ON THE IRREGULARITY STRENGTH AND MODULAR IRREGULARITY STRENGTH OF FRIENDSHIP GRAPHS AND ITS DISJOINT UNION

. For a simple, undirected graph 𝐺 with, at most one isolated vertex and no isolated edges, a labeling 𝑓:𝐸(𝐺) → {1,2,… ,𝑘 1 } of positive integers to the edges of 𝐺 is called irregular if the weights of each vertex of 𝐺 has a different value. The integer 𝑘 1 is then called the irregularity strength of 𝐺 . If the number of vertices in 𝐺 or the order of 𝐺 is |𝐺| , then the labeling 𝜇: 𝐸(𝐺) → {1,2, …, 𝑘 2 } is called modular irregular if the remainder of the weights of each vertex of 𝐺 divided by |𝐺| has a different value. The integer 𝑘 2 is then called the modular irregularity strength of 𝐺 . The disjoint union of two or more graphs, denoted by ‘+’, is an operation where the vertex and edge set of the result each be the disjoint union of the vertex and edge sets of the given graphs. This study discusses about the irregularity and modular irregularity strength of friendship graphs and some of its disjoint union, The result given is 𝑠(𝔽 𝑚 ) = 𝑚 + 1,𝑚𝑠(𝔽 𝑚 ) = 𝑚 + 1 and ms (𝑟𝔽 𝑚 ) = 𝑟𝑚 + ⌈ 𝑟 2 ⌉ , where 𝑟 denotes the


INTRODUCTION
Consider a simple, undirected graph = ( , ) with no loops and at most one isolated vertex [1] [2] [3]. A labeling of is a mapping that maps the elements of the graph to a set of numbers, commonly nonnegative integers or the set of natural numbers [4] [5]. A labeling of is a mapping that maps the elements of the graph to a set of numbers, commonly non-negative integers or the set of natural numbers [4] [5]. A labeling : ( ) → {1,2, … , } of positive integers to the edges of is called an irregular labelling if for every pair of vertices , ∈ , holds ( ) ≠ ( ), or in other words, the weights of each vertex of has a different value [6]. The smallest integer for which the labeling holds is then known as the irregularity strength of and is denoted as ( ) [7]. A labeling : ( ) → {1,2, … , } of positive integers to the edges of is called a modular irregular labelling if for every pair of vertices , ∈ , holds ( ) | | ≠ ( ) | |, or in other words, if the remainder of the weights of each vertex of divided by | | has a different value [6]. The smallest integer for which the labeling holds is then known as the modular irregularity strength of and is denoted as ( ) [7].
The disjoint union of graphs is an operation that combines two or more graphs to form a larger graph. It is analogous to the disjoint union of sets, and is constructed by making the vertex set of the result be the disjoint union of the vertex sets of the given graphs, and by making the edge set of the result be the disjoint union of the edge sets of the given graphs. Any disjoint union of two or more nonempty graphs is necessarily disconnected. The disjoint union is also called the graph sum, and is represented by a plus (+) sign: If 1 , 2 , … are graphs, then 1 + 2 + ⋯ + denotes their disjoint union [8].
A planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints [9]. In other words, it can be drawn in such a way that no edges cross each other [10].
The friendship graph is a planar undirected graph with 2 + 1 vertices and 3m edges [9]. The disjoint union of friendship graphs, denoted by , is defined by where is the friendship graph and is the number of friendship graph copies used in the disjoint union.
Finding the irregularity strength of a graph seems to be hard even for graphs with simple structure, see [11] and [12], Ahmad, et. al. in [13] and Baca,et. al. in [6] has discussed the irregularity strength by dividing the result into edge and vertex strengths. This study will provide a different and unified approach to determine the irregularity strength of the graph like the one discussed in [14].
The result will discuss about the irregularity strength of friendship graphs, denoted by ( ), and the modular irregularity strength of friendship graphs and some of its disjoint union, denoted by ( ) and ( ), respectively.

Research Type
The research described in this paper uses a literature study related to the modular irregularity strength on several types of graphs that have been studied by previous researchers.

Research Materials
The materials used in this research are scientific works, books, scientific journals, papers, and articles related to modular irregularity strength, friendship graphs and disjoint unions by mathematicians.

Research Method
The procedure used in this research are as follows. First is determining the research title, then studying about modular irregularity strength, followed by labeling the edges of friendship graphs, then calculating the weight of the vertices of the friendship graphs, next is determining the pattern of labeling results, then proving the labeling that has been obtained, followed by determining modular irregularity strength, and concluding the research

The general form of the friendship graph and its disjoint union
The general form of the friendship graph and its disjoint union used in this study is given as follows. The general form from Figure 1 is used while reviewing the irregularity strength and modular irregularity strength of friendship graphs, whereas the general form from Figure 2 is used while reviewing the modular irregularity strength of some of the disjoint union of friendship graphs

The irregularity strength of friendship graph
The irregularity strength of is discussed in the following theorem.

Theorem 1.
Let be a friendship graph with petals and 2 + 1 vertices, then for ≥ 1, it holds that The above Theorem will be proven using a Lemma and a labeling. The first one gives us the lower bound of ( ) Lemma 1. Let be a friendship graph with petals and 2 + 1 vertices, then for ≥ 2, it holds that Proof. Consider the general form of the lower bound of ( ), ( ) ≥ max { + −1 | 1 ≤ ≤ Δ }.
( ) = , 1 ≤ ≤ By this labeling, it shows that the largest label is + 1 and the weights of the vertices can be calculated as follows. ( ) = 2 + 2 , respectively. It can be seen that the weights of each vertex of has a different value, meaning that is an irregular labeling. Therefore, it can be concluded that by the labeling , we obtain ( ) = + 1.

The modular irregularity strength of friendship graph
First, consider the following Lemma for the lower bound of ( ).

Lemma 2.
Let be a friendship graph with petals and 2 + 1 vertices, then for ≥ 2, it holds that ( ) ≥ + 1 Proof. To prove the above Lemma, we revisit the following lower bound theorem for modular irregularity strength Because is a connected graph, it only has 1 component. Therefore, it is obtained that ( ) ≥ ( ). By Theorem 1, we have ( ) = + 1 for ≥ 2. So, we can conclude that ( ) ≥ + 1  Next, we define a labeling for 4 different cases to determine the modular irregularity strength of . The 4 cases, are the remainder of the number of petals divided by 4, namely ≡ 0 ( 4), ≡ 1 ( 4), ≡ 2 ( 4) and ≡ 3 ( 4). It can be inferred from the above definition that the largest possible label is + 1, in other words, ( ) ≤ + 1 (5) By this labeling, it also shows that the weights of the vertices can be calculated as follows.

The modular irregularity strength of the disjoint union of friendship graph where ≡ ( )
For the disjoint union, the lowest possible number of copies is = 3, because | | = 2 + 1, causing |2 | = 4 + 2 ≡ 2 ( 4) which makes it impossible for 2 to have a modular irregular labeling.