Exploring a Graph Complement in Quadratic Congruence

Abstract

In this work, we investigate essential definitions, defining $G$ as a simple graph with vertices in ${\mathbb{Z}}_n$ and subgraphs ${\Gamma }_u$ and ${\Gamma }_q$ as unit residue and quadratic residue graphs modulo $n$, respectively. The investigation extends to the degree of $G$, ${\Gamma }_u$, and ${\Gamma }_q$, illuminating the properties of these subgraphs in the context of quadratic congruences.

1. Introduction

Modular arithmetic acts as the foundation for understanding quadratic congruences, a specific class of equations involving quadratic expressions and congruence relations. The modular arithmetic framework deals with numbers that wrap around upon reaching a specific modulus. A congruence relation expresses the equality of remainders when dividing two numbers by a positive integer known as the modulus.

The general form of a quadratic congruence is expressed as $a{x}^2\equiv b (mod n)$, where $a$, $b$, and $n$ are integers with $n>0$, and $x$ represents an unknown integer variable. The solvability of the congruence $x^2\equiv b (mod n)$ depends on the existence of an integer $x$ that satisfies the congruence, referred to as a solution or root.

When exploring quadratic congruences within the ring of integers modulo a prime, the finite field ${\mathbb{Z}}_p$ plays a crucial role. In cases of a non-prime modulus $n$, ${\mathbb{Z}}_n$’s structure significantly influences the solutions to quadratic congruences, particularly in the context of quadratic residues and non-residues modulo $p$.

Within the ring ${\mathbb{Z}}_p$, where $p$ is a prime number, elements fall into distinct categories of idempotents and nilpotents. Idempotents, namely 0 and 1, remain unchanged under repeated operations, while nilpotents eventually become zero after reaching a certain power.

A congruence class $\left[a\right]\in {\mathbb{Z}}_n$ is called a unit if it possesses a multiplicative inverse in ${\mathbb{Z}}_n$, denoted by $\left[b\right]\in {\mathbb{Z}}_n$ such that $\left[a\right]\left[b\right]=\left[1\right]$. The set of units forms a group under multiplication, often denoted as $U\left(n\right)$ or $U_n$. In this group, an element $a\in U_n$ is a quadratic residue mod $n$ if $x^2\equiv a$ for some $x\in U_n$. The set of such quadratic residues is denoted by $Q_n$, forming a group under multiplication, where each residue has a multiplicative inverse within the group.

Quadratic residues, modulo $n$, are counted excluding the trivial case $q=0$. This exclusion ensures that the count is one less than the number of squares modulo $n$. However, some sources include 0 as a quadratic residue, leading to $q$ being termed a quadratic non-residue modulo $n$ in cases where 0 is excluded.

The order of the quadratic residue group $Q_n$ is determined by the number of quadratic residues in the set $\left|U_n\right|$. This order depends on the properties of the modulus $n$. For an odd prime modulus $p$, the order of the group $Q_n$ is $(p-1)/2$, reflecting Euler’s criterion and the fact that exactly half of the residues in the set $U_n$ are quadratic residues. For a composite modulus $n=p_1.p_2.\dots .p_k$, the order of the group is influenced by the factorization of $n$ and cannot be $\varphi \left(n\right)/2$, where $\varphi \left(n\right)$ is Euler’s totient function.

Researchers, in their respective works, have explored various aspects of mathematical symmetry and congruences. Castillo and Mainguez, in their 2022 study on modular arithmetic, delve into the symmetrical patterns within sets of k-units modulo $n$ [1]. Mateen et al., in their 2021 research in graph theory, investigate the symmetrical intricacies within complete graphs involving quadratic and cubic residues [2]. Ali, Bahrami, and Reza’s 2017 contribution uncovers symmetrical properties in Cayley graphs for the ring of Gaussian integers modulo $n$ [3]. Somer and Křížek’s 2006 investigation into digraphs associated with quadratic congruences reveals symmetrical foundations [4]. Mateen and Mahmood, through work spanning 2019, navigate the symmetrical intricacies within power digraphs [5]. Deng and Yuan’s 2012 study provides valuable insights into symmetric digraphs, unraveling symmetrical patterns emerging from powers modulo $n$ [6]. J. Fabrykowski 1994 investigate the estimation of the maximal cardinality of such a set of residues modulo $m$ [7]. Additionally, the study of undirected graphs has been employed in understanding quadratic congruences, with various works relating graphs to the solution of quadratic congruences (e.g., [8,9,10,11]), offering both a visual and structural perspective on the mathematical relationships involved.

