Symmetries and symmetry-breaking in arithmetic graphs

In this paper, we study symmetries and symmetry-breaking of the arithmetic graph of a composite number m, denoted by [Formula: see text]. 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 [Formula: see text] and prove that the automorphism group of [Formula: see text] is isomorphic to the symmetric group [Formula: see text] of n elements, for [Formula: see text]. 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 [Formula: see text] for [Formula: see text] under different conditions on [Formula: see text].