Symmetries and symmetry-breaking in arithmetic graphs

Abstract

In this paper, we study symmetries and symmetry-breaking of the arithmetic graph of a composite number m, denoted by ${\mathbb{A}}_m$. 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 ${\mathbb{A}}_m$ and prove that the automorphism group of ${\mathbb{A}}_m$ 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 ${\mathbb{A}}_m$ for $m={\prod }_{i=1}^n{p}_i^{r_i}$ under different conditions on $r_i$.

1. Introduction

Symmetries of graphs play central role in describing their structures and manipulating objects on graphs using their topologies. Distinguishability and indistinguishability of the vertices of graphs remain important problem for study which provide insights about the vertices and the structure of a graph. For this reason, identification of symmetries, breaking symmetries and distinguishing the vertices of graphs have always been interesting problems and have been well-studied in literature [1], [2], [3]. These concepts find applications in pattern recognition [4], strategies in mastermind games [5], problem of programming a robot for handling of objects [6] and coin weighing problem [7], [8]. Breaking the symmetries of a graph reduces the computational time and unnecessary computations by removing the symmetry-induced redundancies. Formally, each symmetry of a graph is a permutation of its vertex set which preserves the adjacency and non-adjacency relation of vertices in the structure, usually called the automorphism of a graph. To distinguish the vertices of a graph, symmetry, distance and degree related parameters in graphs using their topologies find more attention and these parameters yield interesting insights about graphs [9], [10], [11], [12], [13], [14]. A set sufficient to identify all symmetries of a graph was introduced by Boutin in [15] under the name determining set. If images of all the vertices of a determining set of a graph are known, then it is sufficient to identify the graph's automorphism group. On the other hand, fixing the vertices, i.e. imposing condition on the vertices that they will be mapped onto themselves, destroys symmetries. The minimum number of vertices that are needed to be fixed to destroy all the symmetries of a graph is called the fixing number and such a set of vertices is called a minimum fixing set. Boutin in [15] and Erwin and Harary in [6] independently introduced the terms of the determining number and the fixing number of graphs respectively. Boutin established in [15] that fixing the determining set will yield only identity automorphism which establishes the equivalence of the determining number and the fixing number. Hence, identifying symmetries and destroying symmetries are equivalent problems. The fixing numbers of different families of graphs, including Cartesian products [1], Cayley graphs and Frucht graphs [3] and complete graphs, paths, cycles [6] and as well as automorphism related parameters have been studied in [16], [17], [18], [19], [20], [21], [22]. The concept of symmetry-breaking was introduced by Albertson and Collins [23] using the concept of the distinguishing number of graphs where graphs vertices are labelled using t different labels so that no graph automorphism preserves labels. Minimum such t is called the distinguishing number and in [24], it is shown that the distinguishing number is strongly related to the idea of determining sets.

To distinguish the vertices of a given graph, Harary and Melter [25] and Slater [26] independently introduced the notions of resolving sets and the metric dimension, the minimum number of vertices that enable all other vertices to be uniquely distinguished by their distances from a subset of vertices in the graph. Harary and Melter [25] used the terminology of locating set and the location number whereas Slater [26] used the terms of resolving set and the metric dimension which we will be using wherever needed in this paper. In 2006, it was noticed by Erwin and Harary and also by Boutin that if all the vertices of resolving sets are mapped onto themselves, only trivial automorphism exists and that the minimum cardinality of a resolving set yields an upper bound for the fixing number of graphs. This bound is attained for graphs like paths, cycles, complete graphs and complete bipartite graphs and the metric dimension of trees and wheels is greater than the fixing numbers of these families of graphs. In 2006, Boutin posed the question “Can the difference between a graph's determining number and its smallest distance determining sets size be arbitrarily large?”. The question was addressed in [27] and [28]. Caceres et al. [27] in 2010, studied the difference of the metric dimension and the fixing number of cartesian products of graphs and trees. They showed that there are trees for which this difference is arbitrarily large. Garijo et al. [28], also studied the maximum value of the difference between the fixing number and the metric dimension of a graph. Recently, several authors studied both of these important parameters, the fixing number and the metric dimension of some graphs. In [29], the lower and upper bounds of fixing sets of the twin graphs are studied with some special classes of graphs. Das [30] investigated the fixing number of generalised and double generalised Petersen graphs. The fixing number and the metric dimension of the zero-divisor graph $\mathrm{\Gamma }({\prod }_{i=1}^n{\mathbb{Z}}_2)$ and the co-prime graphs with their properties are studied in [31], [32]. Due to equivalence of the determining number and the fixing number, we will use terms fixing set and the fixing number throughout the paper.

Now, we recall some definitions of graph theory which are necessary for this article. Graphs are used to model relations between elements of sets. Elements of sets are represented by vertices and relations between elements are represented by a set of edges (or links). Let $G=(V(G),E(G))$ be a simple graph where $V(G)$ is a set of elements called vertices and $E(G)$ is a set of relations called edges. The cardinality of $V(G)$ and the cardinality of $E(G)$ are called the order and the size of G, respectively. If there exists a set of edges from vertex x to vertex y, then we will say that there is a path between x and y. We say that G is connected if there is a path between each of its two vertices. The number of edges in the shortest $x-y$ path for a connected graph G is the distance between two vertices x and y, denoted by $d(x,y)$.

An automorphism of a graph G for a graph G is a bijective mapping $\theta :V(G)\to V(G)$ such that $\theta (x)\theta (y)\in E(G)$ if and only if $xy\in E(G)$. The automorphism group of G, formed under the operation of composition of mappings is the set of all automorphisms defined on a graph G and is denoted by $Aut(G)$. The stabilizer of a set F is a subset of $Aut(G)$ and is defined as $Stab(F)=\{\theta \in Aut(G)|y=\theta (y),\forall y\in F\}$. The orbit of a vertex x, denoted by $O(x)$, is the set defined as $\{x\in V(G)|\theta (x)=yforsome\theta \in Aut(G)\}$. If two vertices x and y belongs to the same orbit, then they are similar vertices. A set $F\subseteq V(G)$ is a fixing set of G if the only automorphism that fixes every vertex in F is the identity automorphism and the fixing number of G, denoted by $fix(G)$, is defined as the minimum cardinality of a fixing set. A vertex is fixed in a graph G, if it is fixed by every automorphism of G.

Numerous authors have investigated the connections between number theory and graph theory; for example, see [33], [34], [35]. We notice that graphs associated to different numbers display similar characteristics. Vasumathi et al. [36], introduced the idea of an arithmetic graph ${\mathbb{A}}_m$, studied some of its properties and also discovered an efficient method for creating an arithmetic graph using the given domination parameters. The arithmetic graphs containing the set of all divisors of m (excluding 1), where $m\in \mathbb{N}$ as its vertex set and $m=p_1^{r_1}{p}_2^{r_2}...p_n^{r_n}$, for n distinct primes and $r_i\ge 1$ for $1\le i\le n$. If two different divisors $x,y$ of m have the same prime factors, such as $(x=p_1{p}_{2}andy=p_1^2{p}_2^2)$, then they are said to have the same parity. Then $x\sim y$ if $x,y$ are of different parity and $gcd(x,y)=p_i$ for i, $1\le i\le n$. Also, for any vertex $x={\prod }_{i=1}^n{p}_i^{{\alpha }_i}$ with at least one ${\alpha }_i\ne 0$, $p_i$ is termed as the primary factor whereas $p_i^{{\alpha }_i}$, ${\alpha }_i\ge 2$ is termed as the secondary factor of x. The vertex set of the arithmetic graph ${\mathbb{A}}_m$ for a composite number $m={\prod }_{i=1}^n{p}_i^{r_i}$ is denoted by $V({\mathbb{A}}_m)$ and the cardinality of the vertex set of the arithmetic graphs ${\mathbb{A}}_m$ is given as $|V({\mathbb{A}}_m)|={\prod }_{i=1}^n(r_i+1)-1$. The arithmetic graphs are the twin-free graphs when $m={\prod }_{i=1}^n{p}_i$ and $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $1\le r_i\le 2$. Various authors examined the arithmetic graphs for different graph parameters [37], [38], [39]. In [40], Rao et al. investigated the split domination for arithmetic graphs. Moreover, the annihilator domination has been investigated for the arithmetic graphs in [41]. Rehman et al. [42], studied the arithmetic graphs and computed the results for the metric dimension of the arithmetic graphs ${\mathbb{A}}_m$.

Motivated by the close relationship between the fixing number and the metric dimension, we study symmetries and symmetry breaking of ${\mathbb{A}}_m$ in this paper. For the purpose of this paper, we introduce the term, the skeleton of a vertex $x\in V({\mathbb{A}}_m)$, denoted by $S_x$. The skeleton $S_x$ is the set of all those primes which are used in the prime factorization of x for ${\mathbb{A}}_m$. We partition the vertex set of ${\mathbb{A}}_m$ for $m={\prod }_{i=1}^n{p}_i^{r_{{}_i}}$, into n classes $X_l$, $(1\le l\le n)$, where $X_l=\{x\in V({\mathbb{A}}_m):|S_x|=l\}$. For example, if $n=3$ and $r_i=1$ for $1\le i\le 3$ then $X_1=\{p_1,p_2,p_3\}$, $X_2=\{p_1{p}_2,p_2{p}_3,p_1{p}_3\}$ and $X_3=\{p_1{p}_2{p}_3\}$. For our purpose, we write the canonical representation of $m={\prod }_{i=1}^n{p}_i^{r_i}$ such that $r_1\le r_2\le \dots \le r_n$. For example, if $m=600$ then $p_1=3,p_2=5,p_3=2$ with $r_1=1,r_2=2,r_3=3$.

The paper is organized as follows. In section 2, we study the properties of the arithmetic graphs in which we discuss the distance between any two vertices, the degrees of vertices and give the formula for twin classes in arithmetic graphs. Section 3 is based on the identification of symmetries of the arithmetic graphs and their related properties. We describe symmetries of the arithmetic graph ${\mathbb{A}}_m$ and prove that $Aut({\mathbb{A}}_m)$ is isomorphic to $S_n$ for $m={\prod }_{i=1}^n{p}_i$. We also give the automorphisms of ${\mathbb{A}}_m$ for $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $r_i\ge 1$ and $n\ge 2$ by using permutations of $\{p_1,p_2,\dots ,p_n\}$. In section 4, the connections between the fixing numbers and twin sets of ${\mathbb{A}}_m$ are studied. We find the exact fixing number of the arithmetic graphs where $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $r_i\ge 1$. In the last section, concluding remarks are given.

