Shah Aqsa, Javaid Imran, Rehman Shahid Ur
Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan.
Heliyon. 2023 Sep 7;9(9):e19820. doi: 10.1016/j.heliyon.2023.e19820. eCollection 2023 Sep.
In this paper, we study symmetries and symmetry-breaking of the arithmetic graph of a composite number , denoted by . 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 and prove that the automorphism group of is isomorphic to the symmetric group of elements, for . 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 for under different conditions on .
在本文中，我们研究合数(n)的算术图（记为(\mathcal{AG}(n))）的对称性和对称破缺。我们首先研究一些性质，例如顶点之间的距离、顶点的度数以及算术图中孪生类的数量。我们描述了(\mathcal{AG}(n))的对称性，并证明对于(n\gt1)，(\mathcal{AG}(n))的自同构群同构于(n)个元素的对称群(S_n)。对于对称破缺，我们研究算术图的固定数概念，并给出在(n)的不同条件下算术图(\mathcal{AG}(n))的固定数的精确公式。
