Abstract
A total -labeling is a function from the edge set to first natural number and a function from the vertex set to non negative even number up to , where . A vertex irregular reflexive-labeling of a simple, undirected, and finite graph is total -labeling, if for every two different vertices and of , , where . The minimum for graph which has a vertex irregular reflexive -labeling is called the reflexive vertex strength of the graph , denoted by . In this paper, we determined the exact value of the reflexive vertex strength of any graph with pendant vertex which is useful to analyse the reflexive vertex strength on sunlet graph, helm graph, subdivided star graph, and broom graph.
1. Introduction
We consider a simple and finite graph with vertex set and edge set . We motivate the readers to refer Chartrand et al. [1], for detailed definition of the graph. A topic in graph theory which has grown fast is the labeling of graphs. The concept of graph labeling, firstly, was introduced by Wallis in [2]. He defined a labeling of is a mapping that carries a set of graph elements into a set of integers called labels. By this definition, we can have a vertex label, edge label, or both of them. Baca et al. [3] introduced the total labeling, and they defined the vertex weight as the sum of all incident edge labels along with the label of the vertices. Many types of labeling have been studied by researchers, namely, graceful labeling, magic labeling, antimagic labeling, irregular labeling, and irregular reflexive labeling.
Furthermore, labeling known as a vertex irregular total -labeling and total vertex irregularity strength of graph is the minimum for which the graph has a vertex irregular total -labeling. The bounds for the total vertex irregularity strength are given in [3]. In [4], Tanna et al. identified the concept of vertex irregular reflexive labeling of graphs. In this paper, we continue to study a vertex irregular reflexive labeling as there are still many open problems. By irregular reflexive labeling, we mean a labeling of graph which the vertex labels are assigned by even numbers from and the edge labels are assigned by , where is positive integer. The weight of each vertex, under a total labeling, is determined by summing the incident edge labels and the label of the vertex itself.
A -labeling assigns numbers to the elements of graph. Let be a natural number, a function is called total -irregular labeling. Hinding et al. [5] defined that a total labeling is called vertex irregular total -labeling of graph if the vertex weight is distinct for every two different vertices, for . The minimum for which graph has a vertex irregular total -labeling is called total vertex irregularity strength, denoted by .
The concept of vertex irregular total -labeling extends to a vertex irregular reflexive total -labeling. The definition of total -labeling is a function from the edge set to the first natural number and a function from the vertex set to the nonnegative even number up to , where . A vertex irregular reflexive-labeling of the graph is the total -labeling, for every two different vertices and of , , where . The minimum for graph which has a vertex irregular reflexive -labeling is called the reflexive vertex strength of the graph , denoted by .
Some results related to vertex irregular reflexive labeling have been studied by several researchers. Tanna et al. [4] have studied the vertex irregular reflexive of prism and wheel graphs, Ahmad and Bac̆a [6] have studied the total vertex irregularity strength for two families of graphs, namely, Jahangir graphs and circulant graphs, and Agustin et al. [7] also study the concept of vertex irregular reflexive labeling of cycle graph and generalized friendship. Another results of irregular labeling can be seen on [8–15]. In this paper, we have found the lower bound of vertex irregular reflexive strength of any graph and determined the vertex irregular reflexive strength of graphs with pendant vertex. Our results are started by showing one lemma and theorem which describe a general construction of the existence of vertex irregular reflexive -labeling of graph with pendant vertex.
2. Result and Discussion
The following lemma and theorem will be used as a base construction of analysing the reflexive vertex strength of any graph with pendant vertex, namely, sunlet graph, helm graph, subdivided star graph, and broom graph.
Lemma 1. For any graph of order , the minimum degree , and the maximum degree ,
Proof. Let be a graph of order , the minimum degree , and the maximum degree . The total -labeling which labeling defined and such that if and if , where max. The total -labeling is called a vertex irregular reflexive -labeling of the graph if every two different vertices and , and it holds , where . Furthermore, since we require -minimum for the graph which has a vertex irregular reflexive labeling, the set of a vertex weight should be consecutive, otherwise it will not give a minimum . Thus, the set of a vertex weight is . Since the minimum is the reflexive vertex strength, the maximum possible vertex weight of graph is at most . It implies . Since should be integer and we need a sharpest lower bound, it implies
It completes the proof.
Theorem 1. Let be a graph of order and contains pendant vertex. If , then
Proof. Given that a graph of order is with pendant vertices. The labeling of graph is with respect to two components, namely, the pendant vertices and otherwise vertices. Thus, we will split our proof into two cases.