2. Properties of the arithmetic graphs

In this section, we describe the properties of the arithmetic graphs such as the distance between vertices, the degree of a vertex and the number of twin classes in arithmetic graphs. We begin our study by giving results on the distance between the vertices of ${\mathbb{A}}_m$.

Lemma 2.1

Let $m={\prod }_{i=1}^2{p}_i^{r_i}$ with $r_i\ge 1$ then,

(i) For each $x,y\in X_i$ , $1\le i\le 2$ then $d(x,y)=2$ ,

(ii) For each $x\in X_2$ and $y\in X_1$ , then $d(x,y)=1$ or $d(x,y)=3$ .

Proof

(i) Suppose $x,y\in X_i$, for $1\le i\le 2$ such that $x\nsim y$ then we have the following cases.

Case 1. Let $x,y\in X_1$ and $S_x\ne S_y$ then $x\sim p_1{p}_2\sim y$ gives that $d(x,y)=2$.

Case 2. Let $x,y\in X_2$ and $S_x=S_y$ then $x\sim p_i\sim y$, for any $1\le i\le 2$, gives that $d(x,y)=2$.

$(ii)$ If $x\in X_2$ and $y\in X_1$ such that $x\sim y$ then $d(x,y)=1$. Now suppose that $x=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\in X_2$ for $1\le {\alpha }_i\le 2$ and at least one of ${\alpha }_i\ge 1$ for $1\le i\le 2$ and $y=p_i^{{\beta }_i}\in X_1$, $1\le i\le 2$ and for some ${\beta }_i\ge 2$ such that $x\nsim y$ then $x\sim p_i\sim p_1{p}_2\sim y$ gives that $d(x,y)=3$. □

In the following lemma, we investigate the distance between any two vertices in ${\mathbb{A}}_m$ for $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $n\ge 3$ and $r_i\ge 1$.

Lemma 2.2

Let $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $r_i\ge 1$ and $n\ge 3$ then,

(i) For each $x\in X_n$ and $y\in V({\mathbb{A}}_m)\setminus X_1$ , $d(x,y)=2$ .

(ii) For $x\ne y$ and $x,y\in {\cup }_{i=1}^{n-1}{X}_i$ , $1\le d(x,y)\le 2$ .

(iii) For $x\in X_n$ and $y\in X_1$ , $d(x,y)=1$ or $d(x,y)=3$ .

Proof

(i) Since $S_x=S_m$ so $S_y\subseteq S_x$ and $|S_y|\ge 2$ gives that $gcd(x,y)\ne p_i$ which yield that $x\nsim y$. Let $p_j\in S_y$ then $y\sim p_j$ and $x\sim p_i$ for each $1\le i\le n$ gives that $d(x,y)=2$.

$(ii)$ Now to prove that $1\le d(x,y)\le 2$ for $x\ne y\in {\cup }_{i=1}^{n-1}{X}_i$, we study the following cases:

Case 1. Let $x,y\in X_1$, then $x\nsim y$ and we have the following cases:

Subcase 1. If $S_x\ne S_y$ and $S_x=\{p_i\}$, $S_y=\{p_j\}$ for some $1\le i\ne j\le n$, then $x\sim p_i{p}_j\sim y$ which gives that $d(x,y)=2$.

Subcase 2. If $S_x=S_y=\{p_i\}$ for some i, then $x\sim p_i{p}_j\sim y$ which gives that $d(x,y)=2$.

Case 2. Let $x\in X_1$ and $y\in {\cup }_{i=2}^{n-1}{X}_i$ such that $x\nsim y$, then we have the following cases:

Subcase 1. If $x=p_i$ for some $1\le i\le n$ and $x\nsim y$ so for every $p_j\in S_y$, $x\sim p_i{p}_j\sim y$ which yields that $d(x,y)=2$.

Subcase 2. If $x=p_i^{\alpha }$ where $2\le \alpha \le r_i$ for some i and $x\nsim y$ so we have following possibilities:

(i) Let $p_i\in S_y$. Since $y\in {\cup }_{i=2}^{n-1}{X}_i$ so there exist at least one $p_j\in S_m$ such that $p_j\notin S_y$ then $x\sim p_i{p}_j\sim y$ which gives that $d(x,y)=2$.

$(ii)$ Let $p_i\notin S_y$ and $p_j\in S_y$ such that $x\sim p_i{p}_j\sim y$ which gives that $d(x,y)=2$.

Case 3. Let $x,y\in {\cup }_{i=2}^{n-1}{X}_i$ and $x\nsim y$ then we have following cases:

Subcase 1. Let $S_x\cap S_y=\varnothing$, $p_i\in S_x$ and $p_j\in S_y$ then $x\sim p_i{p}_j\sim y$ which gives that $d(x,y)=2$.

Subcase 2. Let $S_x\cap S_y=\varnothing$ and $p_i\in S_x\cap S_y$ then $x\sim p_i{p}_j\sim y$ which gives that $d(x,y)=2$.

Combining all above cases, we conclude that $1\le d(x,y)\le 2$.

$(iii)$ For $x\in X_n$ and $y\in X_1$, $S_y\subseteq S_x$. If $x=p_1{p}_2...p_n$ and $y=p_i^{r_i}$ for some $1\le i\le n;r_i\ge 1$ then $d(x,y)=1$. For $x={\prod }_{i=1}^n{p}_i^{{\alpha }_i}$ for at least one ${\alpha }_i\ge 2$ and $y=p_i^{{\beta }_i}$ for some $1\le i\le n;{\beta }_i\ge 2$. Then, $gcd(x,y)\ne p_i$ which yields that $x\nsim y$. Therefore, $x\sim p_i\sim p_1{p}_2...p_n\sim y$ which gives that $d(x,y)=3$. □

For $m={\prod }_{i=1}^2{p}_i^{r_i}$ with all $r_i\ge 1$, $1\le i\ne j\le 2$ and $2\le {\alpha }_i,{\alpha }_j\le r_i$, we notice that $deg(p_i)=deg(p_j)=r_i{r}_j$, $deg(p_i{p}_j)=r_i+r_j$, $deg(p_i^{{\alpha }_i})=r_j$, $deg(p_j^{{\alpha }_j})=r_i$, $deg(p_i^{{\alpha }_i}{p}_j)=1+r_j$, $deg(p_i{p}_j^{{\alpha }_j})=r_i+1$ and $deg(p_i^{{\alpha }_i}{p}_j^{{\alpha }_j})=2$. For the degrees of vertices of the arithmetic graph ${\mathbb{A}}_m$ when $m={\prod }_{i=1}^n{p}_i^{r_i}$ with all $r_i\ge 1$ and $n\ge 3$, we introduce the set $X=\{1,2,\dots ,n\}$ as the indexing set of m. For $P=\{p_i|i\in X\}$ and $A\subseteq X$, let $m_A={\prod }_{i\in A}{p}_i^{r_i}$. Moreover, $m_A=0$ if $A=\varnothing$. For a non-identity divisor $x={\prod }_{i=1}^n{p}_i^{{\alpha }_i}$ of m, the collection $I_x=\{i\in X|{\alpha }_i=1\}$, $J_x=\{i\in X|{\alpha }_i\ge 2\}$ and $K_x=\{i\in X|{\alpha }_i=0\}$ gives a partition of X. Note that ${\mathbb{A}}_{m_{K_x}}$ is the induced subgraph of ${\mathbb{A}}_m$ on $m_{K_x}$ vertices and $|V({\mathbb{A}}_{m_{K_x}})|={\prod }_{i\in K_x}(r_i+1)-1$. In the next result, we give the formulae for the degree of x in ${\mathbb{A}}_m$.

Theorem 2.3

Let ${\mathbb{A}}_m$ be the arithmetic graph of $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $n\ge 3$ and $r_i\ge 1$ for each i. For a vertex $x=\prod _{i=1}^n{p}_i^{{\alpha }_i}$ for some $0\le {\alpha }_i\le r_i$ , we have

$$deg(x)=\{\begin{array}{ccc}{r}_j(\prod _{i\in K_x}(r_i+1)-1)\hfill & \mathit{\text{if}}\hfill & x=p_j\hfill \\ \prod _{i\in K_x}(r_i+1)-1\hfill & \mathit{\text{if}}\hfill & x=p_j^{{\alpha }_j},{\alpha }_j\ge 2\hfill \\ (\sum _{i\in I_x}{r}_i+|J_x|)\prod _{i\in K_x}(r_i+1)\hfill & \mathit{\text{if}}\hfill & x\notin X_1.\hfill \end{array}$$

Proof

(i) Suppose $x=p_j$, then $I_x=\{j\}$, $J_x=\varnothing$ and $K_x=X\setminus \{j\}$. By definition of ${\mathbb{A}}_m$, x is not adjacent to any vertex in $X_1$ and for each $r_j\ge {\alpha }_j\ge 1$, x is adjacent to every vertex of the form $p_j^{{\alpha }_j}y$ for each $y\in V({\mathbb{A}}_{m_{K_x}})$ which yields that there are $r_j|V({\mathbb{A}}_{m_{K_x}})|$ number of such vertices in $V({\mathbb{A}}_m)\setminus X_1$. Hence, $deg(x)=r_j|V({\mathbb{A}}_{m_{K_x}})|=r_j({\prod }_{i\in K_x}(r_i+1)-1)$.

$(ii)$ Suppose $x=p_j^{{\alpha }_j}$ with ${\alpha }_j\ge 2$, then $I_x=\varnothing$, $J_x=\{j\}$ and $K_x=X\setminus \{j\}$. By definition of ${\mathbb{A}}_m$, x is not adjacent to any vertex in $X_1$ and x is adjacent to every vertex of the form $p_{j}y$ for each $y\in V({\mathbb{A}}_{m_{K_x}})$ which yields that there are $|V({\mathbb{A}}_{m_{K_x}})|$ number of such vertices in $V({\mathbb{A}}_m)\setminus X_1$. Hence, $deg(x)=|V({\mathbb{A}}_{m_{K_x}})|={\prod }_{i\in K_x}(r_i+1)-1$.

