrocarbons with a plasma torch. The opening angle (apex) of the cones is not arbitrary, according to electron microscopy, but has preferred values of around 20, 40, and 60 degrees. Negative curvature appears when heptagons are included in the hexagonal lattice, as shown in Figure 1, and we call it as one-heptagonal carbon nanocone HCN 7 (q). The single sevenfold in the plain graphene lattice has been examined theoretically, however, this circumstance has yet to be observed in an experiment.
In the recent past, several studies were reported on HCN 7 (q), for instance, Alipour and Ashrafi [2] presented a polynomial using a Java program and the curve fitting method, in order to obtain the Wiener index of HCN 7 (q) for 1 ≤ q ≤ 15. The polynomial so obtained is as follows W (HCN 7 (q)) = bq 5 + cq 4 + dq 3 + eq 2 + fq 1 + gq 0 , where b, c, d, e, f , and g are some fixed values. Further, Ashrafi and Mohammad-Abadi [3], calculated the Wiener index of HCN 7 (q) by using an algorithm by Sandi Klavzar [21]. The proved that W (HCN 7 (q)) = 238 5 q 5 + 238q 4 + 2821 6 q 3 + 917 2 q 2 + 3311 15 q 1 + 42q 0 . Xu and Zhang in [38] generalized the results obtained regarding HCN 7 (q) in [3] and obtained the Hosaya polynomial for it. In HCN 7 (q), carbon atoms and bonds between them represent vertices and edges for our purpose, respectively as shown in Figure 1. In this work, we present results on all the bonds (edges) in HCN 7 (q), due to the fact that no such study has been conducted on the EMD of HCN 7 (q). As a result, we will examine some basic properties as well as EMD of HCN 7 (q).

Suppose
= (V , E) be a graph, which is finite, simple, non-trivial, connected, and undirected. V and E respectively represent the vertex and edge sets for . The distance between a vertex z and an edge ε = pq, denoted by d(ε, z), is defined as d(ε, z) = min{d(p, z), d(q, z)}, where d(p, z) represents the length of the shortest p−z path in . The totality of edges that are incident to a vertex of a graph is known as its valency (degree). A subset Y i of distinct vertices in is refer to be a stable set or independent set, if any pair of distinct vertices in Y i are not adjacent [34].
In order to provide essential mathematical definitions of the concepts investigated, we first discuss the structure of one-heptagonal carbon nanocone HCN 7 (q).

A. ONE-HEPTAGONAL CARBON NANOCONE
By HCN 7 (q), we represent the molecular graph of one-heptagonal carbon nanocone, where q ≥ 2. HCN 7 (q) comprises a single cycle with seven edges and vertices at its core and q denote q − 1 layer of six-sided faces placed around the central seven side cycle as shown in Figure 1. The HCN 7 (q) consists of 7q(q−1) 2 number of hexagonal faces and a single seven-sided face. It has 7(q 2 + q(q−1) 2 ); q ≥ 1 edges (or bonds) and 7q 2 vertices (atoms).
It also consists of 7q and 7q 2 − 7q number of vertices with valency two and three respectively [2]. Next, for our convenience, we write edges p i,j p i,j+1 and p i,j p i+1,k in HCN 7 (q) by p i j,j+1 and p i,i+1 j,k respectively. The vertex and edge set for HCN 7 (q), are denoted by V (HCN 7 (q)) and E (HCN 7 (q)) respectively, where V (HCN 7 (q) and Y v is a resolving set for if Y v resolves every pair of different vertices of . The smallest possible cardinality of a resolving sets Y v on is the metric dimension of , represented by dim v ( ). Y v is called a minimal resolving or metric basis for , if Y v is a resolving set on and |Y v | = dim v ( ) [16], [35].

C. EDGE RESOLVING SET AND EDGE METRIC DIMENSION
Y e ⊆ V in is refer to be an edge resolving set (ERS) if, for any two different edges ε 1 and ε 2 in E, there exists z ∈ Y e such that d(z, ε 1 ) = d(z, ε 2 ). Such an element z is therefore said to distinguish (recognize or resolve) edges ε 1 and ε 2 in . Equivalently, for a subset of different ordered vertices Y e = {n 1 , n 2 , n 3 , . . . , n k } ⊆ V , the edge co-ordinate (edge metric code (EMC)) of ε * ∈ E with respect to Y e is the k-vector r e (ε * ) = r e (ε * |Y e ) = (d(ε * , n 1 ), d(ε * , n 2 ), d(ε * , n 3 ), . . . , d(ε * , n k )). Then, the subset Y e of ordered distinct vertices in is edge resolving, if for any ε 1 , ε 2 ∈ E there exists at least one z ∈ Y e such that d(z, ε 1 ) = d(z, ε 2 ). The smallest possible cardinality of an ERS Y e on is the edge metric dimension of , represented by dim e ( ). Y e is called a minimal edge resolving or edge metric basis for , if Y e is an edge resolving set on and |Y e | = dim e ( ) [19].
A subset Y i e of ordered distinct vertices in is called the independent edge resolving set (IERS) for , if Y i is an (i) edge resolving set and (ii) independent set [37]. P n , K n , C n , and K m,n respectively denote the path graph, complete graph, cycle graph, and the complete bipartite graph with a partition of size m and n. Next, for a simple connected graph on n ≥ 2 vertices, 1 ≤ dim e ( ) ≤ n − 1 where dim e ( ) = n − 1 iff ∼ = K n , and dim e ( ) = 1 iff ∼ = P n [19]. Therefore, for  Recently, Han et al. [15] obtained the following result regarding the metric dimension of HCN 7 (q).