Case 1. Let be a set of pendant vertices and the number of is . A pendant vertex consists of two elements, i.e., a vertex and an edge. The vertex weight on each pendant vertex must be different. Suppose we choose those vertex weights are . Those vertex weights are obtained by summing the vertex and edge labels. To prove the above , let us suppose the maximum vertex weight of . LetDefine an injection as the labels. Since the weight is contributed by one vertex and one edge labels, it will give four possibilities.(1)If is odd number, then and , such that the vertex weight is (2)If is even number, then and , such that the vertex weight is (3)If is odd number, then and , such that the vertex weight is (4)If is even number, then and , such that the vertex weight is The vertex weight of point , and (4) are, respectively, and . It will give all weights are different, whereas, point (3) has a vertex weight of . Since the number of pendants is , we will have at least two vertices which have the same weight. Therefore, for is even and is odd, we need to add 1 for the largest vertex or edge labels. Thus, we will have a different weight for every pendant. Thus, the labels of vertex and edge of the pendant are the following.
From Table 1, it is easy to see that all vertex weights are different.
Case 2. The vertex set apart from pendant vertices must have a degree of at least two. The cardinality of is less than or equal to the cardinality of . It implies that the vertex or edge labels of pendant vertices can be re-used on labels of . Thus, the vertex weight of will be different with the vertices of since it has more combination, namely, .
Based on Case 1 and Case 2, the reflexive vertex strength of graph isIt concludes the proof.
Corollary 1. Let be a sunlet graph, and for every ,
Proof. Moreover, to determine the label of vertices and edge set , we will use (Algorithm 1)It concludes the proof.
For an illustration, see Figure 1.
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Theorem 2. Let be a helm graph, and for every,
Proof. Let be a helm graph with vertex set , and edge set . Helm graph has pendant vertices and one central vertex of degree . Since the central vertex has degree of much greater than the other vertices, it must have a different vertex weight than the others. Based on Theorem 1, we have the following lower bound:Furthermore, we will show the upper bound of vertex irregular reflexive -labeling by defining the injection and in the following:whereBased on the above injection, the overall vertex weight sets areIt is easy to see that the above elements of set are all different. It concludes the proof.
Theorem 3. be a subdivided star graph, and for every ,
Proof. Let be a subdivided star graph with vertex set , and edge set . The maximum degree of is . The graph has one central vertex of degree . Since the central vertex has degree of much greater than the other vertices, it must have a different vertex weight than the others. Based on Lemma 1, we have the following lower bound:For the illustration of the vertex irregular reflexive, -labeling of and can be depicted in Figure 2.
Furthermore, we will show the upper bound of vertex irregular reflexive -labeling by defining the injection and . For , we have the following:For , we have the following function for and :For otherwise , we havewhereBased on the above injection, the overall vertex weight sets of the subdivided star areIt is easy to see that the above elements of the set are all different. It concludes the proof.
Theorem 4. Let be a broom graph, and for every ,
Proof. Let be a broom graph with vertex set , , and edge set .
The Broom graph has pendant vertices and one central vertex of degree . Since the central vertex has a degree much greater than the other vertices, it must have a different vertex weight than the others. Based on Lemma 1, we have the following lower bound:for , and is odd. Since the vertices apart from vertex have degree of at most 2, the labels of are , and the label of edges, which are incident to , are . Thus, the vertex weight is . Furthermore, since the number of vertices of Broom graph is , there must be at least two vertices with the same vertex weight. Thus, we need to add 1 on the sharpest lower bound:Furthermore, we will show that is an upper bound of the reflexive vertex strength of Broom graph . LetDefine an injection and of the vertex irregular reflexive -labeling of Broom graph as follows:Based on the above injection, the overall vertex weight sets of for isMoreover, to determine label of vertices and , we will use Algorithm 2.
It is easy to see that the above elements of set and are all different. It concludes the proof.
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3. Concluding Remark
In this paper, we have studied the construction of the reflexive vertex -labeling of any graph with pendant vertex. We have determined a sharp lower bound of the reflexive vertex strength of any graph in Lemma 1, as well as obtained the exact value the reflexive vertex strength of any graph in Theorem 1. By this lemma and theorem, we finally determined the reflexive vertex -labeling of some families of graph with a pendant vertex. However, we need to find an upper bound of the reflexive vertex strength of any graph and study the reflexive vertex -labeling of other families of graph or some graph operations. Therefore, we propose the following open problems:(1)Determine an upper bound of reflexive vertex strength of any graph to find the gap between lower bound and upper bound, and continue to determine the exact values for reflexive vertex strength of any other special graphs(2)Determine the construction of the reflexive vertex -labeling of any regular graph, planar graph, or some graph operations
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors gratefully acknowledge from the support of CGANT Research Group—the University of Jember, Indonesia of year 2020.