$(iii)$ Now suppose x has at least two primary factors and $x={\prod }_{i=1}^n{p}_i^{{\alpha }_i}$ with $0\le {\alpha }_i\le r_i$. For each $i\in I_x$, the number of vertices adjacent with x in $X_1$ with factor $p_i$ is $r_i$ and for each $j\in J_x$ the number of vertices adjacent with x in $X_1$ with factor $p_j$ is one so the total number of vertices adjacent with x in $X_1$ is ${\sum }_{i\in I_x}{r}_i+|J_x|$. Also for each $i\in I_x$ and $1\le {\alpha }_i\le r_i$, x is adjacent to every vertex of the form $p_j^{{\alpha }_j}y$ for each $y\in V(A_{m_{K_x}})$ and there are ${\sum }_{i\in I_x}{r}_i|V({\mathbb{A}}_{m_{K_x}})|$ number of such vertices in ${\cup }_{i=2}^n{X}_i$. Now for each $j\in J_x$, x is adjacent to every vertex of the form $p_{j}y$ for each $y\in V({\mathbb{A}}_{m_{K_x}})$ and there are $|J_x||V(A_{m_{K_x}})|$ such vertices in ${\cup }_{i=2}^n{X}_i$. Hence, the total number of vertices adjacent with x in ${\cup }_{i=2}^n{X}_i$ is ${\sum }_{i\in I_x}{r}_i|V({\mathbb{A}}_{m_{K_x}})|+|J_x||V({\mathbb{A}}_{m_{K_x}})|$ and the degree of x can be obtained as ${\sum }_{i\in I_x}{r}_i|V({\mathbb{A}}_{m_{K_x}})|+|J_x||V(A_{m_{K_x}})|+{\sum }_{i\in I_x}{r}_i+|J|=({\sum }_{i\in I_x}{r}_i+|J_x|)(|V({\mathbb{A}}_{m_{K_x}})|+1)$. □

In the following corollary, we give the degrees of vertices in ${\mathbb{A}}_m$ when $m={\prod }_{i=1}^n{p}_i$.

Corollary 2.4

For$m={\prod }_{i=1}^n{p}_i$with$n\ge 2$. Then, we have

(i)$deg(p_i)=2^{n-1}-1$,

(ii)$deg(p_{i_1}{p}_{i_2}{p}_{i_3}...p_{i_k})=k{(2)}^{n-k}$, with$2\le k\le n$where$i_j\in \{1,2,...,n\}$and$1\le j\le k$.

Proof

(i) By definition of the arithmetic graph, $p_i$ is adjacent with vertices which have $p_i$ as a factor. Notice that $p_i$ has $\left(\begin{array}{c}n-1\\ 1\end{array}\right)$ neighbours of the form $p_i{p}_{i_1}$, $\left(\begin{array}{c}n-1\\ 2\end{array}\right)$ neighbours of the form $p_i{p}_{i_1}{p}_{i_2}$ and $\left(\begin{array}{c}n-1\\ n-k\end{array}\right)$ neighbours of the form ${\prod }_{j=1}^k{p}_{i_j}$ has $\left(\begin{array}{c}n-1\\ 1\end{array}\right)+\left(\begin{array}{c}n-1\\ 2\end{array}\right)+...+\left(\begin{array}{c}n-1\\ n-1\end{array}\right)$ neighbours. It is a established fact that $\left(\begin{array}{c}n-1\\ 1\end{array}\right)+\left(\begin{array}{c}n-1\\ 2\end{array}\right)+...+\left(\begin{array}{c}n-1\\ n-1\end{array}\right)=2^{n-1}-1$. Hence, $deg(p_i)=2^{n-1}-1$ for $1\le i\le n$.

$(ii)$ By definition of the arithmetic graph, ${\prod }_{j=1}^k{p}_{i_j}$ can not be adjacent with a vertex x if it has two or more primes in common with ${\prod }_{j=1}^k{p}_{i_j}$. Therefore, ${\prod }_{j=1}^k{p}_{i_j}$ has neighbours which have only one prime factor $p_{i_j}$ where $1\le j\le k$ and other factors are from remaining $n-k$ primes. ${\prod }_{j=1}^k{p}_{i_j}$ is adjacent with vertices of the form $p_{i_j}$ where $1\le j\le k$. ${\prod }_{j=1}^k{p}_{i_j}$ has $k(n-k)$ neighbours of the form $p_{i_j}{p}_{l_1}$ where $p_{l_1}\ne p_{i_j}$ for any $1\le j\le k$, $k\left(\begin{array}{c}n-k\\ 2\end{array}\right)$ neighbours of the form $p_{i_j}{p}_{l_1}{p}_{l_2}$ where $p_{l_t}\ne p_{i_j}$ for any $1\le j\le k$ and $t=1,2$ and similarly $k\left(\begin{array}{c}n-k\\ q\end{array}\right)$ neighbours of the form $p_{i_j}{p}_{l_1}{p}_{l_2}...p_{l_q}$ where $p_{l_t}\ne p_{i_j}$ for any $1\le j\le k$ and $1\le t\le q$. So $deg(p_{i_1}{p}_{i_2}{p}_{i_3}...p_{i_k})=k(\left(\begin{array}{c}n-k\\ 0\end{array}\right)+\left(\begin{array}{c}n-k\\ 1\end{array}\right)+\left(\begin{array}{c}n-k\\ 2\end{array}\right)+...+\left(\begin{array}{c}n-k\\ n-k\end{array}\right))=k{(2)}^{n-k}$ which completes the proof. □

From Corollary 2.4(ii), it can be seen that all the vertices of $X_k$ for $2\le k\le n$ have the same degree. It is important to notice that there may exist distinct vertices $x,y\in V({\mathbb{A}}_m)$ such that $deg(x)=deg(y)$ and $x\in X_i$ and $y\in X_j$ where $i\ne j$. For example, when $m={\prod }_{i=1}^3{p}_i$, $deg(p_1)=deg(p_1{p}_2{p}_3)$.

A vertex x of G has the open neighbourhood if $N(x)=\{y\in V(G):xy\in E(G)\}$ and the closed neighbourhood if $N[x]=N(x)\cup \{x\}$. If $N[x]=N[y]$, then two vertices $x,y$ are true twins otherwise, they are false twins if $N(x)=N(y)$. If every pair of vertices in a set of vertices T are twins, the set is referred to as a twin-set. The set of all twins of x is denoted by $T(x)$ and the count of twin classes of graph G is denoted by ${\mathcal{T}}_G$. Note that $T(x)=\{y\in V({\mathbb{A}}_m):N[x]=N[y]\text{or}N(x)=N(y)\}$. A twin class is known as a trivial twin class if $T(x)=\{x\}$ whereas a non-trivial twin class has more than one vertex in it. A twin-free graph is a graph in which $T(x)=\{x\}$ for all $x\in V({\mathbb{A}}_m)$. For $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $n\ge 3$ and $r_i\ge 1$ for each i, $T({\prod }_{j=1}^k{p}_{i_j})=\{{\prod }_{j=1}^k{p}_{i_j}\}$ with $1\le k\le n$ and if $r_{i_j}=2$ then $T({\prod }_{j=1}^k{p}_{i_j}^{{\alpha }_{i_j}})=\{{\prod }_{j=1}^k{p}_{i_j}^{{\alpha }_{i_j}}\}$ with $1\le {\alpha }_{i_j}\le 2$. If $r_{i_j}\ge 3$ then $|T({\prod }_{j=1}^k{p}_{i_j}^{{\alpha }_{i_j}})|\ge 2$ with ${\alpha }_{i_j}\le r_{i_j}$ and at least one ${\alpha }_{i_j}\ge 3$. Notice that for $m={\prod }_{i=1}^n{p}_i$ and $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $n\ge 3$ and $1\le r_i\le 2$ are the twin-free arithmetic graphs.

For $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $n\ge 3$ and $r_i\ge 1$ for each i, let $U=\{i\in X|r_i=1\}$, $V=\{i\in X|r_i=2\}$ and $W=\{i\in X|r_i\ge 3\}$ where $X=\{1,2,...,n\}$ is the indexing set of m. Note that there exists $T_i$, $1\le i\le {\mathcal{T}}_{{\mathbb{A}}_m}$, twin classes in ${\mathbb{A}}_m$ including trivial and non-trivial twin classes. Using these notations, we give the next result on the number of twin classes in ${\mathbb{A}}_m$ where $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $r_i\ge 1$, denoted by ${\mathcal{T}}_{{\mathbb{A}}_m}$.

Lemma 2.5

For $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $n\ge 3$ and $r_i\ge 1$ , then the number of twin classes of ${\mathbb{A}}_m$ is given as,

$${\mathcal{T}}_{{\mathbb{A}}_m}=\sum _{k=0}^{n-|U|}\left(\begin{array}{c}n-|U|\\ k\end{array}\right)2^{n-k}-1,$$

where $U=\{i\in X|r_i=1\}$ and $X=\{1,2,...,n\}$ is an index set.

Proof

For $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $n\ge 3$ and $r_i\ge 1$, let $X=\{1,2,...,n\}$ be an indexing set and U consists of first t indices of primes with power one in the representation of m. We first consider the twin classes of the form ${\prod }_{j=1}^l{p}_{i_j}$ with $1\le l\le n$. Notice that there are $\left(\begin{array}{c}n\\ l\end{array}\right)$ classes of the form ${\prod }_{j=1}^l{p}_{i_j}$. As $1\le l\le n$ so there are ${\sum }_{l=1}^n\left(\begin{array}{c}n\\ l\end{array}\right)$ twin classes. Further, we consider the twin classes of the form ${\prod }_{j=t+1}^s{p}_{i_j}^{{\alpha }_{i_j}}$ with at least one $2\le {\alpha }_{i_j}\le r_{i_j}$ and $t+1\le s\le n$. Notice that there are $\left(\begin{array}{c}n-t\\ k\end{array}\right)$ classes of this form. As $1\le k\le n-t$ so in all there are ${\sum }_{k=1}^{n-t}\left(\begin{array}{c}n-t\\ k\end{array}\right)$ twin classes. Also, there are $\left(\begin{array}{c}n-t\\ k\end{array}\right)(2^{n-k}-1)$ classes of the form ${\prod }_{j^{\prime }=1}^h{p}_{i_{j^{\prime }}}{\prod }_{j=t+1}^s{p}_{i_j}^{{\alpha }_{i_j}}$ for $1\le h\le t$, $t+1\le s\le n$, $2\le {\alpha }_{i_j}\le r_{i_j}$. So, in all there are ${\sum }_{k=1}^{n-|U|}\left(\begin{array}{c}n-|U|\\ k\end{array}\right)2^{n-k}$ twin classes. Hence, summing up we have the total count of twin classes follows as ${\mathcal{T}}_{{\mathbb{A}}_m}={\sum }_{k=0}^{n-|U|}\left(\begin{array}{c}n-|U|\\ k\end{array}\right)2^{n-k}-1$. □

For a composite number $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $n\ge 3$ and $r_i\ge 1$ for each i, let $W^{\prime }=\{i\in X|r_i\le 2\}$, $W=\{i\in X|r_i\ge 3\}$. Note that for $W=\varnothing$, the graph ${\mathbb{A}}_m$ has no non-trivial twin class. Now for $W\ne \varnothing$, let $L_i=\{l_1,l_2,\dots ,l_i\}\subseteq W$, $W_i=W\setminus L_i$ and $m_i={\prod }_{i\in W^{\prime }}{p}_i^{r_i}{\prod }_{k\in W_i}{p}_k$. Using these notations, we give the next result on the number of non-trivial twin classes in ${\mathbb{A}}_m$.

