On the Metric Dimension of Arithmetic Graph of a Composite Number

Shahid Ur Rehman,Muhammad Imran,Imran Javaid

doi:10.3390/sym12040607

Shahid Ur Rehman, Muhammad Imran + Show 1 more

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https://doi.org/10.3390/sym12040607

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Abstract

This paper is devoted to the study of the arithmetic graph of a composite number m, denoted by A m . It has been observed that there exist different composite numbers for which the arithmetic graphs are isomorphic. It is proved that the maximum distance between any two vertices of A m is two or three. Conditions under which the vertices have the same degrees and neighborhoods have also been identified. Symmetric behavior of the vertices lead to the study of the metric dimension of A m which gives minimum cardinality of vertices to distinguish all vertices in the graph. We give exact formulae for the metric dimension of A m , when m has exactly two distinct prime divisors. Moreover, we give bounds on the metric dimension of A m , when m has at least three distinct prime divisors.

Highlights

All graphs considered in this paper are simple, undirected, connected and finite
We prove that there exist different composite numbers for which the arithmetic graphs are isomorphic
For a composite number m with the canonical form m = p1 p2, the arithmetic graph Am is isomorphic to a path graph on three vertices and has metric dimension 1

Summary

Introduction

All graphs considered in this paper are simple, undirected, connected and finite. A graph G consists of two sets, V ( G ) and E( G ), known as the vertex set and the edge set of G, respectively.The elements of V ( G ) and E( G ) are called the vertices and edges of G, respectively. A subset R of the vertices of a graph G satisfying the property that for every two distinct vertices x, y ∈ V ( G ) there exist r ∈ R such that d( x, r ) 6= d(y, r ) is called a resolving set for G and dim( G ) We study the metric dimension of an arithmetic graph associated with a composite number.

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Conclusion