Metric and Edge Metric Dimension of Zigzag Edge Coronoid Fused with Starphene

Abstract

Let $\Gamma =(V,E)$ be a simple connected graph. $d(\alpha ,\epsilon )=min\{d(\alpha ,w),d(\alpha ,d\}$ computes the distance between a vertex $\alpha \in V(\Gamma )$ and an edge $\epsilon =wd\in E(\Gamma )$. A single vertex $\alpha$ is said to recognize (resolve) two different edges ${\epsilon }_1$ and ${\epsilon }_2$ from $E(\Gamma )$ if $d(\alpha ,{\epsilon }_2)\ne d(\alpha ,{\epsilon }_1\}$. A subset of distinct ordered vertices $U_E\subseteq V(\Gamma )$ is said to be an edge metric generator for $\Gamma$ if every pair of distinct edges from $\Gamma$ are recognized by some element of $U_E$. An edge metric generator with a minimum number of elements in it, is called an edge metric basis for $\Gamma$. Then, the cardinality of this edge metric basis of $\Gamma$, is called the edge metric dimension of $\Gamma$, denoted by $edim(\Gamma )$. The concept of studying chemical structures using graph theory terminologies is both appealing and practical. It enables chemical researchers to more precisely and easily examine various chemical topologies and networks. In this article, we investigate a fascinating cluster of organic chemistry as a result of this motivation. We consider a zigzag edge coronoid fused with starphene and find its minimum vertex and edge metric generators.

15 pages, 2 figures