Symmetries and symmetry-breaking in arithmetic graphs
Aqsa Shah, Imran Javaid, Shahid ur Rehman · Heliyon · 2023
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 = ∏ 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 = ∏ i = 1 n p i r i under different conditions on r i .