Theorem 2.6

For $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $n\ge 3$ and $r_i\ge 1$ for each i, let $W^{\prime }=\{i\in X|r_i\le 2\}$ , $W=\{i\in X|r_i\ge 3\}$ and $W\ne \varnothing$ . The number of non-trivial twin classes of ${\mathbb{A}}_m$ can be found by the formula

$$\sum _{i=1}^{|W|}|V({\mathbb{A}}_{m_i})|\left(\begin{array}{c}|W|\\ i\end{array}\right)+2^{|W|}-1,$$

where $m_i={\prod }_{i\in W^{\prime }}{p}_i^{r_i}{\prod }_{k\in W_i}{p}_k$ , $W_i=W\setminus L_i$ and $L_i=\{l_1,l_2,\dots ,l_i\}\subseteq W$ .

Proof

Note that for every nonempty subset $L_i$ of W, the set defined as: ${\prod }_{j\in L_i}{P}_j$ $=\{{\prod }_{j\in L_i}{p}_j^{{\alpha }_j}|2\le {\alpha }_j\le r_j\}$ is a non-trivial class of false twins in ${\mathbb{A}}_m$ and the possible such classes are $2^{|W|}-1$. Now for $L_i\subseteq W$ and $m_i={\prod }_{i\in W^{\prime }}{p}_i^{r_i}{\prod }_{k\in W_i}{p}_k$, for each $x\in V({\mathbb{A}}_{m_i})$ the set defined as: $x{\prod }_{j\in L_i}{P}_j=\{x{\prod }_{j\in L_i}{p}_j^{{\alpha }_j}|2\le {\alpha }_j\le r_j\}$ is a non-trivial class of false twins in ${\mathbb{A}}_m$. Moreover, there are $|V({\mathbb{A}}_{m_i})|\left(\begin{array}{c}|W|\\ i\end{array}\right)$ such non-trivial false twin classes in ${\mathbb{A}}_m$. Hence, the total number of non-trivial twin classes in ${\mathbb{A}}_m$ is ${\sum }_{i=1}^{|W|}|V({\mathbb{A}}_{m_i})|\left(\begin{array}{c}|W|\\ i\end{array}\right)+2^{|W|}-1$. □

3. Symmetries in the arithmetic graphs

In this section, we describe symmetries of the arithmetic graphs ${\mathbb{A}}_m$ and give conditions which induce automorphisms of ${\mathbb{A}}_m$. We begin by proving that for each $p_i\in P$ and $\theta \in Aut({\mathbb{A}}_m)$ then $\theta (p_i)\in P$.

Lemma 3.1

Let$m={\prod }_{i=1}^n{p}_i^{r_i}$with$r_i\ge 1$and$n\ge 3$. If$\theta \in Aut({\mathbb{A}}_m)$, then$\theta (P)=P$where$P=\{p_1,p_2,...,p_n\}$.

Proof

Note that $p_i\sim x$ for all $x\in X_n$. Since θ is automorphism so $\theta (p_i)\sim \theta (x)$ for all $x\in X_n$. If $\theta (p_i)\notin X_1$ then $\theta (p_i)$ can not be adjacent with any vertex of $X_n$. Hence $\theta (p_i)\in X_1$. Further if $\theta (p_i)\notin P$ then $deg(p_i)\ne deg(\theta (p_i))$ which is not possible. Hence $\theta (p_i)\in P$. □

In the next result, we discuss symmetries of m and prove that for each $x\in X_l$ where $1\le l\le n$ and $\theta \in Aut({\mathbb{A}}_m)$, $\theta (x)\subseteq X_l$.

Lemma 3.2

Let$\theta \in Aut({\mathbb{A}}_m)$for$m={\prod }_{i=1}^n{p}_i^{r_i}$, then$\theta (X_l)=X_l$, for each$1\le l\le n$.

Proof

Let $x\in X_l$ with $l\ge 2$ and $S_x=\{p_1,p_2,...,p_l\}$. By definition of the arithmetic graph $p_i\sim x$ for each $i\in \{1,2,...,l\}$. Since θ is an automorphism so $\theta (x)\sim \theta (p_i)$ for each $i\in \{1,2,\dots ,l\}$. Lemma 3.1 gives that $\theta (p_i)\in P$ which yields that $\theta (x)$ has l distinct primary factors and hence $\theta (x)\in X_l$. As $\theta (X_l)=X_l$ for $2\le l\le n$, this yields that $\theta (X_1)=X_1$. □

For a given $m={\prod }_{i=1}^n{p}_i$, we define the set of primes $P=\{p_1,p_2,...,p_n\}$ in the canonical representation of m. We denote by $Sym(P)$, the symmetric group of P, consisting of all the permutations on the set P. Note that $Sym(P)\cong S_n$ where $S_n$ is symmetry group on n elements. For a permutation η on P, we associate an automorphism θ on $x=p_1{p}_2...p_j\in V({\mathbb{A}}_m)$ where $j\le n$ by

$$\theta (x)=\eta (p_1)\eta (p_2)....\eta (p_j),$$ (1)

and $|S_x|=j;1\le j\le l$ for $1\le l\le n$.

Theorem 3.3

For $m={\prod }_{i=1}^n{p}_i$ , $S_n\cong Aut({\mathbb{A}}_m)$ .

Proof

Let $\eta \in Sym(P)$ and $\theta \in Aut({\mathbb{A}}_m)$. We define a function $\phi :Sym(P)\to Aut({\mathbb{A}}_m)$ as $\phi (\eta )=\theta$ and prove that φ is a group isomorphism.

(i) φ is well defined: Let ${\eta }_1,{\eta }_2\in Sym(P)$ be two permutations such that ${\eta }_1\ne {\eta }_2$. Let $\phi ({\eta }_1)={\theta }_1$ and $\phi ({\eta }_2)={\theta }_2$. Since ${\eta }_1\ne {\eta }_2$, therefore ${\eta }_1(p_i)\ne {\eta }_2(p_i)$, for at least one i, $1\le i\le l$, yields $\{{\eta }_1(p_1),{\eta }_1(p_2),{\eta }_1(p_3),...,{\eta }_1(p_l)\}$ $\ne \{{\eta }_2(p_1),{\eta }_2(p_2),{\eta }_2(p_3),...,{\eta }_2(p_l)\}$ $\Rightarrow S_{{\theta }_1(x)}\ne S_{{\theta }_2(x)}\Rightarrow {\theta }_1(x)\ne {\theta }_2(x)$ $\Rightarrow {\theta }_1\ne {\theta }_2$.

$(ii)$φ is injective: Let ${\theta }_1,{\theta }_2\in Aut({\mathbb{A}}_m)$ such that ${\theta }_1\ne {\theta }_2$. Let ${\eta }_1,{\eta }_2\in Sym(P)$ be two permutations such that $\phi ({\eta }_1)={\theta }_1$ and $\phi ({\eta }_2)={\theta }_2$. Since, ${\theta }_1\ne {\theta }_2$, therefore there exists a vertex $x\in X_l$, for some $1\le l\le n$ such that ${\theta }_1(x)\ne {\theta }_2(x)$. Let $S_x=\{p_1,p_2,p_3,...,p_l\}$, then $S_{{\theta }_1(x)}\ne S_{{\theta }_2(x)}$ ⇒ $\{{\eta }_1(p_1),{\eta }_1(p_2),{\eta }_1(p_3),...,{\eta }_1(p_l)\}$ $\ne \{{\eta }_2(p_1),{\eta }_2(p_2),{\eta }_2(p_3),...,{\eta }_2(p_l)\}$ ⇒ ${\eta }_1(p_i)\ne {\eta }_2(p_i)$, for at least one i, $1\le i\le l,\Rightarrow {\eta }_1\ne {\eta }_2$.

$(iii)$φ is surjective: For any automorphism $\theta \in Aut({\mathbb{A}}_m)$, there exists a permutation $\eta \in Sym(P)$ such that $\phi (\eta )=\theta$.

$(iv)$φ is homomorphism: Let ${\eta }_1,{\eta }_2\in Sym(P)$ and ${\theta }_1,{\theta }_2\in Aut({\mathbb{A}}_m)$ such that $\phi ({\eta }_1)={\theta }_1$ and $\phi ({\eta }_2)={\theta }_2$. We claim that $\phi ({\eta }_1{\eta }_2)=\phi ({\eta }_1)\phi ({\eta }_2)={\theta }_1{\theta }_2$. Let $\phi ({\eta }_1{\eta }_2)=\theta$ and $x\in X_l$, for some l, $1\le l\le n$ such that $S_x=\{p_1,p_2,p_3,...,p_l\}$. Then $\theta (x)$ has skeleton $S_{\theta (x)}=\{{\eta }_1{\eta }_2(p_1),{\eta }_1{\eta }_2(p_2),{\eta }_1{\eta }_2(p_3),...,{\eta }_1{\eta }_2(p_l)\}$ $=\{{\eta }_1({\eta }_2(p_1)),{\eta }_1({\eta }_2(p_2)),{\eta }_1({\eta }_2(p_3)),...,{\eta }_1({\eta }_2(p_l))\}=S_{{\theta }_1({\theta }_2(x))}$ $=S_{{\theta }_1{\theta }_2(x)}\Rightarrow \theta ={\theta }_1{\theta }_2\Rightarrow \phi ({\eta }_1{\eta }_2)=\phi ({\eta }_1)\phi ({\eta }_2)$.

