Metric and Fault-Tolerant Metric Dimension of Hollow Coronoid

Ali N A Koam,Mohammed Eltahir Abdelhag,Ali Ahmad,Muhammad Azeem

doi:10.1109/access.2021.3085584

Ali N A Koam, Mohammed Eltahir Abdelhag + Show 2 more

Open Access

https://doi.org/10.1109/access.2021.3085584

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Abstract

Coronoid systems actually arrangements of hexagons into six sides of benzenoids. By nature, it is an organic chemical structure. Hollow coronoids are primitive and catacondensed coronoids. It is also known as polycyclic conjugated hydrocarbons. The mathematical study of chemicals is of great interest to different specialties researchers. While graph theory always played a significant role to make chemical structures understandable and blessed with applications also. After transforming the chemical structure into a graph, one can implement different theoretical and implicative studies on structures. Metric dimension is considered as one of the most studied and implicative parameters of graph theory. In this concept, few suggested vertices are chosen such as the remaining vertices have unique locations or identifications. In this study, we discussed different metric-based parameters for the hollow coronoid structure.

Highlights

Chemical graph theory is considered a varied field and combination of chemistry and mathematics. It is an application of mathematics and it is known as mathematical chemistry. It deals with the study of different chemical structures, networks, and their topologies in the form of a graph
Proof: To show that the graph of hollow coronoid HC (p, q, s), has 4, fault-tolerant metric dimension, by the implementation of method of double inequality, for dimf (HC (p, q, s)) ≤ 4, we are referring the Lemma 2, which is already proved that the fault-tolerant resolving set Rf is a candidate with cardinality 4 and it can be settled as Rf = {a1, b1, b2p−1, e2p−3}
Proof: To show that the graph of hollow coronoid HC (p, q, s), has a candidate for the partition resolving set with cardinality 4, and it is taken as Rp = {Rp1, Rp2, Rp3, Rp4}, where Rp1 = {a1}, Rp2 = {b2}, Rp3 = {b2p−1}, Rp4 = V (HC (p, q, s)) \{a1, b1, b2p−1}

Summary

Introduction

Chemical graph theory is considered a varied field and combination of chemistry and mathematics. INDEX TERMS Hollow coronoid, metric dimension, resolving set, fault-tolerant metric dimension. It deals with the study of different chemical structures, networks, and their topologies in the form of a graph (usually vertex and edge).

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Conclusion