IV. MAIN RESULTS
In CGT, each chemical structure can be represented as a graph, with atoms and bonds representing vertices and edges respectively. The resolvability parameters of a graph are the recently introduced novel concepts, in which the construction of an entire structure is made in such a way that each bond or/and atom has its unique identification. We prove in the next result, that the minimum possible cardinality of an ERS Y e in HCN 7 (q) is three, where the vertices are chosen from all conceivable vertex combinations. Theorem 1: dim e (HCN 7 (q)) ≤ 3, for every q ≥ 2. Proof: In order to obtain that dim e (HCN 7 (q)) ≤ 3, we have to construct an ERS for HCN 7 (q) with cardinality three. Let Y e = {p q,1 , p q,3 , p q,6q−2 } be a set of distinct vertices from HCN 7 (q) (position of these three vertices in HCN 7 (q) are shown in red color in Figure 1). We claim that Y e is an ERS for HCN 7 (q). Now, to obtain dim e (HCN 7 (q)) ≤ 3, we can give EMCs to every member of E(HCN 7 (q)) with respect to the set Y e .
For the edges {p 1,j p 1,j+1 , p 1,7 p 1,1 |1 ≤ j ≤ 6} and the edges joining 1 st and 2 nd -cycle in HCN 7 (q), the EMCs are listed in Table 1: For the edges {p 2,j p 2,j+1 , p 2,21 p 2,1 |1 ≤ j ≤ 20} in HCN 7 (q), the EMCs are listed in Table 2: For the edges joining 2 nd and 3 rd -cycle in HCN 7 (q), the EMCs are listed in Table 3: HCN 7 (q), the EMCs are listed in Table 4: For the edges {p q,j p q,j+1 , p q,14q−7 p q,1 |1 ≤ j ≤ 14q − 6 & i = q} in HCN 7 (q), the EMCs are listed in Table 5: For the edges joining i th and (i + 1) th -cycle (3 ≤ i ≤ q − 2) in HCN 7 (q), the EMCs are listed in Table 6: At last, for the edges joining (q − 1) th and q th -cycle in HCN 7 (q), the EMCs are listed in Table 7: From these EMCs for the edges in HCN 7 (q), we find that no two edges are having the identical EMCs with respect to the set Y e and hence, we have dim e (HCN 7 (q)) ≤ 3.   To complete the proof, we have to show that the lower bound for the EMD of HCN 7 (q) is also three.
Further, by the symmetric property of HCN 7 (q) other possible relations can also be taken, which produces the similar type of contradictions as we observed in the above three cases. Therefore, from the above list of distinct cases, we see that there does not exist any ERS Y e for 1-HCN such that |Y e | = 2. Thus, we have |Y e | ≥ 3 i.e., dim e (HCN 7 (q)) ≥ 3. On combining the above two Theorems, we have the following important result: Theorem 3: dim e (HCN 7 (q)) = 3, for every q ≥ 2. Next, if for HCN 7 (q), its ERS is independent, then we have the following:   (4), as shown in Figure 5, are listed in Table 8 and 9:

V. CONCLUSION
For a given simple connected chemical graph, ERSs and edge metric basis sets contain an essential information for uniquely identifying each edge present in the graph. In this paper, we consider the complex molecular graph of a one-heptagonal carbon nanocone HCN 7 (q) and proved that dim e (HCN 7 (q)) = 3. We observed that the EMD of HCN 7 (q) does not depend upon the number of hexagonal layers and vertices present in HCN 7 (q). Using preliminaries results and Theorem 3, we found that dim e (HCN 7 (q)) = dim v (HCN 7 (q)) = 3. From this, we conclude that the family HCN 7 (q) of a complex molecular graph, is the family with the constant metric and EMD. We further proved that the minimum ERS possesses the property of independence in HCN 7 (q).
KARNIKA SHARMA received the bachelor's degree from the University of Jammu, and the M.Sc. degree in mathematics from Shri Mata Vaishno Devi University, Katra, India, in 2018, where she is currently pursuing the Ph.D. degree with the School of Mathematics. Her research interests include graph theory and combinatorics.
VIJAY KUMAR BHAT received the Ph.D. degree in mathematics from the University of Jammu, India, in 2000. He is currently a Professor of mathematics at Shri Mata Vaishno Devi University. He has presented his work in international conferences held in different countries. He has completed five research projects. He has supervised 30 students at master's level and 12 students at Ph.D. level. He has research collaboration with Mathematicians at international level. He has more than 90 research publications and has authored three books. His research interests include algebra (group theory and skew polynomial rings) and graph theory.
SUNNY KUMAR SHARMA received the bachelor's and M.Sc. degrees in mathematics from the University of Jammu, in 2015 and 2017, respectively. He is currently pursuing the Ph.D. degree with the School of Mathematics, Shri Mata Vaishno Devi University, Katra, India. He qualified the Graduate Aptitude Test (GATE) in mathematical sciences, in 2020. He has published a good number of articles in reputed journals. His research interests include computation of metric dimension, edge metric dimension, mixed metric dimension, and various families of graphs and their applications.