Computing Edge Metric Dimension of One-Pentagonal Carbon Nanocone - Sharma, Sunny Kumar; Raza, Hassan; Bhat, Vijay Kumar (Frontiers in Physics, 2021)
Minimum resolving sets (edge or vertex) have become an integral part in 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 (or distinguishes) two edges $e_{1}$ and $e_{2}$ in a connected graph $G$. A subset of vertices $R$ 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 $u\in R$. 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.