Comments on the Clique Number of Zero-Divisor Graphs of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.344299pt" id="M1" height="16.1743pt" version="1.1" viewBox="-0.0657574 -11.83 18.7954 16.1743" width="18.7954pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-91" d="M669 636L663 650H260C230 650 221 652 208 675H186C177 621 162 548 144 481L173 479C195 534 211 560 226 578C248 604 266 615 373 615H544C460 511 114 112 23 16L31 0H561C577 34 611 135 622 170L594 182C566 125 544 88 515 65C481 38 416 35 319 35C249 35 197 37 152 41C270 184 542 501 669 636Z"/></g><g transform="matrix(.012,0,0,-0.012,11.789,4.134)"><path id="g50-111" d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z"/></g></svg>

Tian, Yanzhao; Li, Lixiang

doi:https://doi.org/10.1155/2022/6591317

Journal of Mathematics

On this page

Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 6591317 | https://doi.org/10.1155/2022/6591317

Comments on the Clique Number of Zero-Divisor Graphs of

Yanzhao Tian^1,2and Lixiang Li^1,2

Academic Editor: Francesca Tartarone

Received12 Jan 2022

Accepted10 Feb 2022

Published29 Mar 2022

Abstract

In 2008, J. Skowronek-kaziw extended the study of the clique number to the zero-divisor graph of the ring , but their result was imperfect. In this paper, we reconsider of the ring and give some counterexamples. We propose a constructive method for calculating and give an algorithm for calculating the clique number of zero-divisor graph. Furthermore, we consider the case of the ternary zero-divisor and give the generation algorithm of the ternary zero-divisor graphs.

1. Introduction

Abstract algebraic systems can be described more directly by studying algebraic systems with graph properties. Some of the results [1–4] showed that nonisomorphic algebraic systems have the same zero-divisor graph, which indicates that there may be some common characteristics among nonisomorphic algebraic systems. The zero-divisor graphs (ZDG) of commutative rings is a very active intersection of commutative algebra and graph theory. In 1988, Beck [5] illustrated the concept of ZDG of a commutative rings and proved that there is a one-to-one mapping between its chromatic number and its clique number . However, Anderson and Naseer [6] gave a counterexample on commutative local rings and denied this correspondence. By modifying the definition of the ZDG, the relationships between ZDG and commutative rings have aroused great interest among scholars [7–10]. Many different problems from bioinformatics and electrical engineering have been modeled by using cliques [11–14]. Tian et al. [15] proposed a password leak detection scheme based on a topological graph, using topological sequence to generate false user’s passwords and enhance the safety of the user’s passwords. Through the analysis of the structure of the topological graph, it was found that the zero-divisor graphs can also be applied to construct the password leak detection scheme. This scheme based on honeywords (false password) is helpful to enhance the security of password files in the system.

Owing to the important properties of ZDG, some scholars have extended the relevant concepts to noncommutative rings, and some interesting results have been obtained [16, 17]. Anderson and Weber [18] investigated the ZDG of , where does not contain an identity. Patil and Waphare [19] characterized the ZDG of a ring with involution. Mulay [20] classified the cycle structure of the ZDG and established some group-theoretic properties of the group of graph-automorphisms. Das et al. [21] proved the upper bound of the oriented relative clique number of a planar graph to 32. Estaji and Khashyarmanesh [22] studied the properties of ZDG on finitely bounded lattices £, which are closely related to Boolean algebra. Patil and Momale [23] proved the relationship between idempotent graphs and zero-divisor graphs over a ring . Bennis et al. [24] characterized powers of zero-divisors graph based on the parametrized family of graphs and investigated the diameter and girth of -extended zero-divisor graphs.

In the commutative ring, Anderson and Livingston [25] studied the general relationship between the graph structure properties of the ZDG and the ring structure properties of . In 2004, Akbari and Mohammadian [26] studied the relationship between the chromatic number of and its degree. Skowronek-kaziw [27] proved the one-to-one relationship between the clique number and the maximum degree of and gave the calculation method, but his result was imperfect.

In this paper, we reconsidered the ZDG of and gave some counterexamples. The method of calculating clique number is constructive. In Theorem 3, the construction method of the vertex set in the maximal cliques is given. On this basis, we introduce the ternary zero-divisor graph and give the calculation formula of three factors. In Section 3, we give an algorithm for calculating the clique number and the generation algorithm of the ternary zero-divisor graph, and the validity of the algorithm can be verified by Python programming.

