Graph labeling is an assignment of numbers to vertices or edges (or both) [1]. Many types of graph labeling have been studied, which can be found in a graph labeling survey by Gallian [2]. In this research, we consider a finite graph that has a set of vertices and a set of edges . Let G be a graph of order n. A graph is called a regular graph if every vertex of has the same degree and an irregular graph if every two distinct vertices of has diferent degrees. It is easy to see that having an irregular simple graph is impossible. Motivated by an irregular graph, Chartrand et al. [3] defined an edge labeling such that the weights for every two distinct vertices are diferent where the weight of vertex is defined by , where is the set of edges in incident with u as an irregular labeling of . The irregularity strength of denoted by , is defined as the smallest integer k such that admits an irregular labeling with k as its maximum label. If there is no possible irregular labeling for then the irregularity strength of is defined to be infinity, that is, . One clear example of a graph without irregular labeling is a graph with isolated edges. Chartrand et al. give the lower bound of this invariant for any connected graph of order greater than 3 as follows.

Theorem 1.1.([3]) Let G be a connected graph of order greater than 3 having vertices of degree i, then

In 2020, Baˇca, Muthugurupackiam, Kathiresan, and Ramya introduced modular irregular labeling [4]. Let be a graph of order n and is an edge k-labeling, . The labeling is called modular irregular labeling of the graph if there exist a bijection defined by (mod n) where is a group of integer modulo . The modular weight of a vertex u is the value of . The modular irregularity strength of denoted by ms(), is defined as the smallest integer k such that admits a modular irregular labeling with k as its maximum label. Similarly to the strength of the irregularity, if G has no modular irregular labeling, the modular irregularity strength of G is defined as infinity, that is, ms the definition of modular irregular labeling, it is easy to see that any modular irregular labeling of is also an irregular labeling for . Hence, we have the following.

Theorem 1.2. ([4]) Let be a graph without isolated edges. Then,

Not all graphs have modular irregular labeling. Characterization such that a graph does not have modular irregular labeling is also shown in [4]. In Theorem 1.3, the characterization is given.

Theorem 1.3. ([4]) If is a graph of order (mod 4), then has no modular irregular labeling, that is, ms .

Several classes of graphs have been shown to have modular irregularity labeling. The modular irregularity strength of several classes of graphs, namely cycle, star, triangular path, and gear graphs have been determined in [4]. Then, as an extension of the previous result of the cycle graph, the modular irregularity strength for the corona product of a cycle graph with a null graph has been determined in [5].

Recently, Barack et al.[6] determined the modular irregularity strength for the corona product of a regular graph containing a 1-factor with a null graph of odd order or with a path graph of order 3.

Theorem 1.4. ([6]) Let be a regular graph of order n containing a 1-factor and be a null graph of odd order p. Then

Theorem 1.5. ([6]) Let be a regular graph of order n containing a 1-factor, and be a path graph of order 3. Then

There are more results on modular irregularity strength of family graphs, such as path graph, cycle graph, star graph, double-star graph, friendship graph, generalized book graph, rose graph, daisy graph, sunflower graph, and firecracker graph [4, 7, 8, 9, 10]. There are also results on the modular irregularity strength of some dense graphs, such as complete graph and complete bipartite graph [11]. For disconnected graph, the modular irregularity strength of some disjoint copies of graphs has been obtained, such as cycle graphs, sun graphs, middle graph of cycle graphs and friendship graphs [12, 13].

Let n be an integer and let be a strictly increasing sequence of positive integers, with . The graph of order n with the vertex set and the edge set is called the circulant graph, denoted by . The circulant graph gives the cycle graph , and the circulant graph gives the complete graph[14].

As the cycle graph is a particular case of a circulant graph, in this research we extend the previous result from ms to ms . As a result of Theorem 1.4, Barack, Sugeng, Semaniˇcov´a-Feˇnovˇc´ıkov´a, and Baˇca have found ms . We now determine ms where G is a regular graph containing a perfect matching G and or These results will complete ms for is a regular graph containing perfect matching and is a graph of order 3. We also extend the previous result of ms with .

