A new graph based on the semi-direct product of some monoids
© Karpuz et al.; licensee Springer 2013
Received: 28 January 2013
Accepted: 1 March 2013
Published: 20 March 2013
In this paper, firstly, we define a new graph based on the semi-direct product of a free abelian monoid of rank n by a finite cyclic monoid, and then discuss some graph properties on this new graph, namely diameter, maximum and minimum degrees, girth, degree sequence and irregularity index, domination number, chromatic number, clique number of . Since graph theoretical studies (including such above graph parameters) consist of some fixed point techniques, they have been applied in fields such as chemistry (in the meaning of atoms, molecules, energy etc.) and engineering (in the meaning of signal processing etc.), game theory and physics.
MSC:05C10, 05C12, 05C25, 20E22, 20M05.
1 Introduction and preliminaries
In this paper, we mainly investigate the interplay between the semi-direct product over monoids and the graph-theoretic properties of the semi-direct product in terms of its relations. In detailed, let us consider a free abelian monoid of rank n and also consider a finite cyclic monoid C. Then, by , we can define the semi-direct product of by C. Moreover, one can also define a new graph associated with this semi-direct product (see Section 1.1 below). Thus the idea in here is to present the interplay between the algebraic semigroup and graph-theoretic properties of this new graph. In fact, by the graph-theoretic properties, we will be interested in the diameter, maximum and minimum degrees, girth, chromatic number, clique number, domination number, degree sequence and irregularity index of the corresponding new graph. In the literature, there are some important graph varieties and works that are related to algebraic and topological structures, namely, Cayley graphs [2–4] or zero-divisor graphs [5–7]. But the graph constructed in here is different from those in the previous studies and is also interesting in terms of using algebraic semi-direct products during the construction of the vertex and edge sets. So, this kind of graph not only provides the classification of algebras (monoids, semigroups), but also solves the problems of normal forms of elements, word problem, rewriting system, embedding theorems, extensions of groups and algebras, growth function, Hilbert series, etc. As is well known, these problems are really important in fixed point results since they have a direct connection to nature sciences.
is a standard monoid presentation for the semi-direct product M. We may refer to [9, 10] for more detailed knowledge about the definition and a standard presentation for the semi-direct product of two monoids.
1.1 A new graph based on semi-direct products
In the following, we define an undirected graph associated with given in (1). Actually, all the results presented in this paper are based on this graph.
The vertex set V consists of the following:
all generators () and x of ,
words of the form () in the presentation (1),
words of the form () in (1). (Here, we omitted the words of the remaining format. In other words, we do not take , , etc. as a vertex in our set),
word of the form if in the presentation (1). Otherwise, i.e., if then we have a relator ; in that case, since x is in the vertex set, there is no need to take as a vertex.
The edge E consists of the following:
connect each vertex to single x for all ,
connect each of the adjacent vertices and for all ,
connect each vertex to the related vertices for all ,
connect each of the vertices and to the vertex from both sides for all ,
connect the unique vertex x to each vertex of the form for all .
In the construction of the semi-direct product, we assume that all rows of the matrix ℳ are different from each other. This affects our matching in the graph as all vertices ’s are distinct.
- (c)As seen in Figure 1, the number of vertex and edge sets depends on the number of generators of the free abelian monoid of rank n. Thus we have
Thus we obtain the graph as drawn in Figure 1.
2 Graph theoretical results over
In this section, by considering the graph drawn in Figure 1, we mainly deal with some graph properties, namely diameter, maximum and minimum degrees, girth, degree sequence, irregularity index, domination number, chromatic number and clique number of .
We thus get the following result.
Theorem 1 The diameter of the graph is 3.
Proof By Figure 1, it is clearly seen that the vertex (if then it exists in the graph) of is pendant and so the diameter can be figured out by considering the distance , where y is one of the other vertices. If in the presentation (1), then the vertex x is pendant of the graph . By Figure 1, we also see that the vertex x is connected with all the vertices except the vertices of the form () in the vertex set. Thus we can reach these vertices by only one edge from the vertices (). So, we get . □
The degree of a vertex v of Γ is the number of vertices adjacent to v. Among all degrees, the maximum degree (or the minimum degree ) of Γ is the number of the largest (or the smallest) degrees in Γ (see ). In our graph, maximum and minimum degrees are obtained as follows.
Proof By Figure 1, it is seen that the vertex x is connected with all vertices of the form (), () and the vertex if in the presentation (1). So, . In the case in the presentation (1), since there is no vertex labeled by , we get .