This work specifically examines the extension of a graph proposed in [12], related to the arbitrary associative group of integers ${\mathbb{Z}}_n$. We explore elementary results regarding the association of this graph with the ring ${\mathbb{Z}}_n$, constructing it based on the elements and units of the ${\mathbb{Z}}_n$ ring. Vertices represent elements, and distinct vertices are connected if and only if their squares are distinct. Additionally, we focus on the degrees of vertices in the graphs of the ring ${\mathbb{Z}}_n$, the subgroup $U_n$, and the subgroup $Q_n$.

2. Results

In this section, we begin by reviewing a fundamental definition crucial to our exploration.

Definition 1. 

Let $n\ge 2$ be a fixed positive integer. We call $G$ a simple graph if its vertex set is $V\left(G\right)={\mathbb{Z}}_n$ such that two distinct vertices $a$ and $b$ are adjacent provided that $a^2\ne b^2 (mod n)$. Additionally, we introduce two subgraphs, ${\Gamma }_u$ and ${\Gamma }_q$:

• We call the subgraph ${\Gamma }_u$ a unit residue graph modulo $n$ if its vertex set is $V \left({\Gamma }_u\right)=U_n$ and $E\left({\Gamma }_u\right)=\{ab: a, b\in U_n and a^2\ne b^2 (mod n\left)\right\}$;
• We call the subgraph ${\Gamma }_q$ a quadratic residue graph modulo $n$ if its vertex set is reduced residue system mod $n$ such that two distinct vertices $a$ and $b$ are adjacent provided that $a^2\ne b^2 (mod n)$. i.e., $V\left({\Gamma }_q\right)=Q_n$ and $E\left({\Gamma }_q\right)=\{ab:a, b\in Q_n and a^2\ne b^2 (mod n\left)\right\}$;

In graph $G$, when considering every pair of vertices $u$ and $v$ in the vertex set $V\left(G\right)$ such that $u+v$ is congruent to 0 modulo $n$, these vertices are disjointed. This property leads to a non-regular graph with vertex degrees less than or equal to $n-2$. Also, it is observed that the vertex labeled as 0 is connected with all vertices except the nilpotent ones, and similarly, the idempotent vertex presented as $n/2$ is connected to all other vertices.

By the definition of graph $G$, an edge $ab\in E\left(G\right)$ is established between vertices $a$ and $b$ provided neither serves as a solution to the quadratic congruence $x^2=c (mod n)$ for a non-zero $c$. However, the solutions to this quadratic congruence may exceed two due to the factorization of $n$. For example, $x^2=4 (mod 15)$ can be factored as $x^2-4=\left(x+2\right)\left(x+13\right)=\left(x+7\right)\left(x+8\right)$, yielding four roots: $x=2, x=7, x=8, x=13$. Noticeably, these vertices do not share edges, indicating a lower degree for the mentioned vertices.

For the integers $n=2, 3, 4, 6, 8, 12, 24$, the graph ${\Gamma }_u$ is empty due to the equality of squares for any pair of vertices. The mentioned values of $n$ will be omitted during the next discussion about the vertex degrees in graphs ${\Gamma }_u$ and ${\Gamma }_q$. However, in the graph $G$, the vertex $v=1$ is adjacent to the vertex $u=0$.

The subsequent Proposition illustrates the vertex degrees within the graph ${\Gamma }_u$.

Proposition 1. 

For any prime $p\ge 5$, and $v\in V\left({\Gamma }_u\right)$, then $\mathrm{deg}\left(v\right)=\varphi \left(p\right)-2$.

Proof. 

Consider any vertex, denoted as $v$, in the set $V\left({\Gamma }_u\right)$. Since $v$ and $p-v$ are the only vertices sharing identical squares, it follows that vertex $v$ is connected to all other vertices in $V\left({\Gamma }_u\right)$ except for the vertex $p-v$. Given that $\left|V\left({\Gamma }_u\right)\right|=\varphi \left(p\right)=p-1$, the number of vertices adjacent to $v$ is $p-2$. □

Corollary 1. 

The graph ${\Gamma }_u$ is regular of order $\varphi \left(p\right)-2$ for any prime number $p\ge 5$.

Proposition 2. 

For any integer $n=2^r$ and $v\in V\left({\Gamma }_u\right)$, then $\mathit{deg}\left(v\right)=2^{r-1}-4$.

Proof. 

