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

Abstract

Let $\Gamma =(V,E)$ be a simple connected graph. A vertex a is said to recognize (resolve) two different elements b₁ and b₂ from $V(\Gamma )\cup E(\Gamma )$ if $d(a,b_1)\ne d(a,b_2\}.$ A subset of distinct ordered vertices $U_M\subseteq V(\Gamma )$ is said to be a mixed metric generator for Γ 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 Γ. Then, the cardinality of this mixed metric basis for Γ is called the mixed metric dimension of Γ, denoted by $\mathit{mdim}(\Gamma ).$ 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 $S{P}_{a,b,c}$ and six-sided hollow coronoid $H{C}_{a,b,c}$ and respectively compute their multiset dimension and mixed metric dimension.

Keywords:

• Multiresolving set
• metric resolving set
• mixed metric resolving set
• hollow coronoid structure
• starphene

MSC (2020):

• 05C12
• 05C90

Acknowledgements

We would like to express our sincere gratitude to the referees for a very careful reading of this paper and for all their insightful comments/criticism, which have resulted in a number of major improvements. We also acknowledge that this manuscript has already been uploaded as arxiv in Cornell University vide link http://arxiv.org/abs/2110.12368

Disclosure statement

No potential conflict of interest was reported by the authors.

Authors’ contributions

All the authors have equally contributed to the final manuscript.

Data availability statement

Data sharing is not applicable to this article as no data set were generated or analyzed during the current study.