Abstract
Let be an ideal of a numerical semigroup . We define an undirected graph with vertex set and edge set . The aim of this article is to discuss the connectedness, girth, completeness, and some other related properties of the graph .
1. Introduction and Preliminaries
In mathematics, graph theory plays an important role in understanding the relationship between pair of objects through graphs. Today, graph theory has a number of applications in the field of engineering, computer science, and other sciences (see [1–3]). The idea of linking graph with algebraic structure gave rise to new research in the field of mathematics. This idea explores the interaction between structures of algebraic objects and graphs by using properties of graphs and algebraic structures. In the past, many researchers associated graphs with different algebraic structures (see [4–11]).
A graph is an ordered pair with the vertex set and the edge set . The order and the size of the graph are the cardinality of its vertex set and edge, respectively. A graph is connected if for any two vertices , there is a path connecting and . The distance between two vertices is the length of the shortest path between them and is denoted by . The length of the longest path in is the diameter of , denoted by . Length of a shortest cycle in the graph is referred to as the girth of the graph. A graph is complete if for any two vertices, there is an edge between them. A subgraph of a graph is a clique if is a complete graph, and the order of maximal clique is called the clique number of , denoted by . A graph is regular if all of its vertices have the same degree. For more undefined terminologies related to graph theory, see [12–14].
Numerical semigroup theory is the study of set of positive integer solutions of diophantine equations (see [15–18]), and this is very useful in the study of algebraic geometry codes (see [19–21]). Let us recall some definitions from the numerical semigroup theory, which we used in the later sections.
A subset of nonnegative integers is called submonoid if the following holds:(1) (2) If , then
A submonoid is called numerical semigroup if is finite. The least positive integer in , denoted by , is known as the multiplicity of the numerical semigroup. The elements of are called the gaps of , and the largest of these gaps is known as the Frobenius number, denoted by . It is well known that every numerical semigroup is finitely generated; that is, there exist such that . Moreover, every numerical semigroup has a unique minimal system of generators. The cardinality of the minimal system of generators is called the embedding dimension of numerical semigroup , denoted by . A subset of numerical semigroup is ideal (integral ideal) of if (that is, and , the element is in ). An ideal is an irreducible ideal if it cannot be written as intersection of two ideals which properly contain it.
A connection of graphs with numerical semigroups can be found in [22]. In this paper, we investigated the interplay between the structure of numerical semigroups and graph theoretic properties of specific graphs gained from numerical semigroups. The sketch of this paper is as follows.
In Section 2, we briefly describe the concept of connectedness, diameter, and girth of . In Section 3, we present a necessary and sufficient condition for a graph associated with an ideal of a numerical semigroup to be complete. In Section 4, we discuss the concept of clique number and cut point of .
2. Connectedness, Diameter, and Girth of
We start this section by showing that is always a connected graph. Also, we show that the diameter of is equal or less than 2 and girth of is 3.
Proposition 1. Let be a numerical semigroup and be an ideal of . Then, is a connected graph of finite order with at least one vertex of degree .
Proof. Choose , the Frobenius number, and , the least element. Then, note that for all . This gives that is always finite, and therefore, is finite. We may assume that is the largest element of not in as is finite. This gives that , . Therefore, is a connected graph of finite order with degree of and is equal to .
Example 1. Let be a numerical semigroup and be an ideal of . Then, and the graph is shown in Figure 1.
Proposition 2. Let be a graph associated with an ideal of a numerical semigroup . Then, . Furthermore, if contains a cycle, then .
Proof. Let be the largest element of not in ; then, has an edge with every vertex , where and . Therefore, . Now, for any two vertices and , , we haveThis implies thatMoreover, if any undirected graph has a cycle, then gr (see [13], Propostion 1.3.2). Therefore, .
Lemma 1. Let be a graph associated with an ideal of a numerical semigroup . If order of is 4, then must contain a cycle of length 3.
Proof. We may assume that . If is complete, then trivially it contains a cycle of length 3; otherwise, there is at least one pair of nonadjacent vertices. We have the following cases: Case_1: if and are nonadjacent, then either or . If , then there are two more possibilities, either or . So, and give . This implies that there is a cycle of length 3. Also, and give . This implies there is a cycle of length 3. Now, if , then clearly , and therefore, there is a cycle of length 3. Case_2: if and are nonadjacent, then , and therefore, there is a cycle of length 3. Case_3: if and are nonadjacent, then , and therefore, there is a cycle of length 3.
Proposition 3. Let be a graph associated with an ideal of a numerical semigroup . If order of , then must contain a cycle of length 3.
Proof. This follows from Lemma 1.
Corollary 1. Let be a graph associated with an ideal of a numerical semigroup . If order of , then is not a bipartite graph.
Corollary 2. Let be a graph associated with an ideal of a numerical semigroup . If order of , then .
3. Completeness of
In this section, we provide a sufficient and necessary condition for to be complete. Moreover, for a given numerical semigroup , we computed the least number of ideals for which graph is complete.
Lemma 2. Any ideal of a numerical semigroup is an intersection of irreducible ideals. Moreover, every irreducible ideal of numerical semigroup is of the form , where , for some .
Proof. For a proof, see [23].
Proposition 4. Let be an irreducible ideal of a numerical semigroup . If order of , then cannot be a complete graph.
Proof. Since is an irreducible ideal, order of for some . We may assume that , where . Then, there must exist for such that . This gives , so there is no edge between and , and therefore, is not a complete graph.