Consider any vertex, denoted as $v$, in the set $V\left({\Gamma }_u\right)$ such that $v^2\equiv a\left(mod n\right)$. Since the congruence $x^2\equiv a\left(mod 2^r\right)$ has either no solution or exactly four incongruent solutions, namely $u_1,u_2,u_3$, $v$ shares only three identical squares. It follows that vertex $v$ is adjacent to all vertices in $V\left({\Gamma }_u\right)$ except for the vertices $u_1,u_2,u_3$. Given that $\left|V\left({\Gamma }_u\right)\right|=\varphi \left(n\right)$, the number of vertices adjacent to $v$ is $\varphi \left(n\right)-4=2^{r-1}-4$. □

Corollary 2. 

Let $n=2^r$, $r>4$ and $v\in V\left({\Gamma }_q\right)$, then $\mathit{deg}\left(v\right)=\frac{\varphi \left(n\right)}{4}-2$.

The task of identifying elements in $Q_n$ is significantly simplified by employing the so-called Legendre symbol; that is, for an odd prime $p$, the Legendre symbol of any integer $a$ is defined as:

$$\left(\frac{a}{p}\right)=\left\{\begin{array}{cc}0& if p|a\\ 1& if a\in Q_p\\ -1& if a\in U_p-Q_p\end{array}\right.$$

The Quadratic Reciprocity Theorem is a key result in number theory. It reveals a significant connection between quadratic residues and non-residues when considering congruences modulo prime numbers. The following is the state of the Quadratic Reciprocity Theorem.

Theorem 1 

([13] The Quadratic Reciprocity Theorem). If $p$ and $q$ are odd primes and $p=q\equiv 3 \left(mod 4\right)$, then $\left(\frac{p}{q}\right)=-\left(\frac{q}{p}\right)$. Otherwise, $\left(\frac{p}{q}\right)=\left(\frac{q}{p}\right)$.

Now, turning our attention to the prime power modulo, we delve into the case of odd modulo after addressing quadratic residues for prime modulo.

Theorem 2 

([14]). Let $p$ be an odd prime, let $m\ge 1$, and let $a\in \mathbb{Z}$. Then $a\in Q_{p^m}$ if and only if $a\in Q_p$.

Combining characterizations of $Q_{p^m}$ for distinct prime powers is facilitated by the following result.

Theorem 3 

([14]). Let $n=p_1^{r_1}{p}_2^{r_2}\dots p_k^{r_k}$ where $p_i$ are primes. Then $a\in Q_n$ if and only if $a\in Q_{p_i^{r_i}}$ for each $i$.

Using the Quadratic Reciprocity Theorem for identifying quadratic residues modulo a prime number, we present the following proposition:

Proposition 3.

For any prime number $p>5$ and any $v\in V\left({\Gamma }_q\right)$, then

$$\mathrm{deg}\left(v\right)=\left\{\begin{array}{cc}\frac{\varphi \left(n\right)}{2}-2& if p\equiv 1 \left(mod 4\right)\\ \frac{\varphi \left(n\right)}{2}-1& if p\equiv 3 \left(mod 4\right)\end{array}\right.$$

Proof.

Consider a vertex $v$ in ${\Gamma }_q$ and a prime number $n$ such that $n=4k+1$ for a positive integer $k$. Applying the Quadratic Reciprocity Theorem, we find that $n-v$ is a quadratic residue modulo $n$. Consequently, $v$ and $n-v$ share identical squares, leading to their status as non-adjacent vertices. As a result, the degree of $v$ is given by $\mathrm{deg}\left(v\right)=\frac{\varphi \left(n\right)}{2}-2$.

Alternatively, when $n$ is a prime number satisfying $n=4k+3$ for some positive integer $k$, the Quadratic Reciprocity Theorem implies that $n-v$ is a non-quadratic residue modulo $n$. Notably, as $Q_n$ is a cyclic group, $v$ is adjacent to all vertices in ${\Gamma }_q$. Thus, the degree of $v$ is determined as $\frac{\varphi \left(n\right)}{2}-1$. □

Corollary 3. 

Let $p$ be any prime number such that $p\equiv 1 mod 4$, then the graph ${\Gamma }_q$ is $K_{\frac{p-1}{2}}$.

Corollary 4. 

Let $p$ be any prime number such that $p\equiv 3 mod 4$, then the graph ${\Gamma }_q$ is a $\left(\frac{p-1}{2}-2\right)$-regular graph.

For prime numbers, $p\equiv 1mod 4$, the subgraph which consists of non-quadratic residues vertices in $V\left({\Gamma }_u\right)$, let us call it ${\Gamma }_{nq}$, is a complete subgraph. However, for some composite integers $n$, ${\Gamma }_{nq}$ is not regular. For example, in $n=15$ and $n=20$, the graphs are not regular, and the degrees of vertices are different, as shown in Figure 1. Nevertheless, the respective subgraphs containing quadratic residues for these specified values of $n$ are empty.

