Abstract
The concept of monogenic semigroup graphs is firstly introduced by Das et al. (2013) based on zero divisor graphs. In this study, we mainly discuss the some graph properties over the line graph of . In detail, we prove the existence of graph parameters, namely, radius, diameter, girth, maximum degree, minimum degree, chromatic number, clique number, and domination number over .
1. Introduction
The history of studying zero-divisor graphs has began over commutative rings by Beck’s paper [1], and then, it is followed over commutative and noncommutative rings by some of the joint papers (cf. [2–4]). After that, DeMeyer et al. [5, 6] studied these graphs over commutative and noncommutative semigroups. Since zero-divisor graphs have taken so much attention, the researchers added a huge number of studies to improve the literature. In [7], the authors introduced monogenic semigroup graphs based on actually zero-divisor graphs. In detail, to define , the authors first considered a finite multiplicative monogenic semigroup with zero as the set:
By considering the definition given in [5], it has been obtained an undirected (zero-divisor) graph associated to as in the following. The vertices of the graph are labeled by the nonzero zero divisors (in other words, all nonzero element) of , and any two distinct vertices and , where , are connected by an edge in case with the rule if and only if . The fundamental spectral properties of graph , such as the diameter, girth, maximum and minimum degree, chromatic number, clique number, degree sequence, irregularity index, and dominating number, are presented in [7]. Furthermore, in [8], first and second Zagreb indices, Randić index, geometric-arithmetic index, and atom-bond connectivity index, Wiener index, Harary index, first and second Zagreb eccentricity indices, eccentric connectivity index, and the degree distance which emphasize the importance of graph have been studied.
It is known that the line graph of is a graph whose vertices are the edges of , and any two vertices are incident if and only if they have a common end vertex in . Although line graphs were firstly introduced by the papers [9, 10], the details of these studies started by Harary [11] and after that by Harary (Chapter 8 in [12]). In fact the line graph is an active topic of research studies at this moment. For example, some topological indices of line graphs have been considered in [13, 14].
In Section 2, we will mainly deal with the special parameters, namely, radius, diameter, girth, maximum and minimum degree, domination number, and chromatic and clique numbers over the line graph of monogenic semigroup graph associated with as given in (1). We note that [15] will be followed for unexplained terminology and notation in this paper.
2. Main Results
In this part, our aim is to reach previously mentioned goals. At this point, we remind that, for any simple graph , the graph properties such as radius, diameter, and girth are obtained by calculating the distance between any two vertices or the total number of whole vertices. So, our proofs will be based on this idea.
The eccentricity of a vertex , denoted by , in a connected graph is the maximum distance between and any other vertex of (for a disconnected graph, all vertices are defined to have infinite eccentricity). It is clear that is equal to the maximum eccentricity among all vertices of . On the contrary, the minimum eccentricity is called the radius [16, 17] of and denoted by
Theorem 1. Let be a monogenic semigroup graph. Then, the radius of the line graph is given by
Proof. We can easily see that the result is true for by Figure 1. So, let us consider , and let us take into account any two vertices and from the graph . By the definition of , we have and , since .
If , then .
If , then since .
So, we obtain for all . Thus, is a 2-self-centered graph for .
Hence, the result is obtained.
It is known that the diameter of is defined by the set
Theorem 2. Let be a monogenic semigroup graph. Then, the diameter of is
Proof. The proof can be obtained in a similar way as in the proof of Theorem 1.
We recall that the girth of a graph is the length of a shortest cycle contained in . Moreover, the girth is defined to be infinity if does not contain any cycle.
Theorem 3. For a monogenic semigroup graph , the girth of line graph is 3.
Proof. Since for , we have . So, we assume that which implies the set is a complete subgraph of . Hence, , as required.
The maximum degree of is the number of the largest degree in , and the minimum degree of is the number of the smallest degree in (see, for instance, [15]). According to these reminders, we can state and prove the following theorem in terms of the line graph.
Theorem 4. Let be a monogenic semigroup graph. Then, .
Proof. We know that for . From definition, the monogenic semigroup graph , we have . Therefore, the vertex of maximum degree in must be the vertex . As a result, we haveas required.
Theorem 5. Let be a monogenic semigroup graph. Then, .
Proof. By definition of the degree sequences of monogenic semigroup graphs in the studies [7, 8], we obtain that the set vertices with minimum degree areTherefore, we haveHence, we get the proof.
A subset of the vertex set of any graph is called a dominating set if every vertex is joined to at least one vertex of by an edge. Additionally, the domination number is the number of vertices in a smallest dominating set for (we may refer [15] for the fundamentals of domination number).
Theorem 6. Let be a monogenic semigroup graph. The domination number of is
Proof. Let us consider the set , where or . In fact, is the domination set in . Case 1: suppose : Case 2: now, suppose :The above steps complete the proof.
Basically, the coloring of any graph is to be an assignment of colors (elements of some set) to the vertices of , one color to each vertex, so that adjacent vertices are assigned distinct colors. If different colors are used, then the coloring is referred to as an - coloring. If there exists an -coloring of , then is called -colorable. The minimum number for which is -colorable is called the chromatic number of and is denoted by .
In addition, there exists another graph parameter, namely, the clique of a graph . In fact, depending on the vertices, each of the maximal complete subgraphs of is called a clique. Moreover, the largest number of vertices in any clique of is called the clique number and denoted by . In general, by [15], it is well known that for any graph . For every induced subgraph of , if holds, then is called a perfect graph [18].
Theorem 7. Let be a monogenic semigroup graph. Then, .
Proof. Due to the definition of , if the vertex is adjacent to the vertex , then . Therefore, the vertex sets of complete graphs in must be the form ofSo, since the vertex with maximum degree in is , the maximum complete subgraph in is the subgraph with vertex set , wheresuch that the number of elements in it is . Thus, , as required.
Theorem 8. For a monogenic semigroup graph , the chromatic number of line graph of is determined by .
Proof. Since the set is the complete subgraph of , we must paint each vertex in this set with a different color. That means we need to use colors for this set . The graph has at least one vertex that is not adjacent to for all of the vertices in , where . Therefore, we can use one of the colors which used in the set for the vertex . Thus, , as required.
Example 1. The graph is given in Figure 2.
Let us consider the semigroup
Now, by considering the graph as drawn in Figure 2, we can list the following results as example:(i) (obtained by Theorem 1)(ii) (obtained by Theorem 2)(iii) (obtained by Theorem 3)(iv) (obtained by Theorem 4)(v) (obtained by Theorem 5)(vi) (obtained by Theorem 6)(vii) (obtained by Theorem 7)(viii) (obtained by Theorem 8)
3. Conclusions
The aim of the study is to investigate the concept of monogenic semigroup graphs , which is firstly introduced by Das et al. [7], based on zero-divisor graphs. We examine the some graph properties over the line graph of . The existences of graph parameters, namely, radius, diameter, girth, maximum degree, minimum degree, chromatic number, clique number, and domination number over are proved, respectively.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.