Metric basis and metric dimension of 1-pentagonal carbon nanocone networks

Zafar Hussain,Junaid Alam Khan,Imtiaz Ahmed,Ashfaq Ahmad,Mobeen Munir,Maqbool Chaudhary

doi:10.1038/s41598-020-76516-1

Abstract

Resolving set and metric basis has become an integral part in combinatorial chemistry and molecular topology. It has a lot of applications in computer, chemistry, pharmacy and mathematical disciplines. A subset S of the vertex set V of a connected graph G resolves G if all vertices of G have different representations with respect to S. A metric basis for G is a resolving set having minimum cardinal number and this cardinal number is called the metric dimension of G. In present work, we find a metric basis and also metric dimension of 1-pentagonal carbon nanocones. We conclude that only three vertices are minimal requirement for the unique identification of all vertices in this network.

Highlights

Resolving set and metric basis has become an integral part in combinatorial chemistry and molecular topology
Other significant applications of resolving sets and metric dimension can be traced in computer network, robot navigation, game theory and signal processing where largely a moving observer in a network system may be located by finding the distance from the point to the collection of sonar stations, which have been properly positioned in the n etwork2
Number of minimal resolving set is called the metric dimension of G, denoted by dim(G)

Summary

Introduction

Resolving set and metric basis has become an integral part in combinatorial chemistry and molecular topology. This particular set of vertices having minimum elements is called a resolving set of the graph space and the cardinal number of this set is called the metric dimension5–7. Number of minimal resolving set is called the metric dimension of G, denoted by dim(G). In24, authors discussed metric dimension of some graphs and proved that it is constant 1 if and only if graph is the path Pn .

Results

Conclusion