Theorem 4. 

Let $n=p^r$and $v\in V\left({\Gamma }_u\right)$, then $\mathit{deg}\left(v\right)=n\left(1-\frac{1}{p}\right)-2$.

Proof .

Since the number of vertices in the graph ${\Gamma }_u$ is determined by $\varphi \left(n\right)=p^{r-1}\left(p-1\right)$, and since the congruence $x^2\equiv a\left(mod p^r\right)$ has either no solution or exactly two incongruent solutions, any vertex $v$ is connected to all vertices in this graph except the vertex $n-v$. Therefore, $\mathrm{deg}\left(v\right)=p^{r-1}\left(p-1\right)-2=n\left(1-\frac{1}{p}\right)-2$. □

Proposition 4. 

For any integer $n=p^r$, and $v\in V\left({\Gamma }_q\right)$, then

$$\mathrm{deg}\left(v\right)=\left\{\begin{array}{cc}\frac{n\left(1-\frac{1}{p}\right)}{2}-1& p\equiv 3 mod 4\\ \frac{n\left(1-\frac{1}{p}\right)}{2}-2& p\equiv 1 mod 4\end{array}\right.$$

Proof. 

Let $p\equiv 3 mod 4$. Since the order of the set $V\left({\Gamma }_q\right)$ is determined by $\varphi \left(n\right)/2=p^{r-1}\left(p-1\right)/2$, and since $n-v$ is a non-quadratic residue, any vertex $v\in V\left({\Gamma }_q\right)$ is connected to all vertices in this graph. Therefore, $\mathrm{deg}\left(v\right)=n\left(1-\frac{1}{p}\right)/2-1$. Similarly, when $p\equiv 1 mod 4$, then $n-v$ is a quadratic residue. Thus, any vertex $v\in V\left({\Gamma }_q\right)$ is connected to all vertices except the vertex $n-v$ in this graph. Therefore, $\mathrm{deg}\left(v\right)=n\left(1-\frac{1}{p}\right)/2-2$. □

Theorem 5. 

Let $n=p_1^{r_1}.p_2^{r_2}\dots . p_k^{r_k}$ are distinct odd primes and $r_i$ are positive integers. Then, the degree of any vertex $v\in {\Gamma }_u$ is $n\prod _{i=1}^k\left(1-\frac{1}{p_i}\right)-2^k$.

Proof. 

Consider $v$ in the set $V\left({\Gamma }_u\right)$. The congruence $x^2\equiv a\left(mod p_i^{r_i}\right)$ has either no solution or precisely two incongruent solutions for each prime power $p_i^{r_i}$; then, by the Chinese remainder theorem, there exist exactly $2^k$ incongruent solutions for the congruence $x^2\equiv a\left(mod n\right)$. So, if the vertex $v$ is one of these solutions, there will be $2^k$ non-adjacent vertices to $v$. Consequently, $\mathrm{deg}\left(v\right)=\varphi \left(n\right)-2^k=n\prod _{i=1}^k\left(1-\frac{1}{p_i}\right)-2^k$. □

Before exploring the discussion of vertex degrees within the quadratic residues group $Q_n$ modulo a composite number $n$, we introduce an integer ${\alpha }_j$ defined as follows:

$${\alpha }_j=\left\{\begin{array}{cc}1& if p_j\equiv 3 mod 4\\ 2& if p_j\equiv 1 mod 4\end{array}\right.$$

Theorem 6. 

Let $n=p_1.p_2\dots . p_k$ are distinct odd primes and $v\in V\left({\Gamma }_q\right)$, then

$$\mathrm{deg}\left(v\right)=\frac{n\prod _{i=1}^k\left(1-\frac{1}{p_i}\right)}{2^k}-{\prod }_{j=1}^k{\alpha }_j$$

Proof. 

We can establish the inclusion $V\left({\Gamma }_q\right)\subset V\left({\Gamma }_u\right)$ based on our previous discussions. According to Theorem 4, the degree of any vertex $v$ in $V\left({\Gamma }_u\right)$ depends on the number of incongruent solutions for the quadratic congruence $x^2\equiv a\left(mod n\right)$, denoted as $2^k$. However, as outlined in Proposition 3, the degree of any quadratic residue is influenced by the specific prime factor $p_j$, where $1\le j\le k$. Therefore, the term $2^k$ is replaced with the integer ${\prod }_{i=1}^k{\alpha }_i$, resulting in the expression for $\mathrm{deg}\left(v\right)=\frac{n\prod _{i=1}^k\left(1-\frac{1}{p_i}\right)}{2^k}-{\prod }_{j=1}^k{\alpha }_j$. □

Proposition 5. 

