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Journal of Applied Mathematics
Volume 2012, Article ID 292895, 10 pages
http://dx.doi.org/10.1155/2012/292895
Research Article

On Simple Graphs Arising from Exponential Congruences

Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan

Received 19 March 2012; Revised 17 June 2012; Accepted 3 September 2012

Academic Editor: Maurizio Porfiri

Copyright © 2012 M. Aslam Malik and M. Khalid Mahmood. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce and investigate a new class of graphs arrived from exponential congruences. For each pair of positive integers and , let denote the graph for which is the set of vertices and there is an edge between and if the congruence is solvable. Let be the prime power factorization of an integer , where are distinct primes. The number of nontrivial self-loops of the graph has been determined and shown to be equal to . It is shown that the graph has components. Further, it is proved that the component of the simple graph is a tree with root at zero, and if is a Fermat's prime, then the component of the simple graph is complete.

1. Introduction

The notion of congruence is intrinsic in number theory. Modular arithmetic has clutched a vital contrivance for most of the number theoretic mathematics. In recent years, studying graphs through congruences is of charismatic and an independent interest of number theorists. It has cemented a novel approach to introduce a premium connection between Number Theory and Graph Theory. On behalf of modular arithmetic, we can manipulate many fascinating features of graphs. We first encounter the ideas used in [15], where digraphs from congruences were discussed. The conditions for regularity and semiregularity of such digraphs are presented in [1, 2]. The necessary and sufficient conditions for the existence of isolated fixed points have been established in [3]. The structures of symmetric digraphs have been studied in [4, 5]. In this paper we discuss the graph arriving from exponential congruences. We assign to each pair of positive integers and an edge of the graph if the congruence is solvable, where is the set of vertices of and is the set of edges of . Then the graph has a loop at a vertex if and only if admits a solution. Since the congruence relation is an equivalence relation, so , always admits a solution . Thus we call a trivial solution and, so far, the loops due to trivial solution are called trivial loops. Thus it becomes interesting to find the number of vertices in such that the congruence admits a nontrivial solution. The loops at vertices corresponding to nontrivial solutions are called nontrivial loops. Since for each , the congruence , for , so the loops at vertices and will be considered as nontrivial loops. We denote the number for nontrivial loops of the graph . Let be the prime power factorization of an integer , where are distinct primes. The number of the graph has been determined and shown to be equal to , where is the Euler’s phi function. It has been shown that the graph has components. We label these components as , where or . The order of each component of the graph has been determined. Further, it is proved that the component of the simple graph is always a tree with root at zero. Also if is a Fermat’s prime, then the component of the simple graph is a complete graph.

2. Preliminaries

A graph is simple if it is free from loops and multiedges. The vertices will constitute a cycle of length if each of the following congruences is solvable: A graph is said to be connected if there is a path from to , for each pair of vertices and . A maximal connected subgraph is called a component [6].

In Figure 1, the simple graph has eight components. A graph is complete if every two distinct vertices of are adjacent. Thus is complete if is solvable for each distinct pair of vertices in . The degree of a vertex in is the number of edges incident with . It is denoted by . If for each vertex of , then is called -regular or regular graph of degree . The simple graph has two 3-regular and two 1-regular components.

292895.fig.001
Figure 1: The simple graph of .

We recall the definition of Euler’s phi function [7], and some of its properties.

Definition 2.1. For , let denote the number of positive integers not exceeding which are relatively prime to . Note that , because . Thus we can write By the definition of Euler’s phi function, it is clear that if , then . Also if and only if is a prime number. We will need the following results of [7], regarding Euler’s phi function, for use in the sequel.

Theorem 2.2. If is a prime and , then Let be an arithmetic function. One recalls that is said to be multiplicative if , .

The following theorem is the generalization of the well-known Fermat’s Little theorem which states that if , then .

Theorem 2.3 (Euler). If and , then .

Definition 2.4 (see [7]). A set of integers is a complete residue system (CRS) modulo if every integer is congruent to exactly one of the modulo . If we delete integers from CRS, which are not prime to , then the remaining integers will constitute the reduced residue system (RRS) modulo .

Theorem 2.5 (counting principle [8]). If a work is distributed over , objects with object occuring ways, object occuring ways, and object occuring ways, then the work can be done in ways.

