All graphs in this paper are considered to be finite, simple, and undirected. The concept of magic labeling based on distance was separately introduced by Vilfred [1] in his doctoral thesis in 1994 and Miller et al. [2] in 2003 with the following definition.

Definition 1.1.Let be a graph of order n. A bijection is called a distance magic labeling if there exists a positive integer k such that , for every . The constant k is referred to as the magic constant of the labeling f. The sum is called the weight of vertex v under f, and is denoted as w(v). If a graph G admits a distance magic labeling, then G is called a distance magic graph, or G is distance magic.

Research on distance magic labeling has been extensively conducted for several families of graphs. Furthermore, studies on distance-based labeling has expanded into several variants, one example of which is graph labeling considering that all weights must be distinct, with the vertex weight defined as the sum of labels of all its closed neighbors. This notion was termed as inclusive distance antimagic labeling, introduced by Dafik et al. [3] with the definition as follows:

Definition 1.2.Let be a graph of order n. A bijection is called an inclusive distance antimagic labeling if any two distinct vertices , where . The set is the closed neighborhod of vertex and is defined as . If the graph G admits such a labeling, then G is said to be an inclusive distance antimagic graph, or G is inclusive distance antimagic.

Observation 1.3.If a graph G is inclusive distance antimagic, then for any two distinct vertices u and v, we have

Conjecture 1.4.A graph G is inclusive distance antimagic if and only does not have two vertices with the same closed neighborhood.

From the notion of inclusive distance antimagic labeling, they investigate the existence of such a labeling for various simple graphs.

Theorem 1.5.[3]The Complete graph is not inclusive distance antimagic.

Theorem 1.6.[3]The Path graph is inclusive distance antimagic, for n

Theorem 1.7.[3]The Cycle graph is inclusive distance antimagic, for n

The term distance-based labeling was generalized by O’Neal and Slater [4, 5] for distance magic labeling, and by Simanjuntak and Wijaya [6] for distance antimagic labeling, by defining the notation for the labeling weight as , where and with denotes the diameter of graph G. If all vertices have the same weight, we call the labeling a D−distance magic labeling, whereas if all vertices have distinct weights, we call the labeling a D−distance antimagic labeling.

If , a D-distance magic labeling is a distance magic labeling and D-distance antimagic labeling is distance antimagic labeling. For , a D-distance antimagic labeling is an inclusive distance antimagic labeling. Ngurah [7] proved that for , if a graph G has a D−distance magic labeling, then G also has a )−distance antimagic labeling. For the case , then this statement can be rewritten in the theorem as follows:

Theorem 1.8.[7]If G is a distance magic graph, then G is inclusive distance antimagic.

Extensive research has been conducted on the existence of distance magic labeling in certain graphs. Miller et al. [2] provided several simple observations for certain graphs that have distance magic labeling, such as path graphs and , cycle graph complete graph , and wheel graph (Wheel graph isomorphic with Additionally, Cichacz and Froncek [8] proved that the circulant graph is distance magic, followed by the circulant graph , for p even, is also distance magic. According to Theorem 1.8, these graphs are also inclusive distance antimagic.

The following corollary represents the contrapositive of Theorem 1.8:

Corollary 1.9. If graph G is not inclusive distance antimagic, then G is not distance magic.

Corollary 1.10.If graph G has pairs of vertices with the same closed neighborhood, then G is not distance magic.

In this paper, we will present the existence of inclusive distance antimagic labeling for circulant graphs, disjoint union graphs, and join graphs. This paper also discusses some examples of graph that have pairs of vertices with the same closed neighborhood. According to Corollary 1.10, these graphs are not distance magic.

As stated in Theorem 1.5, complete graph is not inclusive distance antimagic, mainly because all vertices in has the same closed neighborhood set, which is itself. Since is a regular graph, then Dafik et al. [3] proposed a conjecture regarding the inclusive distance antimagic property in regular graphs.

Conjecture 2.1. Every r-regular graph except complete graph is inclusive distance antimagic.

In this section, several examples of regular graphs will be given that have at least two vertices with the same closed neighborhood set. According to Observation 1.3, these graphs are not inclusive distance antimagic. This also serves as a counterexample to Conjecture 2.1. Circulant graphs are regular graphs where some of them have at least one pair of vertices with the same closed neighborhood set.

Definition 2.2.Let , where n and are positive integers. The Circulant Graph is a regular graph of order n with vertex set and edg set mod and . The numbers are called the generators of the circulant graph.

For example, circulant graph is a graph with vertex set ， and edge set , as shown in Figure 1.

Theorem 2.3.Let are positive integers such that n. Then circulant graph has pairs of vertices with the same closed neighborhood set, which means the graph is not inclusive distance antimagic.

Proof. Suppose graph with vertex set Consider two vertices and . Since , the closed neighborhood of is , and the closed neighborhood of is . However, since it follows that and Therefore, and have the same closed neighborhood set.

Figure 2 below shows the circulant graph , which, according to Theorem 2.3 is not inclusive distance antimagic since v0 and have the same closed neighborhood set

Here we present several corollaries from Theorem 2.3.

Corollary 2.4.Let k is an even number, and are natural numbers with r every , then circulant graph is not inclusive distance antimagic.