Let $n=p_1^{r_1}.p_2^{r_2}\dots . p_k^{r_k}$ are distinct odd primes and $r_i$ are positive integers. Then, the degree of any vertex $v\in {\Gamma }_q$ is $n\prod _{i=1}^k\left(1-\frac{1}{p_i}\right)-{\prod }_{j=1}^k{\alpha }_j$.

Proof. 

The proof follows by applying the Chinese Reminder Theorem on Theorem 6. □

To determine how many square roots an element $a\in Q_n$ can have, we present the following Lemma.

Lemma 1 

([7]). Let $k$ denote the distinct primes dividing $n$. If $a\in Q_n$, then the number $N$ of elements $x\in U_n$ such that $t^2\equiv a mod n$ is given by

$$N=\left\{\begin{array}{cc}{2}^{k+1}& If n\equiv 0 mod 8\\ 2^{k-1}& If n\equiv 2 mod 4\\ 2^k& otherwise\end{array}\right.$$

The count $N$ of square roots is solely determined by the variable $n$ and remains independent of the element $a\in Q_n$. Furthermore, in the presence of a single square root $c$ for a given element $a$, all additional square roots $d=cx$ can be obtained by identifying all solutions of $x^2=1$ through the approach outlined in Example 318 from ref. [14]. Consequently, the order of the group $Q_n$ can be ascertained by dividing $\varphi \left(n\right)$ by $N$, as demonstrated in exercise 7.4 in ref. [14].

Proposition 6. 

Let $n=p_1.p_2\dots . p_k$ are distinct odd primes dividing $n$, then the degree of any vertex $v\in {\Gamma }_q$ is:

$$\mathrm{deg}\left(v\right)=\left\{\begin{array}{cc}\frac{\varphi \left(n\right)}{2^{k+1}}-{\prod }_{j=1}^k{\alpha }_j& If n\equiv 0 mod 8\\ \frac{\varphi \left(n\right)}{2^{k-1}}-{\prod }_{j=1}^k{\alpha }_j& If n\equiv 2 mod 4\\ \frac{\varphi \left(n\right)}{2^k}-{\prod }_{j=1}^k{\alpha }_j& otherwise\end{array}\right.$$

Proof. 

The proof follows from Lemma 1 and Theorem 6. □

Recall that for a composite number $n$, Euler’s totient function can be calculated using the prime factorization, and the formula provides the count of positive integers less than or equal to $n$ that are relatively prime to $n$.

Theorem 7. 

Let $n=2^m{p}_1^{r_1}.p_2^{r_2}\dots . p_k^{r_k}$ are distinct odd primes and $r_i$ are positive integers. Then, the degree of any vertex $v\in {\Gamma }_u$ is:

$$\mathrm{deg}\left(v\right)=\left\{\begin{array}{cc}\varphi \left(n\right)-2^k& m=1\\ \varphi \left(n\right)-2^{k+1}& m=2\\ \varphi \left(n\right)-2^{k+2}& m\ge 3\end{array}\right.$$

Proof. 

Recall that the congruence $x^2\equiv a\left(mod 2^m\right)$ has either no solution or exactly four incongruent solutions for $m\ge 3$. Also, $x^2\equiv a\left(mod 2^2\right)$ has two solutions, and $x^2\equiv a\left(mod 2\right)$ has one solution. Moreover, the congruence $x^2\equiv a\left(mod p_i^{r_i}\right)$ has either no solution or exactly 2 incongruent solutions if $p$ is an odd prime. By applying the Chinese Remainder Theorem, if $m=1$, we determine that $x^2\equiv a\left(mod n\right)$ has either no solution or $2^k$ solutions. Thus, we have $2^k$ non-adjacent vertices. Also, if $m=2$, the congruence $x^2\equiv a\left(mod n\right)$ has either no solution or $2^{k+1}$ solutions. Therefore, there are $2^{k+1}$ non-adjacent vertices. Finally, if $m\ge 3$, the congruence $x^2\equiv a\left(mod n\right)$ has either no solution or $2^{k+2}$ solutions. Consequently, we have $2^{k+2}$ non-shared vertices. □

Corollary 5. 