Example 2. Let be a numerical semigroup. Choose ; then, , and therefore, is an irreducible ideal. As , is not a complete graph (see Figure 2).
Corollary 3. Let is a graph of order , associated with an irreducible ideal of a numerical semigroup . Then, cannot be a regular graph.
Proof. This follows from Proposition 1 and Proposition 4.
Lemma 3. Let , where is the minimal system of generators. Then, for some , if and only if is the multiple of exactly one element .
Proof. Suppose that and is not a multiple of exactly one element of . Then, Case_1: if is a multiple of an element of other than , then we can assume . This gives and therefore , which is not possible. Case_2: if is a linear combination of at least two elements , then for some positive integers and . Note that . This gives , which is not possible.The converse of this statement is straightforward.
Theorem 1. Let be an ideal of a numerical semigroup such that , where is an irreducible ideal and . Then, is complete if and only if order of for each .
Proof. Let , where is the minimal system of generators. As for each , is an irreducible ideal; therefore, we can assume that for some . This gives that the set of vertices of is .
To prove the implication “”, suppose order of for some . Then, from Proposition 4, it follows that is not a complete graph, so there must exist such that ; i.e., there is no edge between and . This gives such that , and therefore, is not complete.
Now, to prove the implication “”, we suppose on contrary that is not a complete graph. Therefore, there exist such that . However, the order of for each ; therefore, either or . This gives that every element of is either an element of the minimal system of generators or twice of some element of (see Lemma 3). Note that cannot be an element of because , so the only possibility is that is twice of some element of , say , for some . This gives , i.e., , which is not possible. Hence, is a complete graph.
Example 3. Let be a numerical semigroup. Consider the irreducible ideals , , , , and , where , , , , and . Let and be two ideals of . Then, by Theorem 1, it follows that must be a complete graph (see Figure 3) while cannot be a complete graph (see Figure 4).
Proposition 5. Let , where be a numerical semigroup. Then, there are at least ideals such that is complete.
Proof. We prove it by induction on . If , then . We have the following cases: Case_1: if is irreducible, then the order of is . From Theorem 1, it follows that is complete if and only if . This gives or 2. If , then , , or , and if , then , , or . But, note that and may or may not exist because there is a possibility that and may be the multiple of more than one element of or these are linear combination of two or more than two elements of . So, in this case, there are at least 4 irreducible ideals for which is complete. Case_2: now, if is not irreducible, then can be written as an intersection of irreducible ideals. We assume , where is an irreducible ideal and . This gives order of as . So, if is complete, then , for all , and therefore, is one of , and . As must exist in irreducible case, the least possibilities for are , , , , , and . This gives that there are at least 7 reducible ideals for which is complete. Moreover, if , then is an empty graph, which is trivially a complete graph. Hence, there are at least ideals for which is complete.Now, suppose that, as the induction hypothesis, and, whenever, a numerical semigroup, say , has as a minimal generating set, then there are at least ideals such that is complete.
Now, let . Note that . Then, induction hypothesis gives that there are at least ideals such that is complete.
4. Clique Number and Cut Point
In this section, we present some results on clique number and cut point of the graph .
Proposition 6. Let be a graph associated with an irreducible ideal of a numerical semigroup . If order of is , then
Proof. Since is irreducible, the order of is exactly , for some . Assume that . We have the following cases: Case_1: if is even, then the cardinality of the set is odd, and by definition of , we know that . This gives that are pairs of nonadjacent vertices. As cardinality of is odd, it must contain an element . Note that every element of the set is equal or greater than , and therefore, the sum of any two different elements is greater than . This gives that the graph corresponding to the set is a complete graph of maximum possible order. Case_2: if is odd, then the cardinality of the set is even. In this case, we have . This gives that are pairs of nonadjacent vertices. Note that every element of the set is greater than , and therefore, the sum of any two different elements is greater than . This gives that the graph corresponding to the set is a complete graph of maximum possible order.
Example 4. Let be a numerical semigroup and be an irreducible ideal of . Then, Proposition 6 gives that the subset of is the maximal clique, and therefore, the clique number of graph is 5. The graph and its corresponding maximal clique are given in Figures 5 and 6, respectively:
Proposition 7. Let be a numerical semigroup, where is the minimal system of generators of and is a graph associated with an irreducible ideal , for some such that the order of . Then, is the only cut point of if and only if for some .
Proof. Assume that there exist some such that , where and . Then, we have the following cases: Case_1: if , then clearly for all ; otherwise, is a multiple of . This gives that for every two elements other than and , we have two different paths and . This implies that is not a cut point of . Case_2: if , then assume that is the largest element such that , where . Note that , where is the smallest element of such that and . Then, , for all and . This gives that for every two elements other than and , we have two different paths, either and or and . This implies that is not a cut point of .Now, if , then clearly and . This gives that has at least two disconnected components and for all . Therefore, is the only cut point.
Example 5. Let be a numerical semigroup and be a irreducible ideal, where . Then, is a cut point of graph (see Figures 7 and 8).
5. Conclusion
In this article, graphs associated with ideals of a numerical semigroup have been studied and it has been proved that these graphs are connected. We investigated some properties such as girth, regularity, clique number, and cut point of these graphs. Also, a necessary and sufficient condition has been given for a graph associated with an ideal of a numerical semigroup to be complete.
Data Availability
No data were used to support this study.
Disclosure
This research was carried out as a part of the employment of the authors.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.