Theorem 2.6 (the inclusion-exclusion principle [9]). Let be finite sets. Then

3. Applications of Euler’s Phi Function

In Graph Theory, a loop or a self-loop is an edge that connects a vertex to itself. A vertex of an undirected graph is called an isolated vertex if it is not the endpoint of any edge. This means that it is not connected with any other vertex of the graph. Thus in our case, the vertex is an isolated vertex if and only if , is not solvable. Now if we omit the condition of simplicity in our graph, then this isolated vertex will contribute twice in the degree as there will be a self-loop at as the congruence is solvable. This leads to the following results.

Theorem 3.1. Let be the prime factorization of , where are distinct, odd primes. Then 0 and are isolated vertices of . That is, the vertices 0 and are not the endpoints of any edge in graph except loop at these vertices.

Proof. The element , where , is an isolated point of the graph if and only if the solvability of implies that . It is trivial that is an isolated point of as is solvable if and only if . Moreover, are distinct, odd primes, so for some integer , is an even integer. Thus and hence is solvable with root . Next, we claim that the congruence , is not solvable. To prove our assertion, let and suppose there exists an integer , such that the congruence, is balanced. Then there exists some integer such that This implies that . But then . Let for some integer . If is even, then and hence by (3.1), , a contradiction as is an odd integer. So let , for some integer . Then and this shows that which is a contradiction against the fact that . Finally, the congruence is not solvable since an odd integer is not divisible by an even integer. Hence 0 and are not adjacent as well.

Corollary 3.2. If is an odd square free integer, then 0 is the only isolated vertex of the graph .

Theorem 3.3. Let be the prime power factorization of a non-square-free integer , where are distinct primes, and . Then 0 and , where , and , are always adjacent vertices in .

Proof. We show that the congruence is solvable for all . Let , be two distinct primes such that . Then . Thus is the solution of the congruence . We follow the fashion explained above. For this, take . Let , then there exist integers such that , for each . Let . Then it is easy to see that This shows that is the solution of the congruence, . Hence, and 0 are the adjacent vertices in

The following results give a formula for finding the number of nontrivial self-loops of the graph .

Lemma 3.4. Let be a prime number. Let , and let be the Euler’s phi function. Then .

Proof. Let , . A vertex of the graph has a self-loop if and only if   (mod n) is solvable. Thus to find the number of self-loops in , we need to count the number of vertices in CRS such that the congruence is solvable. By Euler’s Theorem, if , then is solvable with as its root, whence is solvable with . Since are the integers which are not prime to , so there are integers in CRS for which the congruence is solvable. Also, for each , the vertex 0 has self-loop in . Thus there are self-loops. For the case, if , with as the solution of . Then implies that . This yields a contradiction against the fact that if , then . Hence, there exist no vertex , for which is solvable. Thus are the only vertices for which the graph has self-loops.

The following theorem provides us the cardinality of the set of those vertices of which have nontrivial self-loops, where and .

Theorem 3.5. Let be the prime power factorization of an integer , where are distinct primes, and . Then

Proof. We apply induction on . By Lemma 3.4, result is true for . Suppose result is true for distinct prime factors. That is, if , then . Let . Now, the congruence is solvable if and only if the congruences and are solvable. But by induction, the graph has self-loops and, by Lemma 3.4, the graph has self-loops. Hence, by Theorem 2.5, the graph has self-loops. This through our result. In Figure 2, we depict Theorem 3.5 for .

292895.fig.002
Figure 2: The graph has 15 nontrivial self-loops.

Corollary 3.6. Let be the prime factorization of a square-free integer , where are distinct primes and . Then

Proof. It is easy to see that , for any prime . Thus by Theorem 3.5, corollary follows. That is,

Corollary 3.7. Let and be two non-square-free integers such that . Then,

4. Components and Their Characteristics

Recall that a maximal connected subgraph of a graph is called a component. For instance, the vertices from CRS will constitute a component of the graph if for each , there exist some such that is solvable, for all . The following theorem explores some interesting characteristics of components for the simple graph .

Theorem 4.1 (main theorem). Let be the prime power factorization of an integer , where are distinct primes, and . Then,(a) has components,(b)if , is a prime number, then is a tree with root at 0,(c)if is a Fermat’s prime, then is complete. Let . Then are the possible square-free positive divisors of . We see that the sets of vertices generated by these divisors will constitute components of . We label these components by . Moreover, the set of all those residues of which are prime to will provide us another component. Since this set contains vertices from , so, for the sake of convenience, we label this component by . Before giving the proof of Theorem 4.1, we prove the following results.