Let $n=2^m{p}_1^{r_1}.p_2^{r_2}\dots . p_k^{r_k}$ are distinct odd primes and $r_i$ are positive integers, and $v\in V\left({\Gamma }_q\right)$, then

$$\mathrm{deg}\left(v\right)=\left\{\begin{array}{c}\begin{array}{c}\begin{array}{cc}\frac{\varphi \left(n\right)}{2^k}-{\prod }_{j=1}^k{\alpha }_j& m=1\end{array}\\ \begin{array}{cc}\frac{\varphi \left(n\right)}{2^{k+1}}-{\prod }_{j=1}^k{\alpha }_j& m=2\end{array}\\ \begin{array}{cc}\frac{\varphi \left(n\right)}{2^{k+2}}-{\prod }_{j=1}^k{\alpha }_j& m=3\end{array}\end{array}\\ \begin{array}{cc}\frac{\varphi \left(n\right)}{2^{k+2}}-2{\prod }_{j=1}^k{\alpha }_j& m\ge 4\end{array}\end{array}\right.$$

3. Computer Calculations for Some Integers $3\le \mathit{n}\le 99$

In this section, we present three tables featuring computer-generated computations encompassing various parameters, including factors of $n$, Euler’s totient function ($\varphi \left(n\right)$), idempotents, nilpotents, and the set of vertices $V\left(G\right)$, $V\left({\Gamma }_u\right)$, and $V\left({\Gamma }_q\right)$. Also, the set of degrees of vertices in $V\left(G\right)$ is denoted as D$G$, $V\left({\Gamma }_u\right)$ is denoted as D${\Gamma }_u$, and $V\left({\Gamma }_q\right)$ is denoted as D${\Gamma }_q$. Within this table, degrees in empty graphs ${\Gamma }_u$ and ${\Gamma }_q$ are represented as {}, and highlighted rows draw attention to complete graphs in ${\Gamma }_q$.

Throughout the calculations embedded in

Table 1. Computer calculations for $3\le n\le 46$.

$\mathit{n}$	ϕ(n)	Idempotents	Nilpotents	$\mathit{V}\left(\mathit{G}\right)$	$\mathit{V}\left({\mathit{\Gamma }}_{\mathit{u}}\right)$	$\mathit{V}\left({\mathit{\Gamma }}_{\mathit{q}}\right)$	$\mathbf{D}\mathit{G}$	$\mathbf{D}{\mathit{\Gamma }}_{\mathit{u}}$	$\mathbf{D}{\mathit{\Gamma }}_{\mathit{q}}$
3	2	{0,1}	{0}	3	2	1	{1,2}	{}	{}
4	2	{0,1}	{0,2}	4	2	1	{2}	{}	{}
5	4	{0,1}	{0}	5	4	2	{3,4}	{2}	{}
6	2	{0,1,3,4}	{0}	6	2	1	{4,5}	{}	{}
7	6	{0,1}	{0}	7	6	3	{5,6}	{4}	{2}
8	4	{0,1}	{0,4}	8	4	1	{4,6}	{}	{}
9	6	{0,1}	{0,3,6}	9	6	3	{6,7}	{4}	{2}
10	4	{0,1,5,6}	{0}	10	4	2	{8,9}	{2}	{}
11	10	{0,1}	{0}	11	10	5	{9,10}	{8}	{4}
12	4	{0,1,4,9}	{0,6}	12	4	1	{8,10}	{}	{}
13	12	{0,1}	{0}	13	12	6	{11,12}	{10}	{4}
14	6	{0,1,7,8}	{0}	14	6	3	{12,13}	{4}	{2}
15	8	{0,1,6,10}	{0}	15	8	2	{11,13,14}	{4}	{}
16	8	{0,1}	{0,4,8,12}	16	8	2	{12}	{4}	{}
17	16	{0,1}	{0}	17	16	8	{15,16}	{14}	{6}
18	6	{0,1,9,10}	{0,6,12}	18	6	3	{15,16}	{4}	{2}
19	18	{0,1}	{0}	19	18	9	{17,18}	{16}	{8}
20	8	{0,1,5,16}	{0,10}	20	8	2	{16,18}	{4}	{}
21	12	{0,1,7,15}	{0}	21	12	3	{17,19,20}	{8}	{2}
22	10	{0,1,11,12}	{0}	22	10	5	{20,21}	{8}	{4}
23	22	{0,1}	{0}	23	22	11	{21,22}	{20}	{10}
24	8	{0,1,9,16}	{0,12}	24	8	1	{16,20,22}	{}	{}
25	20	{0,1}	{0,5,10,15,20}	25	20	10	{20,23}	{18}	{8}
26	12	{0,1,13,14}	{0}	26	12	6	{24,25}	{10}	{4}
27	18	{0,1}	{0,9,18}	27	18	9	{21,24,25}	{16}	{8}
28	12	{0,1,8,21}	{0,14}	28	12	3	{24,26}	{8}	{2}
29	28	{0,1}	{0}	29	28	14	{27,28}	{26}	{12}
30	8	{0,1,6,10,15,16,21,25}	{0}	30	8	2	{26,28,29}	{4}	{}
31	30	{0,1}	{0}	31	30	15	{29,30}	{28}	{14}
32	16	{0,1}	{0,8,16,24}	32	16	4	{24,28}	{12}	{2}
33	20	{0,1,12,22}	{0}	33	20	5	{29,31,32}	{16}	{4}
34	16	{0,1,17,18}	{0}	34	16	8	{32,33}	{14}	{6}
35	24	{0,1,15,21}	{0}	35	24	6	{31,33,34}	{20}	{4}
36	12	{0,1,9,28}	{0,6,12,18,24,30}	36	12	3	{30,32}	{8}	{2}
37	36	{0,1}	{0}	37	36	18	{35,36}	{34}	{16}
38	18	{0,1,19,20}	{0}	38	18	9	{36,37}	{16}	{8}
39	24	{0,1,13,27}	{0}	39	24	6	{35,37,38}	{20}	{4}
40	16	{0,1,16,25}	{0,20}	40	16	2	{32,36,38}	{8}	{}
41	40	{0,1}	{0}	41	40	20	{39,40}	{38}	{18}
42	12	{0,1,7,15,21,22,28,36}	{0}	42	12	3	{38,40,41}	{8}	{2}
43	42	{0,1}	{0}	43	42	21	{41,42}	{40}	{20}
44	20	{0,1,12,33}	{0,22}	44	20	5	{40,42}	{16}	{4}
45	24	{0,1,10,36}	{0,15,30}	45	24	6	{39,41,42,43}	{20}	{4}
46	22	{0,1,23,24}	{0}	46	22	11	{44,45}	{20}	{10}

