Multiset and Mixed Metric Dimension for Starphene and Zigzag-Edge Coronoid

TL;DR

This paper computes the multiset and mixed metric dimensions of two chemical graph structures, starphene and coronoid, providing insights into their topological properties relevant to chemical graph theory.

Contribution

It introduces the calculation of multiset and mixed metric dimensions for specific chemical graph structures, expanding the application of metric dimension concepts in chemical topology.

Findings

Determined the mixed metric dimension of starphene $SP_{a,b,c}$.

Calculated the multiset dimension of six-sided hollow coronoid $HC_{a,b,c}$.

Provides formulas for the metric dimensions based on structure parameters.

Abstract

Let $\Gamma=(V,E)$ be a simple connected graph. A vertex $a$ is said to recognize (resolve) two different elements $b_{1}$ and $b_{2}$ from $V(\Gamma)\cup E(\Gamma)$ if $d(a, b_{1})\neq d(a, b_{2}\}$. A subset of distinct ordered vertices $U_{M}\subseteq V(\Gamma)$ is said to be a mixed metric generator for $\Gamma$ if each pair of distinct elements from $V\cup E$ are recognized by some element of $U_{M}$. The mixed metric generator with a minimum number of elements is called a mixed metric basis of $\Gamma$. Then, the cardinality of this mixed metric basis for $\Gamma$ is called the mixed metric dimension of $\Gamma$, denoted by $mdim(\Gamma)$. The concept of studying chemical structures using graph theory terminologies is both appealing and practical. It enables researchers to more precisely and easily examines various chemical topologies and networks. In this paper, we consider two well known chemical structures; starphene $SP_{a,b,c}$ and six-sided hollow coronoid $HC_{a,b,c}$ and respectively compute their multiset dimension and mixed metric dimension.