Corollary 2.5.Let k and n are even number, and are natural numbers with every , then circulant graph is not inclusive distance antimagic.

Theorem 2.6.Suppose the set forms a complete subgraph in graph If for every pair of vertices it holds that , then graph G is not inclusive distance antimagic.

Proof. For two vertices and Since , it follows that .

Let 1 \end{document} ]]> be positive integers such that . The circulant graph with vertex set and all of its generators less than or equal to , is isomorphic to , that is p copies of the complete graph . For a positive integer , by adding generators , circulant graph is an example of the graph that satisfies Theorem 2.6, with one of its sets

Corollary 2.7.Let 1 \end{document} ]]> be positive integers such that . For a positive integer , circulant graph is not inclusive distance antimagic.

In this section, we provide the definition of disjoint union of two or more graphs, then we explore the properties of inclusive distance antimagic labeling of the disjoint union of graphs.

Definition 3.1.The disjoint union between two graphs G and H, denoted as is a disconnected graph with its components are G and H. It means that and . The disjoint union of more than two graphs, , is denoted as . If each graph is isomorphic to graph G, then the disjoint union is denoted as nG.

Theorem 3.2. Let G be an inclusive distance antimagic graph, and H be an h−regular inclusive distance antimagic graph. If , then is also an inclusive distance antimagic graph.

Proof. Let and be the inclusive distance antimagic labeling for G and respectively. Let , and define the labeling f ∪ for as follows:

Let , and be the weight of vertices based on the labeling , and , respectively. Since H is an h−regular graph, then the weight of vertices in graph are:

For each vertex v in graph , it holds that . Since 2 therefore for every vertex v in G and y in H

Hence, under the labeling , every vertex in graph has distinct weight. Thus, is an inclusive distance antimagic graph. □

Theorem 3.3.Let be regular inclusive distance antimagic graphs. Then is an inclusive distance antimagic graph.

Proof. Let be the degree of regular graph for . Without loss of generality, assume . We will use mathematical induction to prove the statement that is inclusive distance antimagic for all k. For , the statement is true since is inclusive distance antimagic. Now assume that the statement holds true for is inclusive distance antimagic. We want to prove that is also inclusive distance antimagic. Note that since , then . By Theorem 3.2, is also inclusive distance antimagic. □

Since , then based on the results by Dafik et al. [3] in Theorems 1.6 and 1.7, and using Theorems 3.2 and 3.3, we can derive the following corollaries:

Corollary 3.4.Graph is inclusive distance antimagic, for m and n

Corollary 3.5.Graph is inclusive distance antimagic, for

In this section, we provide a definition of the join of graphs, followed by presenting several theorems related to the existence of inclusive distance antimagic property of join of graphs.

Definition 4.1.The join between two graphs G and H, denoted by , is a graph with vertex set and edge set as follows:

Dafik et al. [3] provide an example of join graphs that are not inclusive distance antimagic, such as , and . The following theorem summarizes those results more generally.

Theorem 4.2.Let H be any graph, and be a graph that has pairs of vertices with the same closed neighborhood set (meaning graph G is not inclusive distance antimagic). Then is not an inclusive distance antimagic graph.

Proof. If there are two vertices in say u and have same closed neighborhood set in , then u and v also have same closed neighborhood set in . This is because in , every vertex in graph G is connected to every vertex in graph , including u and v.

Corollary 4.3.If any of the graphs has pairs of vertices with the same closed neighborhood set, then is not an inclusive distance antimagic graph.

Theorem 4.4.Let graph of order n be an inclusive distance antimagic graph. Then the graph is an inclusive distance antimagic graph if and only if .

Proof. If , then in graph there exist at least two vertices of degree n. Since graph has order , then these vertices of degree n will have same close neighborhood set. For , let and be the inclusive distance antimagic labeling of G. Let denote the weight of vertex under the labeling Now, let u be the vertex of . Define a bijection for such that for every and . Then, the weights based on bijection are 2 for , and . Since there are no vertices in graph G of degree , then in only vertex u has degree n. Consequently, max , ensuring that each vertex in graph has distinct weight under the bijection .

Aside from paths and cycles, Dafik et al. [3] also provide examples of graphs that are inclusive distance antimagic, such as star , star , broom , and wheel . Among all those graphs, only the wheel graph that satisfies , while the others do not. Hence based on Theorem 4.4, ， and are an inclusive distance antimagic graphs, whereas is not.

Theorem 4.5.Let graph H of order m be an inclusive distance antimagic graph. For , graph is also an inclusive distance antimagic graph.

Proof. Let be the inclusive distance antimagic labeling for , with be the weight of vertices under . Define a bijection such that ， for every , and each vertex in can be labeled arbitrarily within the range . Then, the weight of vertices in graph are:

Obviously, , for every , and 0 \end{document} ]]> , for every . Since , then . Thus, for every , it holds that w _ { H + \overline { { K _ { n } } } } ( v ) \end{document} ]]> . Moreover, it can be observed that for every vertex in and every vertex in , has distinct weights. Therefore, is inclusive distance antimagic. □

There are several examples of graph such as the fan graph which is the join graph and the cone graph which is the join graph . According to Theorem 4.5, the fan graph for , and the cone graph for , are inclusive distance antimagic.