An Inclusive Distance Irregularity Strength of n -ary Tree

ABSTRACT


A. INTRODUCTION
Graph labeling is one of the most popular research areas of graph theory.Graph labeling was first introduced in the mid1960s (Rosa, 1967).In more than 60 years nearly 200 graph labeling techniques have been studied in over 3200 papers (Gallian, 2022).In general, there are two kind of labeling, namely regular and irregular labelings.A regular labeling of graphs means that the labeling satisfying injective function, such as graceful (Wang et al., 2015), harmonious (Lasim et al., 2022), elegant (Elumalai & Sethuraman, 2010), felicitous (Manickam et al., 2012), and so on.In this paper, we focus on irregular labeling.All graphs discussed here are simple, undirected, and finite.For a general terminology of graph-theoretic, we follow (Chartrand & Zhang, 2012) and (Ringel & Hartsfield, 1990).
The distance vertex irregular labeling was introduced by (Slamin, 2017).This labeling was inspired by (Miller et al., 2003) who introduced a distance magic labeling and (Chartrand et al., 1988) who introduced an irregular assignment.Detail survey of distance magic labeling can be studied in (Arumugam et al., 2011).Furthermore, (Bong et al., 2017) generalized the concept of labeling introduced by (Slamin, 2017) to inclusive and non-inclusive vertex irregular ddistance vertex labeling, for any distance d up to the diameter.(Bong et al., 2017) defined the inclusive vertex irregular d-distance vertex labeling.For  = 1, (Bača et al., 2018) called this labeling as an inclusive distance vertex irregular labeling.
An inclusive distance vertex irregular labeling of a graph  is a function : () ⟶ {1, 2, … , } such that each vertex of  has distinct weight.The weight of a vertex  ∈ () under the labeling  is defined as () = () + ∑ ().

𝑢𝑣∈𝐸(𝐺)
Not all graphs can be applied this labeling, namely graph which contains at least two vertices with the same closed neighborhood.For example, a complete graph with three vertices, since all vertices will have the same closed neighborhood.It is not difficult to apply this labeling for a graph.That is why, the problem of this area is find the minimum  for the vertex label such that a graph  admits this labeling is called the inclusive distance vertex irregularity strength of  and is denoted by dis ̂().If such  does not exist we say that dis ̂() = ∞.
The exact value of the inclusive distance vertex irregularity strength of many graphs have been widely studied in the literature (see Bača et al., 2018;Utami et al., 2018;Bong et al., 2020;Halikin et al., 2020;Utami et al., 2020;Susanto et al., 2021;Majid et al., 2023;and Windartini et al., 2014).Furthermore, Cichacz et al. (2021) gave the upper bound of the inclusive distance vertex irregularity strength for a simple graph G on n vertices in which no two vertices have the same closed neighborhood that dis ̂() ≤  2 .Next, Susanto et al. (2022) studied inclusive distance vertex irregularity strength for the join product of graphs.Santoso et al. ( 2022) using genetic algorithm for the inclusive labeling of a graph.Bong et al. (2020) gave a lower bound for the inclusive distance vertex irregularity strength for a graph G of order n, the maximum degree ∆, and the minimum degree δ, Susanto et al. (2021) developed a new lower bound for the inclusive distance vertex irregularity strength of graphs that generalizes the lower bound by (Bong et al., 2020), that a graph G with the maximum degree ∆, the minimum degree δ, where   is the number of vertices of degree r in G for every  ≤  ≤ Δ.In this paper, we give the exact value of inclusive distance vertex irregularity strengths of a complete -ary tree to level two.Denoted by  ,2 , a complete n-ary tree is a rooted tree such that each vertex of degree greater than one has exactly n children and all degree-one vertices are of equal distance (height) to the root (Li et al., 2010).Therefore,  ,2 has ( + 1) vertices and  2 leaves.

B. METHODS
Let  ,2 be a complete n-ary tree.To find the minimum  for the vertex label such that a graph  admits the inclusive distance vertex irregular labeling, are as follows:

C. RESULT AND DISCUSSION
In this section, we discuss the exact value of inclusive distance irregularity strength of a graph -ary tree to level 2. There are two lemmas about the upper bound of  ̂( ,2 ).Indeed the lower bound of  ̂( ,2 ) following the Theorem 1. Lemma 1 Let  ≥ 3 be an odd integer and and  ,2 be an -ary tree to level 2. The upper bound of inclusive distance irregularity strength of  ,2 is  ̂( ,2 ) ≤ ⌈  2 + 1 2 ⌉.
Proof.For prove this by labeling a graph  ,2 using inclusive vertex irregular.For  = 3, an inclusive distance vertex irregular labeling of  3,2 can be seen in Figure 2, where the weight of the vertex shown by a red number.for  = 0, , 1 for  = 1, ( +1 2 ) for  = 2, 3, … ,  − 1. (2) For (3) The illustration of vertex labeling can be seen in Figure 3.We next show that all vertices weight are distinct.For  = 2, 3, … ,  − 1 , we have . We can check easily that ( 1 ) < (  ) < (  ) and also ( 1 ) < ( 0 ) < (  ).Now, we assume that ( 0 ) = (  ), for  = 2, 3, … ,  − 1.Then So,  is not integer, a contradiction.We consider now, the weight of the vertex    .We can see that 2 ≤ ( 1  ) ≤  + 1 <  + 2 ≤ ( 2  ) ≤  2 + 1 <  Lemma 2 Let  ≥ 2 be an even integer and and  ,2 be a complete -ary tree to level 2. The upper bound of inclusive distance irregularity strength of  ,2 is Proof.Let  be an even integer.For  = 2, an inclusive vertex irregular labeling of  2,2 can be seen in Figure 5, where the weight of the vertex is shown by a red number.(5) As an illustration of the formulation of vertex labeling in (4) and ( 5), consider the Figure 6.  4) and ( 5) According to the vertex labeling in (4) and ( 5), we can count the vertex weight, as follow.
Proof.According to Inequality (1) we have The exact value of an inclusive distance irregularity strength of  ,2 given in Theorem 3 attains the greatest lower bound provided by (Bong et al., 2020).

D. CONCLUSION AND SUGGESTIONS
In this research, the exact value of an inclusive distance vertex irregularity strengths of a complete -ary tree up to level two  ,2 is obtained, namely the ceiling of the leaves number of  ,2 plus one divided by two.The conclusion answers the problems of an inclusive distance irregularity strength on a few rooted tree classes.A challenging problem remains: getting the inclusive distance vertex irregularity strength of the complete n-ary tree at a level greater than two and of the not complete n-ary tree at a level greater than one.

Figure 7 .
Figure 7.An inclusive distance vertex irregular labeling of graph  4,2 upper bound of the inclusive distance irregularity strength of  ,2 has been proved by Lemmas 1 and 2. Therefore, we obtain  ̂( ,2