On the other hand, if in the presentation (1), then there is only one edge from the vertex to the vertex x. Thus . Otherwise, since the vertices which are of the form () are connected to the vertices and , we get , as required. □
It is known that the girth of a simple graph Γ is the length of the shortest cycle contained in Γ. However, if the graph does not contain any cycle, then the girth of it is assumed to be infinity. Hence the other result of this section is the following.
Theorem 3 The girth of the graph is 3.
Proof By the edge definition of , the vertex x is connected to the vertices of the form () and (). There also exists an edge between vertices and . So, the length of the shortest cycle contained in the graph is 3. □
There also exists the term degree sequence , which is a sequence of degrees of vertices of the graph Γ. Recently, in , a new parameter for graphs, namely the irregularity index of Γ, has been defined and denoted by . In fact is the number of distinct terms in the set .
Proof Let us consider the case in the presentation (1). In this case, since the vertex is connected with only the vertex x, then we clearly obtain that the degree of is 1. Now we consider the vertices of the form () and (). Since these vertices are connected with the vertices , () and (), x, the degree of them is 2. Thus we have vertices which have degree 2. By Figure 1, we see that the vertices and are connected to , , , x and , , , x, respectively. So, the degree of and is 4. The remaining vertices () are connected to the vertices , , , , and x. So, the degree of them is 6. Since the reminded vertex x is connected with , () and the degree of it is . Now, let us consider the case in the presentation (1). In this case, since the vertex does not exist, we get the degree sequence as depicted in the theorem.
For , we need to consider the number n in the presentation (1). The number of distinct terms in the set depends on the number n. By considering the number n, we can easily obtain the irregularity index , as required. □
A subset D of the vertex set of a graph Γ is called a dominating set if every vertex is joined to at least one vertex of D by an edge. Additionally, the domination number is the number of vertices in the smallest dominating set for the graph Γ (see ).
Proof By considering Figure 1, the vertex x is adjacent to all the vertices except the vertex of the form (). Thus, in the domination set, there must be some vertices of the form adjacent to the vertices of the form . These vertices depend on the number of n in the presentation (1). If n is even, then there are vertices () in the smallest dominating set for . Otherwise, there are vertices. Hence the result. □
Basically, the coloring of a graph Γ is to be an assignment of colors (elements of some set) to the vertices of Γ, one color to each vertex, so that adjacent vertices are assigned distinct colors. If n colors are used, then the coloring is referred to as n-coloring. If there exists an n-coloring of Γ, then Γ is called n-colorable. The minimum number n for which Γ is n-colorable is called the chromatic number of Γ and is denoted by . There exists another graph parameter, namely the clique of a graph Γ. In fact, depending on the vertices, each of the maximal complete subgraphs of Γ is called a clique. Moreover, the largest number of vertices in any clique of Γ is called the clique number and is denoted by . In general, it is well known that for any graph Γ (for instance ).
Theorem 6 The chromatic number is equal to 3.
Proof Let us consider the vertex x in the graph drawn in Figure 1. It is easy to see that x is adjacent to all the other vertices except the vertices of the form (). That means the color used for the vertex x can be used for the vertices . Thus let us suppose that the color for x and is labeled by . Next, let us consider the vertices () in Figure 1. Since these vertices are connected with each other doubly, we have two different colors labeled by and . In other words, if we label the vertex by , then we label the vertex by , by and so on. Now we take account of vertices . Since these vertices are adjacent to the vertices , respectively, they can be labeled by and , respectively. Since the remaining vertex is just connected with the vertex x, it can be labeled by . Hence the result. □
We note that the chromatic number of does not depend on the number of generators of the free abelian monoid of rank n.
Theorem 7 The clique number is equal to 3.
Proof By Figure 1, we have three types of maximal complete subgraphs of . These types consist of the following edges which have three vertices: , and . Hence the result. □
By considering the presentation in (2) and Figure 2, we have the following:
and so .
and so .
and so .
Dedicated to Professor Hari M Srivastava.
The third and fourth authors are partially supported by Research Project Offices (BAP) of Selcuk (with Project Number 13701071) and Uludag (with Project Numbers 2012-12 and 2012-19) Universities, respectively. The second author is supported by the Faculty Research Fund, Sungkyunkwan University, 2012 and Sungkyunkwan University BK21 Project, BK21 Math Modelling HRD Div. Sungkyunkwan University, Suwon, Republic of Korea.
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