2. Main Results

A clique is a subgraph of , any two of its vertices are connected by edges, and the order of the maximum clique in a graph is its clique number [28, 29]. A subgraph with nodes is referred to as a clique with dimension if every two different nodes in the have edges connected. The minimum number of colors that can be used to color the edges of is called the edge chromatic number, which is represented by . The maximum degree of is represented by . The node chromatic number of a graph is the minimum for which has a coloring. The ZDG of the rings is denoted by , and the node-set of the graph is , in which any two nodes and which have connected edges if and only if and . It is found that is a connected graph [30]. The graph is a graph with a vertex set in , where every two vertices and have connected edges if and .

In this section, we give some counterexamples about the clique number of , for example, let us set , and through the analysis of the clique number, we find that contains 30, 42, 70, and 210, so the clique number is 4, and let us set , we find that the vertices that satisfy the definition of are 6, 18, 36, 54, 72, and 90, so the clique number is 6. We give a method for calculating the clique number of . At the same time, we give some examples. Let be the prime power factorization of , where are different primes and , .

Definition 1. The ZDG of the rings is denoted by , and the node-set of the graph is , in which any two nodes and have connected edges if and only if and .

Theorem 1 (see [28],Vizing’s theorem). For every graph , , we have either or .

Theorem 2 (see [27]). The largest degree in has the node , and the largest degree is .

Theorem 3. If is a positive integer, all are equal to 1, then the clique number of the graph is equal to . If (where a is a positive integer, a >1, and all ), the clique number is . Otherwise, the clique number iswhere are even and are odd.

Proof. According to Definition 1, as long as the weight product of any two vertices is a multiple of , the result of the modular operation on is 0, and these vertices are in the set of the zero-divisor graph clique numbers. Based on this consideration, we only need to find that the weight value product of any two vertices is a multiple of . The problem of solving the maximal clique is transformed into the problem of solving the set of any two vertices whose weights are multiples of . Three cases are considered, respectively.(i)If is square-free, let , where are distinct primes. We consider the set , the product of every pair elements of the set is a multiple of , i.e., the elements of is in the vertices set of , and there are no more elements in . Thus, the clique number that satisfies the condition is equal to .(ii)If all are even, we investigate . Obviously, in the set composed of multiples of , the product of any two elements is a multiple of , and the set composed of these elements is the largest, that is, no larger set satisfies this feature. Then, the element and the elements form a clique number of . By the construction of element , it is not difficult to find that the product is the largest multiple of in ring , and the clique number that satisfies the condition is equal to .(iii)If are even and odd numbers, let , () are even, are odd, and satisfies the relation , , , , and satisfy the relation . In the following review of the clique number discussion, we use mathematical induction, and the purpose is to prepare for the later algorithm design.If , , and let . We consider the set . The product of every pair elements of the set is a multiple of , i.e., the elements of are in the vertices set of . By the construction of element , it is not difficult to find that the product is the largest multiple of in ring , and the clique number that satisfies the condition is equal to .
If , , and let . We consider the set . The product of every pair elements of the set is a product of , i.e., the elements of is in the vertices set of . By the construction of element , it is not difficult to find that the product is the largest multiple of in ring , and the clique number that satisfies the condition is equal to .
Repeat the process by using mathematical induction.
If , let , and . Let . We consider the set . The product of every pair elements of the set is a multiple of , i.e., the elements of is in the vertices set of . By the construction of element , it is not difficult to find that the product is the largest multiple of in ring , and the clique number that meets the condition is . Of course, the number .
Through the above analysis, we conclude that the number of elements satisfying the condition is equal toThe proof is complete.

We give some examples of calculating the number of the clique number in the ZDG, and the validity of the algorithm can be verified by Python programming.

In this paper, we present ZDG for by Python. Figure 1 shows the structure of the ZDG of , which is completely symmetric. The clique number of ZDG of is 2, and the sets that satisfy the zero-divisor condition are = \{2, 10\}, = \{6, 10\}, = \{14, 10\}, = \{18, 10\}, = \{4, 10\}, = \{8, 10\}, = \{12, 10\}, = \{16, 10\}, = \{4, 5\}, = \{5, 8\}, = \{5, 12\}, = \{5, 16\}, = \{4, 15\}, = \{8, 15\}, = \{12, 15\}, and = \{15, 16\}. There are 16 sets that satisfy this condition, and these sets are isomorphic. Figure 2 shows the structure of the ZDG of , which is also completely symmetric. The two-way arrow indicates that the condition of ZDG of is satisfied between two points, where the triangle formed by the bold line is the element of the number of clusters of the ZDG of , which include 10, 12, and 15. Figure 3 shows the structure of the ZDG of , which include 7, 14, 21, 28, 35, and 42. Figure 4 shows the structure of the ZDG of . The number of elements which satisfying the condition is 4, which include 6, 30, 45, and 60.

