## 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.

Original language | English |
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Article number | 607 |

Journal | Symmetry |

Volume | 12 |

Issue number | 4 |

DOIs | |

Publication status | Published - Apr 1 2020 |

## Keywords

- Arithmetic graph
- Isomorphism
- Resolving set

## ASJC Scopus subject areas

- Computer Science (miscellaneous)
- Chemistry (miscellaneous)
- General Mathematics
- Physics and Astronomy (miscellaneous)