Thus, φ is an isomorphism between $Sym(P)$ and $Aut({\mathbb{A}}_m)$. □

In the next result, we give the automorphisms of ${\mathbb{A}}_m$ for $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $r_i\ge 1$ and $n\ge 2$ by using permutations of $\{p_1,p_2,\dots ,p_n\}$.

Theorem 3.4

Let $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $n\ge 2$ and $\eta \in Sym(P)$ then $\theta :V({\mathbb{A}}_m)\to V({\mathbb{A}}_m)$ defined as $\theta (p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}...p_n^{{\alpha }_n})=\eta {(p_1)}^{{\alpha }_1}\eta {(p_2)}^{{\alpha }_2}...\eta {(p_n)}^{{\alpha }_n}$ for each $p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}...p_n^{{\alpha }_n}\in V({\mathbb{A}}_m)$ , where $0\le {\alpha }_i\le r_i$ , $\forall i$ with at least one ${\alpha }_i\ne 0$ for some i, is an automorphism of ${\mathbb{A}}_m$ .

Proof

Let $\theta :V({\mathbb{A}}_m)\to V({\mathbb{A}}_m)$ be a function defined as $\theta (p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}...p_n^{{\alpha }_n})=\eta {(p_1)}^{{\alpha }_1}\eta {(p_2)}^{{\alpha }_2}...\eta {(p_n)}^{{\alpha }_n}$ for each $p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}...p_n^{{\alpha }_n}\in V({\mathbb{A}}_m)$ and we prove that θ is an automorphism of ${\mathbb{A}}_m$. Let $x=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}...p_n^{{\alpha }_n}$ and $y=p_1^{{\beta }_1}{p}_2^{{\beta }_2}...p_n^{{\beta }_n}$ with $0\le {\alpha }_i,{\beta }_i\le r_i$ for each i and at least one ${\alpha }_i\ne 0$ and one ${\beta }_i\ne 0$ for some $1\le i\le n$.

(i) θ is well defined: For $x=y$, we have ${\alpha }_i={\beta }_i$ and $\eta \in Sym(P)$ gives that $\theta (x)=\theta (y)$.

(ii) θ is injective: Suppose $\theta (x)=\theta (y)$ $\Rightarrow \eta {(p_1)}^{{\alpha }_1}\eta {(p_2)}^{{\alpha }_2}...\eta {(p_n)}^{{\alpha }_n}=\eta {(p_1)}^{{\beta }_1}\eta {(p_2)}^{{\beta }_2}...\eta {(p_n)}^{{\beta }_n}$ gives that ${\alpha }_i={\beta }_i$ for each i so $x=y$.

(iii) θ is surjective: For $v=p_1^{{\prime }^{{\alpha }_1}}{p}_2^{{\prime }^{{\alpha }_2}}...p_n^{{\prime }^{{\alpha }_n}}\in V({\mathbb{A}}_m)$ with $p_i^{\prime }\in P$ and $\eta \in Sym(P)$. Let $p_i\in P$ such that $\eta (p_i)=p_i^{\prime }$ for each $1\le i\le n$ then $u=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}...p_n^{{\alpha }_n}\in V({\mathbb{A}}_m)$ such that $\theta (u)=v$.

(iv) θ is homomorphism: Next, we assume $x\sim y$ then x and y have different parity and $gcd(x,y)=p_i$ for some $1\le i\le n$ and $S_x\cap S_y=p_i$.

Clearly, $\eta (p_i)=S_{\theta (x)}\cap S_{\theta (y)}$. Now $gcd(x,y)=p_i$ gives that ${\alpha }_i=1$ or ${\beta }_i=1$ or ${\alpha }_i={\beta }_i=1$ which gives that $gcd(\theta (x),\theta (y))=\eta (p_i)$. Hence, $\theta (x)\sim \theta (y)$.

Now, suppose that $x\nsim y$, then we consider following cases:

Case 1. If x and y have the same parity then $S_x=S_y$ and $\eta \in Sym(P)$ gives that $S_{\theta (x)}=S_{\theta (y)}$ which implies $\theta (x)\nsim \theta (y)$.

Case 2. If $S_x\cap S_y=\varnothing$ then $S_{\theta (x)}\cap S_{\theta (y)}=\varnothing$ because $\eta \in Sym(P)$ which yields that $\theta (x)\nsim \theta (y)$.

Hence, θ is an automorphism of ${\mathbb{A}}_m$. □

In the following corollary, using equation (1) and above theorem, we give the form of $\theta (x)={\prod }_{i\in S_x}\eta (p_i)$ for all $x\in V({\mathbb{A}}_m)$ if θ is an automorphism.

Corollary 3.5

For$m={\prod }_{i=1}^n{p}_i$with$n\ge 3$, a mapping$\theta :V({\mathbb{A}}_m)\to V({\mathbb{A}}_m)$is an automorphism of${\mathbb{A}}_m$if and only if there exists a permutation$\eta \in Sym(P)$such that θ has the form$\theta (x)={\prod }_{i\in S_x}\eta (p_i)$for all$x\in V({\mathbb{A}}_m)$.

4. Fixing number of the arithmetic graphs

In this section, we will give results on the fixing number of the arithmetic graph ${\mathbb{A}}_m$ for a given m. For a given $m={\prod }_{i=1}^n{p}_i^{r_i}$ with all $r_i\ge 1$ and $n\ge 2$ with indexing set $X=\{1,2,3,\dots ,n\}$, let $U=\{i\in X|r_i=1\}$, $V=\{i\in X|r_i=2\}$ and $W=\{i\in X|r_i\ge 3\}$ be the collection of subsets of X which give a partition of X. Note that for $n=2$, at least one of U, V and W is empty set. Notice that for $m={\prod }_{i=1}^2{p}_i^{r_i}$ with $r_i\ge 1$, $p_1,p_2$ are adjacent with all vertices of the form ${\prod }_{i=1}^2{p}_i^{{\alpha }_i}$ with $1\le {\alpha }_i\le r_i$. This yields that $N(p_1)=N(p_2)$ and $deg(p_1)=deg(p_2)=r_1{r}_2$ which implies that ${\mathbb{A}}_m$ has non-trivial symmetries and $fix({\mathbb{A}}_m)\ge 1$. This motivates us to study the fixing number of the arithmetic graph for $m={\prod }_{i=1}^2{p}_i^{r_i}$ where $r_i\ge 1$ in the following theorem.

Theorem 4.1

For a composite number $m={\prod }_{i=1}^2{p}_i^{r_i}$ with $r_i\ge 1$ , we have

$$fix({\mathbb{A}}_m)=\{\begin{array}{ccc}1\hfill & \mathit{\text{if}}\hfill & U\ne \varnothing ,W=\varnothing \hfill \\ 2\hfill & \mathit{\text{if}}\hfill & U=\varnothing ,V\ne \varnothing ,W=\varnothing \hfill \\ 1+2(r_i-2)\hfill & \mathit{\text{if}}\hfill & U\ne \varnothing ,V=\varnothing ,i\in W\hfill \\ 1+3(r_i-2)\hfill & \mathit{\text{if}}\hfill & U=\varnothing ,V\ne \varnothing ,i\in W\hfill \\ 2\sum _{i=1}^2(r_i-2)+\prod _{i=1}^2(r_i-1)\hfill & \mathit{\text{if}}\hfill & U=\varnothing =V,W\ne \varnothing .\hfill \end{array}$$

Proof

We consider the following cases:

(i) For $U\ne \varnothing ,V=\varnothing =W$, we have $m=p_1{p}_2$ and ${\mathbb{A}}_m$ is isomorphic to $P_3$ so $fix({\mathbb{A}}_m)=1$.

Also, for $U\ne \varnothing$, $V\ne \varnothing$ and $W=\varnothing$, the arithmetic graph ${\mathbb{A}}_m$ has order 5 and $p_1,p_2$ are false twins, moreover, degrees of the remaining three vertices are different which yields that $\{p_1\}$ is a minimum fixing set for ${\mathbb{A}}_m$.

$(ii)$ For $U=\varnothing ,V\ne \varnothing ,W=\varnothing$, we have $m=p_1^2{p}_2^2$ where the order of ${\mathbb{A}}_m$ is 8. We claim that $F=\{p_1,p_1^2\}$ is a fixing set for ${\mathbb{A}}_m$. We also note that $p_1,p_2$ are false twins in ${\mathbb{A}}_m$ and $N(p_1^2)=\{p_1{p}_2,p_1{p}_2^2\}$ and $N(p_2^2)=\{p_1{p}_2,p_1^2{p}_2\}$ which implies that $\theta (p_1^2)=p_2^2$, $\theta (p_2^2)=p_1^2$ and $\theta (p_1^2{p}_2)=p_1{p}_2^2$, $\theta (p_1{p}_2^2)=p_1^2{p}_2$ which shows that $p_1^2$ can be mapped onto $p_2^2$ if $p_1{p}_2^2$ mapped onto $p_1^2{p}_2$ and vice versa and hence $fix({\mathbb{A}}_m)\ge 2$. For this, we only need to prove that the set $F=\{p_1,p_1^2\}$ is a fixing set for ${\mathbb{A}}_m$. Consider $\theta \in Aut({\mathbb{A}}_m)$ such that $\theta (s)=s$ for each $s\in F$, we will prove that it is an identity automorphism. Assume contrary that θ is a non-trivial, i.e. there exist $x\ne y\in V({\mathbb{A}}_m)$ such that $\theta (x)=y$. Note that $V({\mathbb{A}}_m)\setminus F=\{p_2,p_2^2,p_1{p}_2,p_1^2{p}_2,p_1{p}_2^2,p_1^2{p}_2^2\}$ and $deg(p_2)=r_1{r}_2$, $deg(p_2^2)=r_1$, $deg(p_1{p}_2)=r_1+r_2$, $deg(p_1^2{p}_2)=1+r_2$, $deg(p_1{p}_2^2)=r_1+1$ and $deg(p_1^2{p}_2^2)=2$ which yields that θ acts trivially on $V({\mathbb{A}}_m)$. Hence, $fix({\mathbb{A}}_m)=2$.