Lemma 4.2. (a) Let , where is a prime number. Then has components, namely, and .
(b) Let and , . Then the order of the component is

Proof. Recall that an integer such that is called a primitive root modulo only if is of order . Let be a primitive root of . Then the smallest integer is called the index of with respect to if . Moreover, it is well known that the exponential congruence is solvable if and only if , where . We divide the numbers into two sets, one of which is and the other is RRS modulo . We claim that these two sets constitute independent components of . Let where each is prime to , the set of vertices of the component . Then every vertex of must have some index with respect to primitive root . Thus the exponential congruence is solvable since and is divisible by , where, . This means that each vertex of the set is adjacent with 1. So it must contribute as one component of the graph . Moreover, for any integer , and hence is solvable with as its root. Thus each vertices of the set is adjacent with 0. Then will form another component of the graph . Next we claim that none of the vertex is adjacent with any of the vertex . To prove our assertion, let and be two vertices of such that and . Then there exist integers and such that Let , then there exist integers and such that . Hence by (4.2), we get Now, if and are adjacent, then there must be some integer such that . Then by (4.3), we obtain, which is absurd. This is through the claim. Hence and are the only components of . The part is the direct consequence of Theorem 2.6.

Corollary 4.3. For .

Two graphs or two components of a graph are said to be isomorphic to each other if there is an isomorphism between their vertex sets. Thus, if and are isomorphic, then . Moreover if the vertices and are adjacent in , then and must be adjacent in , where is an isomorphism between their sets of vertices. The following corollary can easily be proved by using Lemma 4.2 and Corollary 3.7.

Corollary 4.4. If and , are prime divisors of , then and , are nonisomorphic components of .

Proof. Since both and are prime numbers, so by Lemma 4.2 (b), the order of components and must be different. Hence both can never be isomorphic. For instance, take , , and . Then by Lemma 4.2 (b), Hence by (4.4), the components and are not isomorphic in . However, . Thus the isomorphism can be established between the components and in if they possess the same structure as well. This leads to the following corollary.

Corollary 4.5. Let and . If and are the components of , then .

Proof of Main Theorem. (a) It is well known that is solvable if and only if the congruence is solvable for each . Then the proof is analogous to Theorem 2.5 together with Lemma 4.2.
(b) It is easy to see that each element of in is connected with 0 since for each , , is satisfied. This shows that the congruence is solvable for each in . To show that in is a tree, we claim that the nonzero vertices in are not connected to each other. To prove our assertion, let for and . Then by Cancelation Law of congruences, we obtain, . As and , so by the simple divisibility rules, . But , thus we arrive at a contradiction. This through our claim and finally the component in is a tree with root at zero.
(c) It is wellknown that , is always composite. Thus the only Fermat’s primes are 3, 5, 17, 257, and 65537. Let us write , . Then , since is prime. To show that the component is complete, we need to show that the congruence is solvable for each . By [9], every odd prime has a primitive root. Since Fermat’s numbers are always odd, so each of the Fermat’s prime has a primitive root. Let it be . Then the congruence is solvable if and only if the congruence is solvable, where . As , , so the later congruence reduced to the linear congruence , where and . Then the simple graph will be complete if the congruence , or the congruence , is solvable. To prove our assertion, it is easy to see that either or , where . For the first case, the congruence , is solvable and has a unique solution. To discuss the rest of the case, we let and , , . If , then , so the linear congruence is solvable and has solutions. But if , we replace and and solve the congruence , which is solvable and has solutions. Thus in either case, we see that there is an edge , , in component of the simple graph when is a Fermat’s prime.

The following corollaries are the direct consequences of Theorem 4.1.

Corollary 4.6. Let , where is square free integer. Then is a tree with root at zero.

Corollary 4.7. If is Fermat prime, then is regular of degree .

5. Conclusions

This piece of work describes the relationship of exponential congruences with graphs. It has been explored that an exponential congruence yields a set of graphs. Also certain components of the graphs form trees if self-loops are suppressed. The types of graphs and trees were hence yielded form a pattern based on the nature of variables within the congruence. The major results formed show that the component of the simple graph is complete if is Fermat’s prime and also that the component , where is a square free integer, is always a tree. Intuitively, the results formed find their place in various applications of number theory, encryption, and algorithms. In various problems of data decryption using brute force method it is desirable to find the nature of a number and establish if it is divisible by some other prime number or not. Such problems can be tackled by forming a graph of its exponential congruence and determine the pattern formed by its components. Moreover these exponential congruences can be established as a succinct representation of graphs of a specific nature for their applications in various computer algorithms.

Acknowledgment

The authors are very thankful to the referees for their valuable comments and advices. This has made the paper more interesting and informative.

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