Symmetries and symmetry-breaking in arithmetic graphs

Aqsa Shah; Imran Javaid; Shahid Ur Rehman

doi:10.1016/j.heliyon.2023.e19820

Symmetries and symmetry-breaking in arithmetic graphs

Heliyon. 2023 Sep 7;9(9):e19820. doi: 10.1016/j.heliyon.2023.e19820. eCollection 2023 Sep.

Authors

Aqsa Shah¹, Imran Javaid¹, Shahid Ur Rehman¹

Affiliation

¹ Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan.

Abstract

In this paper, we study symmetries and symmetry-breaking of the arithmetic graph of a composite number m, denoted by $A_{m}$ . We first study some properties such as the distance between vertices, the degree of a vertex and the number of twin classes in the arithmetic graphs. We describe symmetries of $A_{m}$ and prove that the automorphism group of $A_{m}$ is isomorphic to the symmetric group $S_{n}$ of n elements, for $m = \prod_{i = 1}^{n} p_{i}$ . For symmetry-breaking, we study the concept of the fixing number of the arithmetic graphs and give exact formulae of the fixing number for the arithmetic graphs $A_{m}$ for $m = \prod_{i = 1}^{n} p_{i}^{r_{i}}$ under different conditions on $r_{i}$ .

Keywords: 05A18; 05C12; 05C30; Arithmetic graph; Automorphism; Fixing number; Symmetry-breaking; Twins.