To further investigate the properties of the zero-divisor graphs, we generalize the number of zero-divisor graphs satisfying Definition 1 to three, namely, all zero-divisor graphs satisfying equation , where is a positive natural number; the product of any three positive natural numbers is a multiple of ; after the product takes the modulus of , the result is zero, which we call zero-divisor. The three zero-divisor elements represent the weight values of vertices and edges in graph , respectively.

Definition 2. The zero-divisor graph of the rings is denoted by , and the node-set of the graph is , in which any three nodes , , and have connected edges if and only if and , , and . In the triple , we specify that the first element and the third element represent two vertices of the zero-divisor graph, and the second element represents the edge weight values of the zero-divisor graph.

Theorem 4. Suppose , , and are positive integers, where , , and are different from each other and satisfy the following 3-tuples congruence equation:When is taken of different values, the number of triples formed by the solutions to equation (3) is given byFor the sake of convenience, the symbol represents the set of solutions for all the positive integers satisfying the 3-tuples congruence equation, represents the number of elements in the set , represents the set that contains , , represents the set that contains , , , and represents the set that contains multiple of .

Proof. If , according to Definition 2, it is not difficult to find that set is empty. is a prime number, and then set is empty.
The following assumes that is a composite number, according to the fundamental arithmetic theorem, and we have the standard decomposition of which iswhere are different primes and , .
We discuss the value of in the following situations, and the method discussed is similar when takes other values. According to the prime factorization of integer , we divide the elements in the interval into two disjoint sets and select the elements from different sets to generate the multiple of . Obviously, the product of elements satisfying the multiple of also satisfies equation (1). Based on this idea, we have the following discussion:(i) The number of elements is . Since , , the number of elements is , and the number of elements is . By the permutation and combination formula, we can get In this case, is calculated from equation (6).(ii) The number of elements is . Since , , the number of elements is , and the number of elements is . By the permutation and combination formula, we can get In this case, is calculated from equation (7).(iii) The number of elements is , the number of elements is , the number of elements is , and the number of elements is . By the permutation and combination formula, we have In this case, is calculated from equation (8).(iv) The number of elements is , the number of elements is , the number of elements is , the number of elements is , the number of elements is , the number of elements is , the number of elements is , the number of elements is , the number of elements is , the number of elements is , the number of elements is . For the sake of writing conveniently, is abbreviated as , represents the set , represents the set , represents the set , represents the set , represents the set , represents the set , represents the set , represents the set , represents the set , represents the set , represents the set , represents the set , represents the set , represents the set , represents the set , represents the set , and represents the set . By the permutation and combination formula, we have In this case, is calculated from equation (9).(v) The number of elements is , the number of elements is , the number of elements is , the number of elements is , the number of elements is , the number of elements is , and the number of elements is . represents the set represents the set , represents the set , represents the set , represents the set , represents the set , represents the set , and represents the set . The number of elements is . By the permutation and combination formula, we have In this case, is calculated from equation (10). This completes the proof.

In Figures 5–8, the zero-divisor graphs corresponding to different are given. With the increase of , the number of edges of the zero-divisor graphs shows a rapid increase trend, but the number of the corresponding maximum clique does not change significantly, which indicates that the zero-divisor graphs are characterized by sparsity. The clique number of is 5. The vertices set of maximal clique is . The set of maximal clique elements is . The clique number of is 4. The vertices set of maximal clique is . The set of maximal clique elements is . The clique number of is 3. The vertices set of maximal clique is . The set of maximal clique elements is . The clique number of is 5. The vertices set of maximal clique is . The set of maximal clique elements is .

3. The Clique Number Algorithm and Some Examples

3.1. Clique Number Algorithm

The Clique number algorithm is detailed in Algorithm 1. The speed at which the clique number on the ring is solved depends on the prime factorization of the integer. If the integer is a composite number, according to the discussion of Theorem 3, the clique number is solved in three cases, in one case, the indexes are all 1, in the second case, the indexes are all even, and in the third case, the number of odd and even indexes is at least 1. It is not difficult to find that the number of the clique number is closely related to the prime factors and indicators of integers. The zero-divisor graph of three factors is given by Algorithm 2, and the formula of the solution of the ternary is given by Theorem 4. We find that the complexity of solving zero-divisor is increasing rapidly with the difference of the number and index of prime factors.

