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.
Highlights
Carbon nanocones (CNC) made their first appearance in 1968, or perhaps earlier, on the surface of graphite occurring naturally [25]
We show that the minimum edge resolving set for 1-PCNC has cardinality three, with atoms/vertices chosen from all possible atom/vertex combinations
Edge metric generators for a given connected chemical graph contain crucial information required for the identification of each bond present in the graph, uniquely
Summary
Carbon nanocones (CNC) made their first appearance in 1968, or perhaps earlier, on the surface of graphite occurring naturally [25]. A two-dimensional planar graph of a 1PCNC is constructed, with carbon atoms representing vertices and bonds representing edges between them (see Figure 2). 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. Bultheel and Ori [9], analyzed topological modeling techniques used to study 1-PCNC and obtained significant findings about the chemical reactivity and desired sizes They addressed the topological roundness and efficiency of CNC5[m] as the long-range topological potential whose local minima correspond to magic sizes of nanocones with a greater percentage of formation.
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