AUTHOR=Sharma Sunny Kumar , Raza Hassan , Bhat Vijay Kumar 

TITLE=Computing Edge Metric Dimension of One-Pentagonal Carbon Nanocone

JOURNAL=Frontiers in Physics

VOLUME=9

YEAR=2021

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2021.749166

DOI=10.3389/fphy.2021.749166

ISSN=2296-424X

ABSTRACT=<p>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 <italic>e</italic> = <italic>cz</italic> and a vertex <italic>u</italic> is defined by <italic>d</italic>(<italic>e</italic>, <italic>u</italic>) = <italic>min</italic>{<italic>d</italic>(<italic>c</italic>, <italic>u</italic>), <italic>d</italic>(<italic>z</italic>, <italic>u</italic>)}. If <italic>d</italic>(<italic>e</italic><sub>1</sub>, <italic>u</italic>) ≠ <italic>d</italic>(<italic>e</italic><sub>2</sub>, <italic>u</italic>), then we say that the vertex <italic>u</italic> resolves (distinguishes) two edges <italic>e</italic><sub>1</sub> and <italic>e</italic><sub>2</sub> in a connected graph <italic>G</italic>. A subset of vertices <italic>R</italic><sub><italic>E</italic></sub> in <italic>G</italic> is said to be an edge resolving set for <italic>G</italic>, if for every two distinct edges <italic>e</italic><sub>1</sub> and <italic>e</italic><sub>2</sub> in <italic>G</italic> we have <italic>d</italic>(<italic>e</italic><sub>1</sub>, <italic>u</italic>) ≠ <italic>d</italic>(<italic>e</italic><sub>2</sub>, <italic>u</italic>) for at least one vertex <italic>u</italic> ∈ <italic>R</italic><sub><italic>E</italic></sub>. An edge metric basis for <italic>G</italic> is an edge resolving set with minimum cardinality and this cardinality is called the edge metric dimension <italic>edim(G</italic>) of <italic>G</italic>. 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.</p>