$(iii)$ For $U\ne \varnothing ,V=\varnothing$ and $i\in W$ without loss of generality suppose $m=p_1{p}_2^{r_2}$ with $r_2\ge 3$. Now the sets $\{p_1,p_2\}$, $\{p_2^{\beta }|2\le \beta \le r_2\}$ and $\{p_1{p}_2^{\beta }|2\le \beta \le r_2\}$ are non-trivial false twin classes in ${\mathbb{A}}_m$ and hence $fix({\mathbb{A}}_m)\ge 1+2(r_2-2)$. Now we only need to prove that the set $F=\{p_1,p_2^3,...,p_2^{r_2},p_1{p}_2^3,..,p_1{p}_2^{r_2}\}$ is a fixing set for ${\mathbb{A}}_m$. Consider $\theta \in Aut({\mathbb{A}}_m)$ such that $\theta (s)=s$ for each $s\in F$, we will prove that it is an identity automorphism. Assume contrary that θ is a non-trivial, i.e. there exist $x\ne y\in V({\mathbb{A}}_m)$ such that $\theta (x)=y$. Note that $V({\mathbb{A}}_m)\setminus F=\{p_2,p_2^2,p_1{p}_2,p_1{p}_2^2\}$ and $deg(p_2)=r_2$, $deg(p_2^2)=1$, $deg(p_1{p}_2)=1+r_2$, $deg(p_1{p}_2^2)=2$ which yields that θ acts trivially on $V({\mathbb{A}}_m)$. Hence, $fix({\mathbb{A}}_m)=1+2(r_2-2)$.

$(iv)$ For $U=\varnothing ,V\ne \varnothing$ and $W\ne \varnothing$ without loss of generality suppose $m=p_1^2{p}_2^{r_2}$ with $r_2\ge 3$. Now the sets $\{p_1,p_2\}$, $\{p_2^{\beta }|2\le \beta \le r_2\}$, $\{p_1{p}_2^{\beta }|2\le \beta \le r_2\}$ and $\{p_1^2{p}_2^{\beta }|2\le \beta \le r_2\}$ are non-trivial false twin classes in ${\mathbb{A}}_m$ and hence $fix({\mathbb{A}}_m)\ge 1+3(r_2-2)$. Now we only need to prove that the set $F=\{p_1,p_2^3,\dots ,p_2^{r_2},p_1{p}_2^3,\dots$, $p_1{p}_2^{r_2},p_1^2{p}_2^3,\dots$, $p_1^2{p}_2^{r_2}\}$ is a fixing set for ${\mathbb{A}}_m$. Consider $\theta \in Aut({\mathbb{A}}_m)$ such that $\theta (s)=s$ for each $s\in F$, we will prove that it is an identity automorphism. Assume contrary that θ is a non-trivial, i.e. there exist $x\ne y\in V({\mathbb{A}}_m)$ such that $\theta (x)=y$. Note that $V({\mathbb{A}}_m)\setminus F=\{p_2,p_2^2,p_1{p}_2,p_1^2{p}_2,p_1{p}_2^2,p_1^2{p}_2^2\}$ and $deg(p_2)=2r_2$, $deg(p_2^2)=2$, $deg(p_1{p}_2)=2+r_2$, $deg(p_1{p}_2^2)=3$, $deg(p_1^2{p}_2)=1+r_2$, $deg(p_1^2{p}_2^2)=2$ which yields that θ acts trivially on $V({\mathbb{A}}_m)$. Hence, $fix({\mathbb{A}}_m)=1+3(r_2-2)$.

(v) For $U=V=\varnothing$, we have $m=p_1^{r_1}{p}_2^{r_2}$ with $r_1,r_2\ge 3$ and the sets $\{p_1,p_2\},\{p_1^{\alpha }|2\le \alpha \le r_1\},\{p_2^{\beta }|2\le \beta \le r_2\}$, $\{p_1^{\alpha }{p}_2|2\le \alpha \le r_1\},\{p_1{p}_2^{\beta }|2\le \beta \le r_2\},\{p_1^{\alpha }{p}_2^{\beta }|2\le \alpha \le r_1,2\le \beta \le r_2\}$ are non-trivial false twin classes in ${\mathbb{A}}_m$. Therefore, $fix({\mathbb{A}}_m)\ge 1+2(r_1-2)+2(r_2-2)+(r_1-1)(r_2-1)-1$ and we need to prove that the set $F=\{p_1,p_1^3,...,p_1^{r_1},p_2^3,...,p_2^{r_2},p_1{p}_2^3,...,p_1{p}_2^{r_2},p_1^3{p}_2,...,p_1^{r_1}{p}_2,p_1^2{p}_2^3,...,p_1^{r_1}{p}_2^{r_2}\}$ is a fixing set for ${\mathbb{A}}_m$. $V({\mathbb{A}}_m)\setminus F=\{p_2,p_1^2,p_2^2,p_1{p}_2,p_1{p}_2^2,p_1^2{p}_2,p_1^2{p}_2^2\}$, where $deg(p_1^2)=deg(p_2^2)$ and $deg(p_1^2{p}_2)=deg(p_1{p}_2^2)$. All other vertices in $V({\mathbb{A}}_m)\setminus F$ have distinct degrees. Let $\theta \in Aut({\mathbb{A}}_m)$ such that $\theta (s)=s$ for all $s\in F$ then we have two cases:

Case 1. Suppose $\theta (p_1^2)=p_2^2$ now $\theta (p_1{p}_2^3)=p_1{p}_2^3$ and $p_2^2\nsim p_1{p}_2^3$ but $p_1^2\sim p_1{p}_2^3$, hence for being θ an automorphism of ${\mathbb{A}}_m$, we have $\theta (p_1^2)=p_1^2$ and $\theta (p_2^2)=p_2^2$.

Case 2. Suppose $\theta (p_1{p}_2^2)=p_1^2{p}_2$, also $\theta (p_1^3)=p_1^3$ and $p_1^3\sim p_1{p}_2^2$ but note that $\theta (p_1{p}_2^2)\nsim \theta (p_1^3)$ hence for θ be an automorphism $\theta (p_1{p}_2^2)=p_1{p}_2^2$.

Concluding above discussion, θ is an identity automorphism hence F is a fixing set and $fix({\mathbb{A}}_m)=2\sum _{i=1}^2(r_i-2)+\prod _{i=1}^2(r_i-1)$. □

From Theorem 4.1, note that for $m_1=p_1{p}_2$ and $m_2=p_1^{r_1}{p}_2^{r_2}$ with at least one $r_i\ge 2$, $1\le i\le 2$ then $fix({\mathbb{A}}_{m_1})=1$ and $fix({\mathbb{A}}_{m_2})\ge 2$ except for $m_2=p_1{p}_2^2,p_1^2{p}_2$ as its $fix({\mathbb{A}}_{m_2})=1$. Therefore, we have the following result.

Corollary 4.2

For $m_1=p_1{p}_2$ and $m_2=p_1^{r_1}{p}_2^{r_2}$ with at least one $r_i\ge 2$ and $1\le i\le 2$ , then $fix({\mathbb{A}}_{m_1})\le fix({\mathbb{A}}_{m_2})$ .

In the following Fig. 1, the arithmetic graph for $m=p_1{p}_2{p}_3^2$ is shown.

Figure 1.

An arithmetic graph when $m=p_1{p}_2{p}_3^2$.

In the following theorem, the fixing number is given for $m={\prod }_{i=1}^n{p}_i^{r_i}$, with $n\ge 3$ and $W=\varnothing$.

Theorem 4.3

For $m={\prod }_{i=1}^n{p}_i^{r_i}$ , with $n\ge 3$ and $W=\varnothing$ , we have

$$fix({\mathbb{A}}_m)=\{\begin{array}{ccc}n-1\hfill & \mathit{\text{if}}\hfill & U\ne \varnothing ,V=\varnothing \hfill \\ 2n-|U|-3\hfill & \mathit{\text{if}}\hfill & U\ne \varnothing ,V\ne \varnothing \hfill \\ 2n-2\hfill & \mathit{\text{if}}\hfill & U=\varnothing ,V\ne \varnothing .\hfill \end{array}$$

Proof

(i) For $U\ne \varnothing$ and $V=\varnothing$, we have $m={\prod }_{i=1}^n{p}_i$. To prove that $fix({\mathbb{A}}_m)=n-1$, we first prove that $F=\{p_1,p_2,\dots ,p_{n-1}\}$ is a fixing set for ${\mathbb{A}}_m$. Let $\theta \in Aut({\mathbb{A}}_m)$ such that $\theta (p_i)=p_i$ for $1\le i\le n-1$ as by Lemma 3.1, $\theta (p_n)=p_n$. Now suppose that $\theta (x)\ne x$ for some $x\in V({\mathbb{A}}_m)\setminus X_1$ then $S_x\ne S_{\theta (x)}$, because there exists at least one $p_i\in X_1$ such that $p_i\in S_x$ and $p_i\notin S_{\theta (x)}$. Now $p_i\in S_x$ and $x\in V({\mathbb{A}}_m)\setminus X_1$ so $x\sim p_i$ but $\theta (x)\nsim \theta (p_i)=p_i$ because $p_i\notin S_{\theta (x)}$ which gives that θ is not an automorphism. Hence, $\theta (x)=x$ for each $x\in V({\mathbb{A}}_m)$ and F is a fixing set for ${\mathbb{A}}_m$.

Now suppose that $F\subseteq V({\mathbb{A}}_m)$ such that $|F|=n-2$, then $|F\cap X_1|\le n-2$ then there exists an automorphism θ in $Aut({\mathbb{A}}_m)$ such that $\theta (p_{n-1})=p_n$ and $\theta (p_n)=p_{n-1}$ which yields that F is not a fixing set for ${\mathbb{A}}_m$. Hence, $fix({\mathbb{A}}_m)=n-1$.

$(ii)$ For $U\ne \varnothing$ and $V\ne \varnothing$, suppose that $|U|=t>0$ and $V=n-t>0$. In particular suppose that $m={\prod }_{i=1}^t{p}_i{\prod }_{j=t+1}^n{p}_j^2$, where set $P_1=\{p_1,p_2,...,p_t\}$, $P_2=\{p_{t+1},p_{t+2},...,p_n\}$ and $P_2^2=\{p_{t+1}^2,p_{t+2}^2,...,p_n^2\}$ such that $X_1=P_1\cup P_2\cup P_2^2$. We first prove that $F=X_1\setminus \{p_t,p_n,p_n^2\}$ is a fixing set for ${\mathbb{A}}_m$. Let $\theta \in Aut({\mathbb{A}}_m)$ such that $\theta (p_i)=p_i$ for $1\le i\le t-1$, $i\ne t$, $\theta (p_j)=p_j$ and $\theta (p_j^2)=p_j^2$ for $t+1\le j\le n-1$. Note that by Lemma 3.1, $\theta (P)=P$ so $\theta (p_t)=p_t$ because $deg(p_t)\ne deg(p_n)$. By Lemma 3.2, $\theta (X_l)=X_l$ for $1\le l\le n$ yields that $\theta (X_1)=X_1$ so $\theta (p_n)=p_n$ and $\theta (p_n^2)=p_n^2$ as $deg(p_n)\ne deg(p_n^2)$. Thus, $\theta (x)=x\forall x\in X_1$.