	Input: integer
Output: the clique number
(1)	Pollard Rho Prime Factorization ( is prime, is index).
(2)	if is a composite number then
(3)	if = 1(i = 1 to n) then
(4)	The clique number is the number of prime factors .
(5)	else if mod 2 0 (i = 1 to ) then
(6)	The clique number is .
(7)	else if mod 2 0 ( =1 to ) and mod 2 1 ( =1 to ) and and then
(8)	The clique number is ( is the number of prime factors that are odd, and is the number of prime factors that are even).
(9)	else
(10)	Integer is prime, and the clique number does not exist.
(11)	end if
(12)	end if

	Input: the sets and produced by equation (1)
Output: zero-divisor graph .
(1)	Initialization: set the operations (intersection, union and difference, symmetric difference, etc.) on sets and to generate the vertex set .
(2)	for to do
(3)
(4)	for to do
(5)	if weight weight () then
(6)	Generate the edge set: is the number of vertices connected to , is the set of vertices with the smallest labeling.
(7)	else
(8)	,
(9)	return Step 6.
(10)	end if
(11)	end for
(12)	end for
(13)	return.

3.2. Examples

(1)If , based on Theorem 3, the clique number of satisfying the condition is 3. The vertices set of is .(2)If , based on Theorem 3, the clique number of that meets the criteria is 3. The vertices set of is .(3)If , then based on Theorem 3, the clique number of that meets the criteria is 141.(4)If , then based on Theorem 3, the clique number of satisfying the condition is 106.(5)If , then based on Theorem 3, the clique number of satisfying the condition is 2. The vertices set of is .

4. Conclusions

Through the induction and classification discussion methods, we simplified the calculation of the clique number and constructed an algorithm to calculate the clique number in the ZDG. Through analysis, it is found that the ZDG satisfies the properties of high symmetry and degree centrality. The so-called degree centrality refers to that a very large scale of edges are connected by very few vertices, at the same time, other edges are joined by very few vertices. Furthermore, we discussed the calculation of the ternary zero-divisor graph and gave the algorithm of the ternary zero-divisor graph. By analyzing Theorem 4, we found that the number of zero-divisor solutions is given by fractional polynomials, and the complexity of solving zero-divisor solutions increases rapidly with the increase of the number of prime factors.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported in part by the National Key Research and Development Program of China (Grant no. 2020YFB1805403), the National Natural Science Foundation of China (Grant nos. 62032002, 61972051), and the Natural Science Foundation of Beijing Municipality (Grant no. M21034).

