Modular irregularity strength of disjoint union of cycle-related graph

. Let 𝐺𝐺 = ( 𝑉𝑉 , 𝐸𝐸 ) be a graph with a vertex set 𝑉𝑉 and an edge set 𝐸𝐸 of 𝐺𝐺 , with order 𝑛𝑛 . Modular irregular labeling of a graph 𝐺𝐺 is an edge 𝑘𝑘 -labeling 𝜑𝜑 : 𝐸𝐸 → {1, 2, ⋯ , 𝑘𝑘 } such that the modular weight of all vertices is all different. The modular weight is defined by 𝑤𝑤𝑡𝑡 𝜑𝜑 ( 𝑢𝑢 ) = ∑ 𝜑𝜑 ( 𝑢𝑢𝑢𝑢 ) 𝑣𝑣∈𝑁𝑁 ( 𝑢𝑢 ) (mod 𝑛𝑛 ) . The minimum number 𝑘𝑘 such that a graph 𝐺𝐺 has modular irregular labeling with the largest label 𝑘𝑘 is called modular irregularity strength of 𝐺𝐺 . In this research, we determine the modular irregularity strength for a disjoint union of cycle graph, ms( 𝑚𝑚𝐶𝐶 𝑛𝑛 ) = 𝑚𝑚𝑛𝑛 2 + 1 for 𝑛𝑛 ≡ 0 (mod 4) , a disjoint union of sun graph, ms( 𝑚𝑚 ( 𝐶𝐶 𝑛𝑛 ⊙ 𝐾𝐾 1 )) = ∞ for 𝑛𝑛 and 𝑚𝑚 even and ms( 𝑚𝑚 ( 𝐶𝐶 𝑛𝑛 ⊙ 𝐾𝐾 1 )) = 𝑚𝑚𝑛𝑛 otherwise, and a disjoint union of middle graph of cycle graph, ms �𝑚𝑚𝑚𝑚 ( 𝐶𝐶 𝑛𝑛 ) � = ∞ for 𝑛𝑛 and 𝑚𝑚 both odd numbers and ms �𝑚𝑚𝑚𝑚 ( 𝐶𝐶 𝑛𝑛 ) � = 𝑚𝑚𝑛𝑛 2 + 1 otherwise.


Introduction
Graph labeling is a mapping from a set of numbers to elements of a graph , usually the vertices or the edges [1].Over the years, many graph labelings have been introduced; among them are irregular labeling and modular irregular labeling.
In [2], Chartrand et al. introduced irregular labeling.An edge labeling : () → {1,2, … , }, where  is a positive integer, such that the weighst of the vertices are all different, is called irregular labeling.The weight of a vertex  ∈ () is defined by   () = ∑ () ∈() , where () denotes the set of neighbors of  in .The irregularity strength of , notated as s() is the minimum number  for which the graph  has irregular labeling with label at most .Chartrand et al. also give the lower bound of the irregularity strength of a graph as follows.
Theorem 1 [2].Let  be a connected graph of order ≥ 3 containing   vertices of degree , then, In [3] Bača et al. introduced modular irregular labeling as a variation of irregular labeling.Modular irregular labeling of a graph , with order , is an edge -labeling :  → {1, 2, ⋯ , } such that the modular weights of all vertices are all different.The modular weight is defined by   () = ∑ () ∈() (mod ), where () denotes the set of neighbors of  in .The minimum number  such that a graph  has modular irregular labeling with the largest label  is called modular irregularity strength of .If no modular irregular labeling for the graph  could be found, it is defined as () = ∞.The relation between ms() and s() is provided by Bača et al. in the following theorem.
Bača et al. also characterized a sufficient condition such that a graph has no modular irregular labeling.
There are some classes of graphs for which the modular irregularity strength has been found.Bača et al. [3] determined the modular irregularity strength of some graphs, namely path, star, triangular, cycle, and gear graphs.Muthugurupackiam and Ramya determined the modular irregularity strength of tadpole and double-cycle graphs [4].Bača et al. also determined the modular irregularity strength of the fan graph [5] and wheel [6].Then Sugeng et al. determined the modular irregularity strength of double-star and friendship graphs [7].Tilukay determined the modular irregularity strength of the triangular book graph [8].Then Hinding et al. determined the modular irregularity strength of the dodecahedralmodified generalization graph [9].Dewi determined the modular irregularity strength of   ⊙  1 [10].Recently, Sugeng determined the modular irregularity strength of (  ) and several other flower-type graphs [11].
A sun graph   ⊙  1 is the graph obtained from a cycle graph   by adding a pendant edge to every vertex in the cycle [13].Thus,   ⊙  1 has 2 vertices.
The middle graph () of a connected graph  = ((), ()) is defined as a graph where �()� = () ∪ () and two vertices  and  are adjacent if: 1.  and  are adjacent edges of , or 2.  is a vertex of , and  is an edge that incident with it , or vice versa [14].The graph union of two graphs  1 = ( 1 ,  1 ), and  2 = ( 2 ,  2 ) is the graph  =  1 ∪  2 whose vertex-set is the disjoint union of the vertex-sets of  1 and  2 , that is () =  1 ∪  2 , and the edge-set is the disjoint union of the edge-set of  1 and  2 , that is () =  1 ∪  2 .The iterated union  ∪  ∪ … ∪  of  disjoint copies of the graph  notated as  [15].
The ms(  ), ms(  ⊙  1 ), and ms((  )) have been determined [3,10,11].In this research, we determine the modular irregularity strength for a disjoint union of the cycle graph, a disjoint union of the sun graph, and a disjoint union of the middle graph of the cycle graph.