,

Table 2. Computer calculations for $47\le n\le 85$.

$\mathit{n}$	ϕ(n)	Idempotents	Nilpotents	$\mathit{V}\left(\mathit{G}\right)$	$\mathit{V}\left({\mathit{\Gamma }}_{\mathit{u}}\right)$	$\mathit{V}\left({\mathit{\Gamma }}_{\mathit{q}}\right)$	$\mathbf{D}\mathit{G}$	$\mathbf{D}{\mathit{\Gamma }}_{\mathit{u}}$	$\mathbf{D}{\mathit{\Gamma }}_{\mathit{q}}$
47	46	{0,1}	{0}	47	46	23	{45,46}	{44}	{22}
48	16	{0,1,16,33}	{0,12,24,36}	48	16	2	{40,44}	{8}	{}
49	42	{0,1}	{0,7,14,21,28,35,42}	49	42	21	{42,47}	{40}	{20}
50	20	{0,1,25,26}	{0,10,20,30,40}	50	20	10	{45,48}	{18}	{8}
51	32	{0,1,18,34}	{0}	51	32	8	{47,49,50}	{28}	{6}
52	24	{0,1,13,40}	{0,26}	52	24	6	{48,50}	{20}	{4}
53	52	{0,1}	{0}	53	52	26	{51,52}	{50}	{24}
54	18	{0,1,27,28}	{0,18,36}	54	18	9	{48,51,52}	{16}	{8}
55	40	{0,1,11,45}	{0}	55	40	10	{51,53,54}	{36}	{8}
56	24	{0,1,8,49}	{0,28}	56	24	3	{48,52,54}	{16}	{2}
57	36	{0,1,19,39}	{0}	57	36	9	{53,55,56}	{32}	{8}
58	28	{0,1,29,30}	{0}	58	28	14	{56,57}	{26}	{12}
59	58	{0,1}	{0}	59	58	29	{57,58}	{56}	{28}
60	16	{0,1,16,21,25,36,40,45}	{0,30}	60	16	2	{52,56,58}	{8}	{}
61	60	{0,1}	{0}	61	60	30	{59,60}	{58}	{28}
62	30	{0,1,31,32}	{0}	62	30	15	{60,61}	{28}	{14}
63	36	{0,1,28,36}	{0,21,42}	63	36	9	{57,59,60,61}	{32}	{8}
64	32	{0,1}	{0,8,16,24,32,40,48,56}	64	32	8	{56,60}	{28}	{6}
65	48	{0,1,26,40}	{0}	65	48	12	{61,63,64}	{44}	{8}
66	20	{0,1,12,22,33,34,45,55}	{0}	66	20	5	{62,64,65}	{16}	{4}
67	66	{0,1}	{0}	67	66	33	{65,66}	{64}	{32}
68	32	{0,1,17,52}	{0,34}	68	32	8	{64,66}	{28}	{6}
69	44	{0,1,24,46}	{0}	69	44	11	{65,67,68}	{40}	{10}
70	24	{0,1,15,21,35,36,50,56}	{0}	70	24	6	{66,68,69}	{20}	{4}
71	70	{0,1}	{0}	71	70	35	{69,70}	{68}	{34}
72	24	{0,1,9,64}	{0,12,24,36,48,60}	72	24	3	{60,64,66,68}	{16}	{2}
73	72	{0,1}	{0}	73	72	36	{71,72}	{70}	{34}
74	36	{0,1,37,38}	{0}	74	36	18	{72,73}	{34}	{16}
75	40	{0,1,25,51}	{0,15,30,45,60}	75	40	10	{65,70,71,73}	{36}	{8}
76	36	{0,1,20,57}	{0,38}	76	36	9	{72,74}	{32}	{8}
77	60	{0,1,22,56}	{0}	77	60	15	{73,75,76}	{56}	{14}
78	24	{0,1,13,27,39,40,52,66}	{0}	78	24	6	{74,76,77}	{20}	{4}
79	78	{0,1}	{0}	79	78	39	{77,78}	{76}	{38}
80	32	{0,1,16,65}	{0,20,40,60}	80	32	4	{72,76}	{24}	{}
81	54	{0,1}	{0,9,18,27,36,45,54,63,72}	81	54	27	{72,75,79}	{52}	{26}
82	40	{0,1,41,42}	{0}	82	40	20	{80,81}	{38}	{18}
83	82	{0,1}	{0}	83	82	41	{81,82}	{80}	{40}
84	24	{0,1,21,28,36,49,57,64}	{0,42}	84	24	3	{76,80,82}	{16}	{2}
85	64	{0,1,35,51}	{0}	85	64	16	{81,83,84}	{60}	{12}

