Computing Edge Metric Dimension of One-Pentagonal Carbon Nanocone
Sunny Kumar Sharma, Hassan Raza, Vijay Kumar Bhat
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 ( 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.