References

K. E. Smith and M. Van den Bergh, “Simplicity of rings of differential operators in prime characteristic,” Proceedings of the London Mathematical Society, vol. 75, no. 1, pp. 32–62, 1997.
View at: Publisher Site | Google Scholar
L. W. Christensen, D. A. Jorgensen, H. Rahmati, J. Striuli, and R. Wiegand, “Brauer-Thrall for totally reflexive modules,” Journal of Algebra, vol. 350, no. 1, pp. 340–373, 2012.
View at: Publisher Site | Google Scholar
A. Patil, A. Khairnar, and P. S. Momale, “Zero-divisor graph of a ring with respect to an automorphism,” Soft Computing, vol. 26, pp. 1–13, 2022.
View at: Publisher Site | Google Scholar
M. Behboodi, R. Beyranvand, and H. Khabazian, “Strong zero-divisors of non-commutative rings,” Journal of Algebra and Its Applications, vol. 8, no. 4, pp. 565–580, 2009.
View at: Publisher Site | Google Scholar
I. Beck, “Coloring of commutative rings,” Journal of Algebra, vol. 116, no. 1, pp. 208–226, 1988.
View at: Publisher Site | Google Scholar
D. D. Anderson and M. Naseer, “Beck’s coloring of a commutative ring,” Journal of Algebra, vol. 159, no. 2, pp. 500–514, 1993.
View at: Publisher Site | Google Scholar
G. Aalipour and S. Akbari, “On the cayley graph of a commutative ring with respect to its zero-divisors,” Communications in Algebra, vol. 44, no. 4, pp. 1443–1459, 2016.
View at: Publisher Site | Google Scholar
D. F. Anderson, R. Levy, and J. Shapiro, “Zero-divisor graphs, von neumann regular rings, and boolean algebras,” Journal of Pure and Applied Algebra, vol. 180, no. 3, pp. 221–241, 2003.
View at: Publisher Site | Google Scholar
P. Livingston, “Structure in zero-divisor graphs of commutative rings,” University of Tennessee, Knoxville, TN, USA, 1997, Master’s Thesis.
View at: Google Scholar
S. P. Redmond, “An ideal-based zero-divisor graph of a commutative ring,” Communications in Algebra, vol. 31, no. 9, pp. 4425–4443, 2003.
View at: Publisher Site | Google Scholar
A. Ben-Dor, R. Shamir, and Z. Yakhini, “Clustering gene expression patterns,” Journal of Computational Biology, vol. 6, no. 3, pp. 281–297, 1999.
View at: Publisher Site | Google Scholar
I. Hamzaoglu and J. H. Patel, “Test set compaction algorithms for combinational circuits,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 19, no. 8, pp. 957–963, 2000.
View at: Publisher Site | Google Scholar
V. Spirin and L. A. Mirny, “Protein complexes and functional modules in molecular networks,” Proceedings of the National Academy of Sciences, vol. 100, no. 21, pp. 12123–12128, 2003.
View at: Publisher Site | Google Scholar
A. Tanay, R. Sharan, and R. Shamir, “Discovering statistically significant biclusters in gene expression data,” Bioinformatics, vol. 18, no. suppl 1, pp. S136–S144, 2002.
View at: Publisher Site | Google Scholar
Y. Tian, L. Li, H. Peng, and Y. Yang, “Achieving flatness: graph labeling can generate graphical honeywords,” Computers & Security, vol. 104, Article ID 102212, 2021.
View at: Publisher Site | Google Scholar
S. Akbari and A. Mohammadian, “Zero-divisor graphs of non-commutative rings,” Journal of Algebra, vol. 296, no. 2, pp. 462–479, 2006.
View at: Publisher Site | Google Scholar
S. P. Redmond, “The zero-divisor graph of a non-commutative ring,” International Journal of Commutative Rings, vol. 1, pp. 203–211, 2002.
View at: Google Scholar
D. F. Anderson and D. Weber, “The zero-divisor graph of a commutative ring without identity,” International Electronic Journal of Algebra, vol. 23, no. 23, pp. 176–202, 2018.
View at: Publisher Site | Google Scholar
A. Patil and B. Waphare, “The zero-divisor graph of a ring with involution,” Journal of Algebra and Its Applications, vol. 17, no. 3, pp. 1–17, 2018.
View at: Publisher Site | Google Scholar
S. B. Mulay, “Cycles and symmetries of zero-divisors,” Communications in Algebra, vol. 30, no. 7, pp. 3533–3558, 2002.
View at: Publisher Site | Google Scholar
S. Das, S. Prabhu, and S. Sen, “A study on oriented relative clique number,” Discrete Mathematics, vol. 341, no. 7, pp. 2049–2057, 2018.
View at: Publisher Site | Google Scholar
E. Estaji and K. Khashyarmanesh, “The zero-divisor graph of a lattice,” Results in Mathematics, vol. 61, no. 1, pp. 1–11, 2012.
View at: Publisher Site | Google Scholar
A. Patil and P. S. Momale, “Idempotent graphs, weak perfectness, and zero-divisor graphs,” Soft Computing, vol. 25, no. 15, pp. 10083–10088, 2021.
View at: Publisher Site | Google Scholar
D. Bennis, B. Alaoui, B. Fahid, M. Farnik, and R. L’hamri, “The i-extended zero-divisor graphs of commutative rings,” Communications in Algebra, vol. 49, no. 11, pp. 1–18, 2021.
View at: Publisher Site | Google Scholar
D. F. Anderson and P. S. Livingston, “The zero-divisor graph of a commutative ring,” Journal of Algebra, vol. 217, no. 2, pp. 434–447, 1999.
View at: Publisher Site | Google Scholar
S. Akbari and A. Mohammadian, “On the zero-divisor graph of a commutative ring,” Journal of Algebra, vol. 274, no. 2, pp. 847–855, 2014.
View at: Google Scholar
J. Skowronek-Kazi w, “Some digraphs arising from number theory and remarks on the zero-divisor graph of the ring ,” Information Processing Letters, vol. 108, no. 3, pp. 165–169, 2008.
View at: Google Scholar
P. Zhang and G. Chartrand, Introduction to Graph Theory, Tata McGraw-Hill Education, New York, NY, USA, 2006.
H. P. Yap and H. Y. Yap, Some Topics in Graph Theory, Cambridge University Press, Cambridge, MA, USA, vol. 108, 1986.
F. R. DeMeyer, T. McKenzie, and K. Schneider, “The zero-divisor graph of a commutative semigroup,” Semigroup Forum, vol. 65, no. 2, pp. 206–214, 2002.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2022 Yanzhao Tian and Lixiang Li. 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.

PDF Download Citation

Download other formats

Order printed copies