Now we suppose that $\theta (x)\ne x$ and $\theta (x)=y$ where $x,y\in X_l,2\le l\le n$. This implies that $deg(x)=deg(y)$ then either $S_x\ne S_y$ or $S_x=S_y$. If $S_x\ne S_y$ then there exist at least one $p_i\in P$ such that $p_i\in S_x$ but $p_i\notin S_y$ and if $S_x=S_y$ then there exists at least one $p_j^2\in X_1$ such that $p_j^2\sim x$ but $p_j^2\nsim y$ so $x\sim p_i$ but $\theta (x)\nsim \theta (p_i)=p_i$ which gives that θ is not an automorphism. Hence, $\theta (x)=x$ for each $x\in V({\mathbb{A}}_m)$ and F is a fixing set for ${\mathbb{A}}_m$.

Now suppose $|F|\le 2n-|U|-4$ and $|X_1﹨F|\ge 4$ then there exist two vertices $x,y$ in $P_1,P_2$ or $P_2^2$ which yield a non-trivial automorphism and hence F is not a fixing set. Hence, $fix({\mathbb{A}}_m)=2n-|U|-3$.

$(iii)$ For $U=\varnothing ,V\ne \varnothing$ and we have $m={\prod }_{i=1}^n{p}_i^{r_i}$. To prove that $fix({\mathbb{A}}_m)=2n-2$. For $m=p_1^2{p}_2^2...p_n^2$, we define the set $P_2=\{p_1,p_2,...,p_n\}$ and $P_2^2=\{p_1^2,p_2^2,...,p_n^2\}$. Now, we consider the set $F=\{p_1,...,p_{n-1},p_1^2,...,p_{n-1}^2\}=X_1\setminus \{p_n,p_n^2\}\subset V({\mathbb{A}}_m)$. We first prove that F is a fixing set for ${\mathbb{A}}_m$. Let $\theta \in Aut({\mathbb{A}}_m)$ such that $\theta (p_i)=p_i$ and $\theta (p_i^2)=p_i^2$ for $1\le i\le n-1$. Note that as $\theta (P)=P$ by Lemma 3.1 and $\theta (X_1)=X_1$ by using Lemma 3.2 so $\theta (p_n)=p_n$ and $\theta (p_n^2)=p_n^2$ as $p_n$ and $p_n^2$ have different degrees.

Now we suppose that $\theta (x)\ne x$ and $\theta (x)=y$ where $x,y\in X_l,2\le l\le n$. This implies that $deg(x)=deg(y)$ then either $S_x\ne S_y$ or $S_x=S_y$. If $S_x\ne S_y$ then there exists at least one $p_i\in P$ such that $p_i\in S_x$ but $p_i\notin S_y$ and if $S_x=S_y$ then there exists at least one $p_i^2\in X_1$ such that $p_i^2\sim x$ but $p_i^2\nsim y$ so $x\sim p_i$ but $\theta (x)\nsim \theta (p_i)=p_i$ which gives that θ is not an automorphism. Hence, $\theta (x)=x$ for each $x\in V({\mathbb{A}}_m)$ and F is a fixing set for ${\mathbb{A}}_m$.

Now suppose that $|F|\le 2n-3$ and $|X_1﹨F|\ge 3$ then there exist two vertices $x,y$ in $P_2$ or $P_2^2$ which yield a non-trivial automorphism. Hence, $fix({\mathbb{A}}_m)=2n-2$. □

The following result for the fixing number of the arithmetic graphs when $W=\varnothing$ is established by using the Theorem 4.3.

Corollary 4.4

For$m_1={\prod }_{i=1}^n{p}_i$and$m_2={\prod }_{i=1}^n{p}_i^{r_i}$with$n\ge 3$and$1\le r_i\le 2$, then the following are true:

(i)$fix({\mathbb{A}}_{m_1})\ge fix({\mathbb{A}}_{m_2})$where$m_2$has at most one$r_i=2$,

(ii)$fix({\mathbb{A}}_{m_1})<fix({\mathbb{A}}_{m_2})$where$m_2$has at least three$r_i^{\prime }s$equal to 2,

(iii)$fix({\mathbb{A}}_{m_1})=fix({\mathbb{A}}_{m_2})$if$m_2$has exactly any two$r_i^{\prime }s$equal to 2.

Let ${\mathcal{T}}_{{\mathbb{A}}_m}$ be number of twin classes and $T_1,T_2,...,T_{{\mathcal{T}}_{{\mathbb{A}}_m}}$ be the twin classes in ${\mathbb{A}}_m$. We consider $T_i^{\prime }=T_i\setminus \{u_i\}$ for some fixed $u_i\in T_i$ and define $F^1={\cup }_{i=1}^{\xi }{T}_i^{\prime }$. Note that $|F^1|=|V({\mathbb{A}}_m)|-{\mathcal{T}}_{{\mathbb{A}}_m}$. For $W=\{i\in X|r_i\ge 3\}$, we define $W_{t_j}=\{i\in W|r_i=t_j\}$, $3\le t_j$ and $1\le j\le k$ where $t_k$ is the highest power of any prime in decomposition of m. $W_{t_1},W_{t_2},...,W_{t_k}$ form a partition of W and for $1\le j\le k$, we define $F_{t_j}=\{p_{\alpha }|\alpha \in W_{t_j}\setminus \beta \}$ where β is some fixed index in $W_{t_j}$. Let $F^2={\cup }_{j=1}^k{F}_{t_j}$ and $|F^2|=\sum _{j=1}^k(|W_{t_j}|-1)$. Using the definitions of $F^1$ and $F^2$ and their cardinalities, we give formula for the fixing number of ${\mathbb{A}}_m$ where $m={\prod }_{i=1}^n{p}_i^{r_i}$, with $n\ge 3$, $r_i\ge 1$ and $W\ne \varnothing$.

It was established in [28] that for given graph G with ${\mathcal{T}}_G$ twin classes, $fix(G)\ge |V(G)|-{\mathcal{T}}_G$. In the following theorem, we prove that this bound is attained for $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $3\le r_1<r_2<...<r_n$. Further, for $W\ne \varnothing$, we present exact formula for the fixing number of ${\mathbb{A}}_m$ for different cases of U and V.

Theorem 4.5

For $m=\prod _{i=1}^n{p}_i^{r_i}$ with $n\ge 3$ , $r_i\ge 1$ and $W\ne \varnothing$ , we have

(i) $fix({\mathbb{A}}_m)=\sum _{j=1}^k(|W_{t_j}|-1)+|V({\mathbb{A}}_m)|-{\mathcal{T}}_{{\mathbb{A}}_m}$ if $U=\varnothing ,V=\varnothing$ ,

(ii) $fix({\mathbb{A}}_m)=|U|-1+\sum _{j=1}^k(|W_{t_j}|-1)+|V({\mathbb{A}}_m)|-{\mathcal{T}}_{{\mathbb{A}}_m}$ if $U\ne \varnothing ,V=\varnothing$ ,

(iii) $fix({\mathbb{A}}_m)=2|V|-2+\sum _{j=1}^k(|W_{t_j}|-1)+|V({\mathbb{A}}_m)|-{\mathcal{T}}_{{\mathbb{A}}_m}$ if $U=\varnothing ,V\ne \varnothing$ ,

(iv) $fix({\mathbb{A}}_m)=2(|U|+|V|)-|U|-3+\sum _{j=1}^k(|W_{t_j}|-1)+|V({\mathbb{A}}_m)|-{\mathcal{T}}_{{\mathbb{A}}_m}$ if $U\ne \varnothing ,V\ne \varnothing$ .

Proof

We will prove that every fixing set of ${\mathbb{A}}_m$ for $W\ne \varnothing$ must contains the set $F=F^1\cup F^2$. Note that all vertices of non-trivial twin classes belong to F except a fixed $x_i\in T_i$. As all vertices of F are mapped onto themselves so $\theta (x_i)=x_i\forall x_i\in T_i$. Hence, $F^1$ fixes every vertex of the non-trivial twin classes for any choice of m. Now, we need to fix vertices of ${\mathbb{A}}_m$ to destroy any possible symmetries induced by trivial twin classes which are vertices of the form ${\prod }_{j=1}^k{p}_{i_j}^{{\alpha }_{i_j}}$ with $1\le {\alpha }_{i_j}\le 2$. Now we have following four cases:

(i) Case 1. For $U=\varnothing$ and $V=\varnothing$ and $|W_{t_j}|=1$ ∀ $j,1\le j\le k$, degrees of all the vertices in trivial twin classes are different which yields that the vertices in the trivial twin classes are fixed which implies that $fix({\mathbb{A}}_m)\le |V({\mathbb{A}}_m)|-{\mathcal{T}}_{{\mathbb{A}}_m}$ and by using $fix(G)\ge |V(G)|-{\mathcal{T}}_G$, result is established with $\sum _{j=1}^k(|W_{t_j}|-1)=0$.

Case 2. Now, for at least one $|W_{t_j}|\ge 2$, then $\sum _{j=1}^k(|W_{t_j}|-1)\ne 0$. This means that there exist vertices of the form $p_i$ and $p_j$ with the same degree and by Lemma 3.1, there exist $\theta \in Aut({\mathbb{A}}_m)$ such that $\theta (p_i)=p_j$. By fixing $F^2$ with cardinality $\sum _{j=1}^k(|W_{t_j}|-1)$, all such automorphisms are destroyed. Note that the other vertices of trivial twin classes are of the form $p_{i_1}{p}_{i_2}...p_{i_k}$ with $1\le k\le n$ are fixed as all the vertices of P in ${\mathbb{A}}_m$ are fixed by using the same arguments as in Theorem 4.3(i). Hence, $fix({\mathbb{A}}_m)=\sum _{j=1}^k(|W_{t_j}|-1)+|V({\mathbb{A}}_m)|-{\mathcal{T}}_{{\mathbb{A}}_m}$.

