On Total Vertex Irregularity Strength of Hexagonal Cluster Graphs

Abstract

For a simple graph with a vertex set and an edge set , a labeling is called a vertex irregular total of if for any two different vertices and in we have where The smallest positive integer such that has a vertex irregular total is called the total vertex irregularity strength of , denoted by . The lower bound of for any graph have been found by Baca et. al. In this paper, we determined the exact value of the total vertex irregularity strength of the hexagonal cluster graph on cluster for . Moreover, we show that the total vertex irregularity strength of the hexagonal cluster graph on cluster is .

1. Introduction

A graph labeling is an assignment of integers from 1 to , for the vertices, edges, or both. Graph labelings have been used in many applications like communication network addressing, software testing, information security, technology and sports tournament scheduling, and coding theory problems including the design of good radar location codes, missile guidance codes, and convolution codes.

We consider the finite undirected graph without loops and multiple edges with vertex set and edge set . The degree of a vertex is the number of edges that have as an endpoint, and the set of neighbors of is denoted by . If the domain of the labeling function is the vertex set or the edge set, the labeling is called, respectively, vertex labeling or edge labeling. If the domain is , then we call the labeling a total labeling.

A labeling is called an edge of . The associated vertex weight of a vertex under an edge is defined as . Chartrand et al. [1] introduced an edge of a graph such that for all vertices with Such labelings were called irregular assignments, and the irregularity strength of a graph is known as the minimum for which has an irregular assignment using labels at most . The irregularity strength can be interpreted as the smallest integer for which can be turned into a multigraph by replacing each edge by a set of at most parallel edges, such that the degrees of the vertices in are all different.

In this paper, we consider for a total of , that is, . The associated vertex weight of a vertex under a total is defined as . A total is defined to be a vertex irregular total of if for every two different vertices and of , . The minimum positive integer for which has a vertex irregular total is called the total vertex irregularity strength of , denoted by . In Figure 1, there are a two total labeling of , one of is a vertex irregular total 3-labeling of and the other is a vertex irregular total 2-labeling of . However, does not have a vertex irregular total vertex 1-labeling. Such that the minimum positive integer for which has a vertex irregular total is 2, or the total vertex irregularity strength of is 2.

(a)

(b)

Figure 1 

A vertex irregular total k-labeling of for and .

Baca et al. [2] in 2007 started to investigate the total vertex irregularity strength of a graph, an invariant analogous to the irregularity strength for total labelings. There are not many graphs for which the exact values of their total vertex irregularity strength are known. Baca et al. [2] have determined the total vertex irregularity strengths for some classes of graphs, namely, cycles, stars, and prisms. Nurdin et al. have determined the total vertex irregularity strengths of a disjoint union of copies of a path [3], tree graphs [4], and caterpillar graph [5]. Nurdin and Kim have determined the total vertex irregularity strength of splitting graphs of stars [6].

In this paper, we determine exact value of the total vertex irregularity strength of the hexagonal cluster graph with cluster for .

2. Hexagonal Cluster Graph

In this section, we give the definition of hexagonal cluster graphs. The hexagonal cluster graph with cluster, denoted by , where isomorphs to , is obtained by adding as many as cycle on the outer path of . Figure 2 demonstrates hexagonal cluster graphs , , and .

Figure 2 

Hexagonal cluster graphs , , and .

Some interconnection networks are designed, and some are borrowed from nature. For example, hypercubes, complete binary trees, butterflies, and torus networks are some of the designed architectures. Grids, hexagonal networks, honeycomb networks, and diamond networks, for instance, bear resemblance to atomic or molecular lattice structures. They are called natural architectures. The advancement of large scale integrated circuit technology has enabled the construction of complex interconnection networks. Besides that, the hexagonal cluster graphs have been studied as models of organic compounds build up entirely from benzene rings, social networks, and wireless sensor networks. Graph theory provides a fundamental tool for designing and analyzing such networks [7–11].

In [2], Baca et al. studied the lower bound of for any graph as follows.

Theorem 1. If is the number of vertices of any graph , is the minimum degree of vertex, and is the maximum degree of vertex of , then

3. Results and Discussion

In this paper, we have proved that the total vertex irregularity strength of the hexagonal cluster graph (network) with cluster is equal to its lower bound in Theorem 1.