and

Table 3. Computer calculations for $86\le n\le 99$.

$\mathit{n}$	ϕ(n)	Idempotents	Nilpotents	$\mathit{V}\left(\mathit{G}\right)$	$\mathit{V}\left({\mathit{\Gamma }}_{\mathit{u}}\right)$	$\mathit{V}\left({\mathit{\Gamma }}_{\mathit{q}}\right)$	$\mathbf{D}\mathit{G}$	$\mathbf{D}{\mathit{\Gamma }}_{\mathit{u}}$	$\mathbf{D}{\mathit{\Gamma }}_{\mathit{q}}$
86	42	{0,1,43,44}	{0}	86	42	21	{84,85}	{40}	{20}
87	56	{0,1,30,58}	{0}	87	56	14	{83,85,86}	{52}	{12}
88	40	{0,1,33,56}	{0,44}	88	40	5	{80,84,86}	{32}	{4}
89	88	{0,1}	{0}	89	88	44	{87,88}	{86}	{42}
90	24	{0,1,10,36,45,46,55,81}	{0,30,60}	90	24	6	{84,86,87,88}	{20}	{4}
91	72	{0,1,14,78}	{0}	91	72	18	{87,89,90}	{68}	{16}
92	44	{0,1,24,69}	{0,46}	92	44	11	{88,90}	{40}	{10}
93	60	{0,1,31,63}	{0}	93	60	15	{89,91,92}	{56}	{14}
94	46	{0,1,47,48}	{0}	94	46	23	{92,93}	{44}	{22}
95	72	{0,1,20,76}	{0}	95	72	18	{91,93,94}	{68}	{16}
96	32	{0,1,33,64}	{0,24,48,72}	96	32	4	{80,88,92}	{24}	{2}
97	96	{0,1}	{0}	97	96	48	{95,96}	{94}	{46}
98	42	{0,1,49,50}	{0,14,28,42,56,70,84}	98	42	21	{91,96}	{40}	{20}
99	60	{0,1,45,55}	{0,33,66}	99	60	15	{93,95,96,97}	{56}	{14}

for the induced values of $n$, we make note of the following observations:

• The graph ${\Gamma }_u$ is regular but not complete.
• The graph ${\Gamma }_q$ is regular for values of $n\equiv 1(mod 4)$ and is not complete. However, for values of $n\equiv 3(mod 4)$, it is complete. Notably, it is complete for certain composite values of $n$ such as $n=9,14,18,21,22,27,28,33.$
• The graph $G$ is not regular, thus it is not complete for any induced value of $n$.
• In general, the variation of degrees of vertices in $V\left(G\right)$ is not influenced by idempotent and nilpotent properties.

4. Conclusions

This scientific exploration contributes to the understanding of quadratic congruences and their graphical representations, offering valuable insights into the structure and properties of associated graphs. The presented results and analyses pave the way for further investigations in graph theory and number theory.