Computing Edge Metric Dimension of One-Pentagonal Carbon Nanocone

Abstract

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(e1, u) ≠ d(e2, u), then we say that the vertex u resolves (distinguishes) two edges e1 and e2 in a connected graph G. A subset of vertices RE in G is said to be an edge resolving set for G, if for every two distinct edges e1 and e2 in G we have d(e1, u) ≠ d(e2, u) for at least one vertex u ∈ RE. 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.

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

“…The totality of edges that are incident to a vertex of a graph Γ is known as its valency (degree). A subset Y i of distinct vertices in Γ is refer to be a stable set or independent set, if any pair of distinct vertices in Y i are not adjacent [34].…”

“…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).…”

Vertex-Based Resolvability Parameters for Identification of Certain Chemical Structures

“…Siddiqui and Imran examined the metric dimension of H-naphthalenic and VC5C7 nanotubes. The concept of EMD for the same nanocone network was applied by Sharma and Bhat . For networks such as the silicate star network, the metric dimension was determined by Simonraj and George…”

On Degree-Based Topological Indices of Carbon Nanocones

“…Klein et al divided nanocones into eight groups based on the definite signed curvature. These structures have been categorized by Brinkmann et al The expander constants and boundaries of these nanocone triangular patches were determined by Justus et al Furthermore, carbon nanocones attracted the researchers’ interest due to their unusual features and potential applications in a variety of new fields, including energy and hydrogen storage …”

“…These structures have been categorized by Brinkmann et al 4 The expander constants and boundaries of these nanocone triangular patches were determined by Justus et al 5 Furthermore, carbon nanocones attracted the researchers' interest due to their unusual features and potential applications in a variety of new fields, including energy and hydrogen storage. 6 The structural and computational properties of carbon nanocones have been the topic of several studies over the years, 7,8 many of which have been prompted by the introduction of such nanomaterials and fullerenes. 9 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.…”