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

Abstract

Let Γ = Γ(V, E) be a simple connected graph, where V and E represent the set of vertices and edges respectively. The distance between a vertex z and an edge ε = pq, denoted by d(z, ε), is defined as d(z, ε) = min{d(p, z), d(q, z)}, where d(p, z) represents the length of a shortest p−z path in Γ. A subset Y e ⊆ V of ordered distinct vertices is refer to be an edge resolving set (ERS) for Γ, if for any pair of different edges ε 1 and ε 2 in E, we have d(z, ε 2 ) = d(z, ε 1 ) for at least one vertex z in Y e . An edge metric basis for Γ is the ERS with the smallest cardinality and this smallest cardinality is called the edge metric dimension (EMD) of Γ, represented by dim e (Γ). 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. In this paper, we determine the edge metric basis and EMD of the complex molecular graph of a one-heptagonal carbon nanocone (HCN 7 (q)). We prove that only three non-adjacent vertices are the minimum requirement for the identification of all the edges in HCN 7 (q), uniquely.INDEX TERMS Connected graph, edge metric basis, edge metric dimension, independent set, oneheptagonal carbon nanocone, resolving set.

Mathematical analysis of the structure of one-heptagonal carbon nanocone in terms of its basis and dimension

“…Negative curvature, as shown in figure 2, is completely due to the addition of heptagons to the hexagonal lattice. Furthermore, the existence of the single sevenfold in the ordinary graphene lattice has only been studied theoretically, it is known as the one-heptagonal carbon nanocone, denoted by HCN s , and this situation has not yet been verified experimentally, it has been proposed that it may exist [9].…”

“…The solution to such a problem for several complex graph structures is hard to get, but significant information can be provided by it, in order to tackle many complicated situations [7]. For instance, the small set U can assist robots navigating in a physical space, identifying intersection points on roads, uniquely specifying a group of people by assigning a UIN (unique identification number), or tracing disease transmission between distinct regions [9,20]. This might be employed as well for more general activities like recognizing a source of lies and deceitful information in a network of people, comparing distinct architectures of networks, chemical structure categorization, or quantitatively encoding symbolic data [19].…”

Structural investigation of carbon nanocone through topological coindices

“…The introduction of nanomaterials and fullerenes were generated because of its structural and computational properties of CNCs [17]. Mortazavi et al work on the first principles based simulations conducted by the chemical vapor deposition to examine the mechanical properties, lattice thermal conductivity, piezoelectric, and photocatalytic features of monolayers [18].…”

“…The introduction of nanomaterials and fullerenes were generated because of its structural and computational properties of CNCs [17]. [24,25].…”

On Degree-Based Topological Indices of Carbon Nanocones

“…The structural and computational properties of carbon nanocones have been the topic of several studies over the years, , many of which have been prompted by the introduction of such nanomaterials and fullerenes . Mortazavi et al motivated by the chemical vapor deposition experiment conducted extensive first-principles-based simulations to explore the stability, mechanical properties, lattice thermal conductivity, piezoelectric and flexoelectric response, and photocatalytic and electronic features of MA 2 Z 4 (M = Cr, Mo, W; A = Si, Ge; Z = N, P) monolayers.…”