Results and discussion
In this section, we determined the modular irregularity strength a disjoint union of cycle graph, a disjoint union of sun graph, and a disjoint union of middle graph of cycle graph.

Disjoint union of cycle graph
Lemma 1.Let   be a cycle graph with  ≡ 0 (mod 4) and   be the disjoint union of  copies of   ,  ≥ 1.Then, Proof.According to Theorem 1, we obtain: Then, based on Theorem 2, we obtain: The weights of the vertices of   under the labeling  are {2,3, … ,  + 1}.
Then the modular weights of the vertices of   under the labeling  are {0,1, … ,  − 1}, under the modulo .Therefore,  fulfill the requirement of a modular irregular labeling for   and we can conclude that, Hence, according to Equation ( 1) and Lemma 1, we can conclude that, The example of Theorem 4 for modular irregular labeling on the disjoint union of three cycle graph  8 is shown in Fig. 1.  .The proof will be divided into 2 cases:
• Case 2:  odd and  even or  even We define  as the following edge labelling.
The pendant vertices must have different weights, so the minimum number of labels must be , that is, ms((  ⊙  1 )) ≥ .

∎
The example of Theorem 5 for modular irregular labeling on the disjoint union of four sun graph  6 ⊙  1 is shown in Fig. 2. Proof.According to Theorem 1, we obtain: Then, from Theorem 2, we obtain: The weights of the vertices of (  ) under the labeling  are {2,3, … ,2 + 1}.

Conclusion
In this paper, we proved the modular irregularity strength of three classes of graphs.A disjoint union of the cycle graph has ms(  ) =

𝑚𝑚𝑛𝑛 2 + 1
for  ≡ 0 (mod 4).A disjoint union of the sun graph has ms((  ⊙  1 )) = ∞ for  and  even and ms((  ⊙  1 )) =  otherwise.Lastly, a disjoint union of the middle graph of the cycle graph has ms�(  )� = ∞ for  and  both odd numbers, and ms�(  )� = Let (  ) be a middle graph of cycle graph with  ≥ 3 and let (  ) be the disjoint union of  copies of (  ),  ≥ 1.Then, ≤  ≤ ; 1 ≤  ≤ } be the vertex set and edge set of the graph ( (  )) respectively, where    are the vertices from the edges of cycle graph   and    are the vertices from the vertices of cycle graph   .Let  +1 =  1 .The proof will be divided into 3 cases:• Case 1:  even We define  as the following edge labelling, The weight set of the vertices of (  ) under the labeling  is {2,3, … ,2 + 1}.Then the modular weights of the vertices of (  ) under the labeling  are {0,1, … ,2 − 1}, under the modulo 2.Therefore,  fulfill the requirement of a modular irregular labeling for (  ), and we can conclude that,