Theorem 2. For , we have

Proof. Since the number of vertices of is , the minimum degree of vertex is 2 and the maximum degree of vertex is 3; by using (1) in Theorem 1, we found thatTo find that , we have to construct a vertex irregular total on where as follows.
Note that there are layers in . Layer is denoted by . For illustration, in Figure 3, there are 3 layers in .
In , there is an outer cycle of which consists of vertices of degree 2 denoted by vertices of degree 3 denoted by , and edges denoted by
Now, consider all vertices and all edges in the outer cycle of sequentially.
To label some of vertices and all edges in we use Algorithm 1 as follows.
To label all of edges in for and for defineand use Algorithm 2 as follows.
All of edges of are already labeled, but vertices have not labeled yet. Next, label all of vertices using the following method. The temporary weight of a vertex is the number of label of all edges incident to, denoted by . For example, the temporary weight of a vertex in Figure 4 is 8.
Order the temporary weight of all vertices and rename them by such that For , definewhere is the total weight of vertex .
Based on equation (6), we have thatBesides that, from Algorithms 1 and 2, we can see thatBased on Statements (7) and (8), we conclude that the function construction with Algorithms 1 and 2 is a total vertex irregular on where . This shown as follows:Based on equations (3) and (9), we find equation (2), i.e.,
As illustration, we shall use Algorithms 1 and 2 to construct a total vertex irregular 25-labeling on . The first step, all the vertices and all the edges on the outer cyrcle of are named sequentially as in Figure 5.
Now, we use Algorithm 3 to label some of vertices and all edges in as follows.
For , using equations (4) and (5), we have and .
Then, use Algorithm 4 to label all edges in and , as follows.

Figure 3 

Hexagonal cluster graphs with their layers , and .

Step 1. For where , label , , and with where and are edges incident to vertex   Step 2. For where , label edges between and by except for the labeling edges in Step 1   Step 3. Label all edges between and by   Step 4. Label the remaining edges of by

Algorithm 1 

Algorithm for labeling all edges of the outer cycle of L1.

K2t−1=(S2t−1+2−K2t−2)/3,  Step A. Label all of edges of the outer cycle in for by where   Step B. Label all of the remaining edges from for by   Step C. Label all of the remaining edges in by

Algorithm 2 

Algorithm for labeling all other edges in HC(n).

Figure 4 

The temporary weight of is 8.

Figure 5 

The name of elements of the outer cycle of in .

Step 1. For where , label , , and with where and are edges incident to vertex (see Figure 6)   Step 2. For where , label edges between and by except for the labeling edges in Step 1   Step 3. Label all edges between and by except for the labeling edges in Step 1 (see Figure 7)   Step 4. Label the remaining edges of by 13 (see Figure 8)

Algorithm 3 

Algorithm for labeling all edges of the outer cycle of L1 of HC(4).

Figure 6 

The labels of some vertices and edges of in .

Figure 7 

The labels of some vertices and edges of in .

Figure 8 

The labels of some vertices and edges of in .

Step A. Label all edges of the outer cycle in by using Algorithm 2, i.e., where i.e., (see Figure 9) and all edges of the outer cycle in by (see Figure 10)   Step B. Label all of the remaining edges from by using Algorithm 2, i.e., (see Figure 11)   Step C. Label all of the remaining edges in by (see Figure 12)

Algorithm 4 

Algorithm for labeling all other edges in HC(4).

Figure 9 

The labels of all edges of the outer cycle of in .

Figure 10 

The labels of all edges of the outer cycle of in .

Figure 11 

Layer 2 with edge labels.

Figure 12 

Layer 4 with edge labels.

4. Conclusions

In this paper, we obtained the precise values for the total vertex irregularity strength of the hexagonal cluster graphs , for all Furthermore, we show that the hexagonal cluster graphs is an example that the lower bound in \cite{BJMR07} is sharp. In the future, we are interested in computing the exact value for the total vertex (or edge) irregularity strength of grids, hexagonal networks, and honeycombs networks.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The author(s) declare that they have no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research was supported by the Basic Science Research Program, National Research Foundation of Korea, Ministry of Education, (NRF-2018R1D1A1B07049584) and Basic Research Superior College, Directorate of Research and Community Service, Ministry of Research, Technology and Higher Education, Republic of Indonesia (007/SP2H/AMD/LT/DRPM/2020).