$(ii)$ For $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $n\ge 3$, $r_i\ge 1$, $U\ne \varnothing ,V=\varnothing$ and $W\ne \varnothing$, we define $P_1=\{p_1,p_2,...,p_t\}$ and $|U|=t$. Consider the set $F=\{p_1,p_2,...,p_{t-1}\}\cup F^1\cup F^2$. We will prove that F is a fixing set for ${\mathbb{A}}_m$. As $F^1\cup F^2\subseteq F$ so $fix({\mathbb{A}}_m)\ge \sum _{j=1}^k(|W_{t_j}|-1)+|V({\mathbb{A}}_m)|-{\mathcal{T}}_{{\mathbb{A}}_m}$ by using the same arguments as in Theorem 4.5(i). Please note that $\{p_1,p_2,...,p_{t-1}\}$ fixes all vertices which have $p_1,...,p_t$ in their canonical representation and $\{p_1,p_2,...,p_{t-1}\}\cup F^2$ fixes all the vertices of the form $p_{i_j}$ for $1\le i_j\le n$. Using $\theta (P)=P$ and arguments of Theorem 4.3(i), we conclude that vertices of ${\prod }_{j=1}^k{p}_{i_j}^{{\alpha }_{i_j}}$ with $1\le k\le n$ for some $0\le {\alpha }_{i_j}\le 1$ are mapped onto themselves which yields that F is a fixing set for ${\mathbb{A}}_m$. Hence, $fix({\mathbb{A}}_m)=|U|-1+\sum _{j=1}^k(|W_{t_j}|-1)+|V({\mathbb{A}}_m)|-{\mathcal{T}}_{{\mathbb{A}}_m}$.

$(iii)$ For $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $n\ge 3$, $r_i\ge 1$, $U=\varnothing ,V\ne \varnothing$ and $W\ne \varnothing$, we define $P_2=\{p_1,p_2,...,p_s\}$, $P_2^2=\{p_1^2,p_2^2,...,p_s^2\}$ and $|V|=s$ and we consider $F=\{p_1,p_2,...,p_{s-1}\}\cup \{p_1^2,p_2^2,...,p_{s-1}^2\}\cup F^1\cup F^2$. We will prove that F is a fixing set for ${\mathbb{A}}_m$. As $F^1\cup F^2\subset F$ so $fix({\mathbb{A}}_m)\ge \sum _{j=1}^k(|W_{t_j}|-1)+|V({\mathbb{A}}_m)|-{\mathcal{T}}_{{\mathbb{A}}_m}$ by using the same arguments as in Theorem 4.5(i). Please note that $\{p_1,p_2,...,p_{s-1}\}\cup \{p_1^2,p_2^2,...,p_{s-1}^2\}$ fixes all the vertices which have $p_1,...,p_s,p_1^2,p_2^2,...,p_s^2$ in their canonical representation. Using $\theta (P)=P$ and using arguments of Theorem 4.3(iii), we conclude that vertices of the form ${\prod }_{j=1}^k{p}_{i_j}^{{\alpha }_{i_j}}$ with $1\le k\le n$ for some $0\le {\alpha }_{i_j}\le 2$ are mapped onto themselves which yields that F is a fixing set for ${\mathbb{A}}_m$. Hence, $fix({\mathbb{A}}_m)=2|V|-2+\sum _{j=1}^k(|W_{t_j}|-1)+|V({\mathbb{A}}_m)|-{\mathcal{T}}_{{\mathbb{A}}_m}$.

$(iv)$ For $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $n\ge 3$, $r_i\ge 1$, $U\ne \varnothing ,V\ne \varnothing$ and $W\ne \varnothing$, we define $P_1=\{p_1,p_2,...,p_t\}$, $P_2=\{p_{t+1},p_{t+2},...,p_s\}$, $P_2^2=\{p_{t+1}^2,p_{t+2}^2,...,p_s^2\}$ and $|U|=t$, $|V|=s-t$, we consider a set F consisting of $\{p_1,p_2,...,p_{t-1}\}\cup \{p_{t+1},p_{t+2},...,p_{s-1}\}\cup \{p_{t+1}^2,p_{t+2}^2,...,p_{s-1}^2\}\cup F^1\cup F^2$. We will prove that F is a fixing set for ${\mathbb{A}}_m$. As $F^1\cup F^2\subset F$ so $fix({\mathbb{A}}_m)\ge \sum _{j=1}^k(|W_{t_j}|-1)+|V({\mathbb{A}}_m)|-{\mathcal{T}}_{{\mathbb{A}}_m}$ by using the same arguments as in Theorem 4.5(i). Please note that $\{p_1,p_2,...,p_{t-1}\}\cup \{p_{t+1},p_{t+2},...,p_{s-1}\}\cup \{p_{t+1}^2,p_{t+2}^2,...,p_{s-1}^2\}$ fixes all the vertices which have $p_1,...,p_t,p_{t+1},p_{t+2},...,p_s,p_{t+1}^2,p_{t+2}^2,...,p_s^2$ in their canonical representation. Using $\theta (P)=P$ and by using arguments of Theorem 4.3(ii), we conclude that vertices of ${\prod }_{j=1}^k{p}_{i_j}^{{\alpha }_{i_j}}$ for some $0\le {\alpha }_{i_j}\le 2$ are mapped onto themselves which yields that F is a fixing set for ${\mathbb{A}}_m$. Hence, $fix({\mathbb{A}}_m)=2(|U|+|V|)-|U|-3+\sum _{j=1}^k(|W_{t_j}|-1)+|V({\mathbb{A}}_m)|-{\mathcal{T}}_{{\mathbb{A}}_m}$. □

Based on Theorem 4.5, we have the following corollary which gives inequality on the fixing number of the arithmetic graphs associated with $m_1={\prod }_{i=1}^n{p}_i^{r_i}$ and $m_2={\prod }_{j=1}^n{p}_j^{s_j}$.

Corollary 4.6

For $m_1={\prod }_{i=1}^n{p}_i^{r_i}$ and $m_2={\prod }_{j=1}^n{p}_j^{s_j}$ with $n\ge 3$ , then $fix({\mathbb{A}}_{m_1})<fix({\mathbb{A}}_{m_2})$ where $U\ne \varnothing ,W\ne \varnothing$ for $m_1$ and $V\ne \varnothing ,W\ne \varnothing$ for $m_2$ .

5. Conclusions

Symmetries obtained through automorphisms provide useful insights about a network or a graph. Symmetries can be used to identify relationships among the vertices of graphs. It is important to observe that symmetries of graphs are identified by the similarities of vertices and therefore by identification of symmetries, one can break the symmetries of graph which have vast applications in the field of networking.

In this paper, we have studied symmetries and symmetry-breaking of the arithmetic graphs with some of their associated properties. We have firstly described the properties of the arithmetic graphs such as the distance between vertices, the degree of any vertex and the number of twin classes in arithmetic graphs. We have identified the automorphism group of ${\mathbb{A}}_m$ for $m={\prod }_{i=1}^n{p}_i$ and have proved that it is isomorphic to the symmetric group of n elements. Moreover, we have studied the conditions on permutations of $\{p_1,p_2,\dots ,p_n\}$ of the arithmetic graphs which induce automorphisms of ${\mathbb{A}}_m$ for $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $r_i\ge 1$. Therefore, it is interesting to examine the non-trivial symmetries of the arithmetic graphs, which lead to investigate the fixing numbers of the arithmetic graphs ${\mathbb{A}}_m$ of $m={\prod }_{i=1}^n{p}_i^{r_i}$ with $r_i\ge 1$.

Moreover, we have investigated the twin-free arithmetic graphs with their fixing numbers when m is the form of ${\prod }_{i=1}^n{p}_i$ and ${\prod }_{i=1}^n{p}_i^{r_i}$ with $1\le r_i\le 2$. We have also noted the difference between the metric dimension and the fixing number of the arithmetic graphs and noticed that for $p_1{p}_2$, $p_1{p}_2^{r_2},r_2\ge 3$, $p_1^2{p}_2^{r_2},r_2\ge 3$, $p_1^{r_1}{p}_2^{r_2},r_1,r_2\ge 3$, the metric dimension and the fixing number of ${\mathbb{A}}_m$ are equal.

Table 1 shows a comparison of the metric dimension proved by Rehman et al. [42] and the fixing numbers of the arithmetic graphs ${\mathbb{A}}_m$. Please note that the results coincide for some composite numbers m.

Table 1.

Comparison of the fixing number and the metric dimension of m of the arithmetic graphs.

m	$dim({\mathbb{A}}_m)$	$fix({\mathbb{A}}_m)$
$\prod _{i=1}^2{p}_i$	1	1
$\prod _{i=1}^2{p}_i^{r_i},r_2\ge 3$	1 + 2(r2 − 2)	1 + 2(r2 − 2)
$\prod _{i=1}^2{p}_i^2$	3	2
$\prod _{i=1}^2{p}_i^{r_i},r_2\ge 3$	2 + 3(r2 − 2)	2 + 3(r2 − 2)
$\prod _{i=1}^2{p}_i^{r_i},r_i\ge 3$	2(r1 − 2)+2(r2 − 2)+(r1 − 1)(r2 − 1)	2(r1 − 2)+2(r2 − 2)+(r1 − 1)(r2 − 1)
$\prod _{i=1}^n{p}_i$	n	n − 1
$\prod _{i=1}^n{p}_i^2$	2n	2n − 2
$\prod _{i=1}^t{p}_i\prod _{i=t+1}^n{p}_i^2$ and n ≥ 4	2n − t	2n − t − 3

In [28], “Garijo et al. posed the question that whether the difference between the metric dimension and the fixing number of a graph can be arbitrarily large of order n?” and studied the stated problem. In [42], Rehman et al. computed the metric dimension of the arithmetic graphs and it was interesting to note the difference between the metric dimension and the fixing number of the arithmetic graphs ${\mathbb{A}}_m$. It is an open and interesting question that for $m=\prod _{i=1}^n{p}_i^{r_i}$ with $n\ge 3$, $r_i\ge 1$ and $W\ne \varnothing$, can the difference between the metric dimension and the fixing number of ${\mathbb{A}}_m$ be arbitrarily large? The answer seems positive if the trivial twin classes of vertices are increased.

CRediT authorship contribution statement

Aqsa Shah: Conceived and designed the experiments; Performed the experiments; Wrote the paper.

Imran Javaid: Conceived and designed the experiments; Analyzed and interpreted the data.

Shahid Ur Rehman: Contributed reagents, materials, analysis tools or data.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data included in article/supp.material/referenced in article.