On The Edge Irregularity Strength of Firecracker Graphs F 2, m

Let 𝐺 = (𝑉, 𝐸) be a graph and k be a positive integer. A vertex k -labeling 𝑓 ∶ 𝑉(𝐺) → {1,2, ⋯ , 𝑘} is called an edge irregular labeling if there are no two edges with the same weight, where the weight of an edge uv is 𝑓(𝑢) + 𝑓(𝑣) . The edge irregularity strength of G , denoted by es ( G ), is the minimum k such that 𝐺 has an edge irregular k -labeling. This labeling was introduced by Ahmad, Al-Mushayt, and Bacˇa in 2014. An (n,k) -firecracker is a graph obtained by the concatenation of n k-stars by linking one leaf from each. In this paper, we determine the edge irregularity strength of fireworks graphs F 2, m .


Introduction
Graph labeling was first introduced by Sadlàčk (1964), then Stewart (1966), Kotzig and Rosa (1970). The process of labeling a graph includes assigning values (labels), represented by a set of positive integers to vertices, edges, or both. These numbers are called labels [1]. There are several types of labeling on graphs, including gracefull labeling, harmony labeling, total irregular labeling, magic labeling, and anti-magic labeling. The concept of irregular labeling on a graph was first introduced by Chartrand et al. in 1986 [2].
In 2014, Ahmad et al. [3] introduced edge irregular labeling of graphs, namely edge irregular labeling. For an integer , a total labeling : ( ) → {1, 2, ⋯ } is called an edge irregularlabeling of if every two distinct edges 2 and 2 in satisfy ( 1 ) ≠ ( 2 ), where ( 1 = ) = ( ) + ( ). As an example, we have a graph 2 ∪ 3 in the Figure 1.  By the labeling in the Figure 2, the weight of edges of the graph are 3, 7, 8, and 9 and the maximum label used in this labeling is 5. Besides that, there are no two edges with the same weight, so the labeling is an edge irregular k-labeling of 2 ∪ 3 with k=5.
The minimum for which a graph has an edge irregular -labeling, denoted by ( ), is called the edge irregularity strength of . For example, given an edge irregular 3-labeling of 2 ∪ 3 in the Figure 3. The labeling of the Figure 3 is in edge irregular 3-labeling of 2 ∪ 3 because there are no two edges with the same weight and the maximum label used is 3. It is impossible to have an edge irregular klabeling of 2 ∪ 3 with maximum label 2. So, 3 is the minium k for which 2 ∪ 3 has an edge irregular k-labeling. We conclude that the edge irregularity strength of 2 ∪ 3 is 3, denoted by To get the exact value of es of a graph G, we would previously determine a lower bound and an upper bound of es(G) before. A lower bound on es(G) is obtained by using the theorem from Martin Baca and Ali Ahmad in 2014. [3] as follows. Other results about computing the edge irregularity strength of graphs are given by Imran et al. in [4]. In the paper, Imran et al. determined the egde irregularity strength of caterpillars, n-star graphs, kite graphs, cycle chains and friendship graphs.
Tarawneh et al. [5], determined he edge irregularity strength of corona product of cycle with isolated vertices. In [6], Tarawneh et al. determined he exact value of edge irregularity strength for triangular grid graph , zigzag graph and Cartesian product 2 . In 2017, Ahmad et al determined the edge irregularity strengths of some chain graphs and the join of two graphs. They also introduced a conjecture and open problems for researchers for further research [7]. Ahmad et al. [6], gave computing of the edge irregularity strength of bipartite graphs and wheel related graphs. In [8] , Asim et al. gained an edge irregular k-labeling for several classes of trees. Asim et al also gained the edge irregularity strength of disjoint union of star graph and subdivision of star graph [9].
In 2020, Ahmad et al performed a computer based experiment dealing with the edge irregularity strength of complete bipartite graphs. They also presented some bounds on this parameter for wheel related graphs [6]. In [10], Tarawneh et al. gave the edge irregularity strength of some classes of plane graphs.
In this paper, we determined the exact value of es of firecracker graphs F2,m with arbitrary m. A firecracker is a graph obtained by the concatenation of stars by linking one of leaf from each. If the number of stars is and the number of leaves in each star is , then the firecracker is denoted by , .

Methods
The method we use in this research is analytical method. To get the exact value of of firecracker graph, we consider a lower bound and an upper bound of es( , ). Theorem 1 is used to have a lower bound of es( , ). Besides that, an upper bound of es( , ), we construct an edge irregular-k labeling with minimum k.

Result and Discussion
The main result of our research is the edge irregularity strength of firecracker graphs , is + . The result written in Theorem 2.
In the Figure 5, we can see an illustration of firecracker graph F2,m So, we have ( 2, ) ≥ + 1 Next, we give an edge irregular −labeling with k = + 1 to get ( 2, ) ≤ + 1 as follows.
From the edge weight formula (4), there are no two edges with the same weight. The maximum label used in the labeling f is + 1. So, f is an edge irregular-( + 1) of , . So, we can conclude that ( 2, ) ≤ + 1 From inequalities (1) and (2), we have an equality ( 2, ) = + 1.
For an illustration, in the Figure 6, we can see the edge irregular labeling f of firecracker graph 2, = 8. In the Figure 7, we can see the weight of edges of 2,8 under the labeling in Figure 6.

Conclusion
By the research, we have a lower bound of ( 2, ) is + 1, which is also an upper bound of ( 2, ). So that, we can conclude the exact value of the edge irregularity strength of firecracker graphs 2, is + 1 for ≥ 2.