Theorem 2.1.Let be a circulant graph of order 4 \end{document} ]]> and let be a null graph of order p. (mod 4), then ms

Proof. Let

be the vertex set and edge set of , where are the vertices of the circulant graph and are the pendant vertices. It is evident that . Therefore, according to Theorem 1.3, if (mod 4), ms . Let , and . For the case where (mod 4), the proof will be divided into cases, for odd and even For odd define the edge labeling as follows.

Then, for the labeling of the edge and , we further divide into 3 subcases.

(1) For

(2) For ),

(3) For ),

Observe that in each case we obtain that is a pn-labeling.

Since the vertex is a pendant vertex, the weight of is the same as the edge label that incident with , that is:

The edges that are incident with the vertex in are the edges that are incident with in and the edges . Hence, the modular weight of vertex under the labeling is:

We find that the vertex weights are all distinct, and the set of all modular weights of vertex

Hence, is a modular irregular labeling for . and we can conclude that:

For even define an edge labeling as follows.

Then, for the labeling of the edge and , we divide furthermore into 3 subcases.

(1) For

Observe that for this subcase we obtain that is a pn-labeling.

Since the vertex is a pendant vertex, then the weight of is the same as the label of the edge that incident with it, that is:

The edges that incident with the vertex in are the edges that are incident with in and the edges Hence the modular weight of vertex under the labeling is:

We have that the vertex weights are all distinct, and the set of all modular weights of

Hence, for this subcase, is a modular irregular labeling for and we can conclude that:

(2) For n ≡ 1 (mod 4),

Observe that for this subcase, we obtain that is a pn-labeling.

Since the vertex is a pendant vertex, then the weight of is the same as the label of the edge that incident with it, that is:

The edges that incident with the vertex in are the edges that are incident with in and the edges Hence, the modular weight of vertex under the labeling is:

We have the vertex weights are all distinct, and the set of all modular weights of

Hence, for this subcase, is a modular irregular labeling for 2 and we can conclude that:

(3) For (mod 4),

Observe that for this subcase, we find that is a pn-labeling.

Since the vertex is a pendant vertex, the weight of is the same as the label of the edge that incident with it, that is:

The edges that are incident with the vertex in are the edges that are incident with in and the edges Hence, the modular weight of vertex under the labeling is:

We have the vertex weights are all distinct, and the set of all modular weights of :

Hence, for this subcase, is a modular irregular labeling for and we can conclude that:

The modular irregularity strength of is at least because has pendant vertices, and each pendant vertex must have a diferent weight, that is:

From (1),(2),(3),(4),and (5), we can conclude that:

As the circulant graph is also a regular graph containing a perfect matching whenever n is even, for the case where n is even and p is odd, it is a corollary of Theorem 1.4. The example of Theorem 2.1 for modular irregular labeling in some corona product of circulant graph and complement of a complete graph is shown in Figure 1 below.

Theorem 2.2.Let be a d-regular graph of order n with 1 \end{document} ]]> containing a perfect matching and be a cycle graph of order then

Proof. Let be a d-regular graph of order 1 \end{document} ]]> containing a perfect matching . Let be the vertices of G and be the vertices of an i-th copy of , adjacent to . For simplicity, let

We define an edge labeling as follows.

Observe that the maximum value of the labeling τ is . Hence, we see that is a -labeling.

The weight of the vertex can be calculated as follows

Hence, we get the set of modular weights of vertex , which is

The edges that are incident with the vertex in are the edges that are incident with in and the edges .. Therefore, the modular weight of the vertex under the labeling τ is:

We get

And the set of modular weight of the vertex is

From (6) and (7) , we have the set of all modular weights of :

Therefore, is a modular irregular labeling of , and we can conclude that

Using Theorem 1.1 and Theorem 1.2 gives us

Since modular irregularity strength of a graph must be an integer, we have

From (8) and (9), we can conclude that

Theorem 2.3.Let be a regular graph of order n containing a perfect matching and be a disjoint union of path graph of order 2 with path graph of order 1. Then

Proof. Let be a d-regular graph of order n containing a perfect matching Let be the vertices of be the vertex of an i-th copy of adjacent to and be the vertices of an i-th copy of adjacent to an . For simplicity, let

We define an edge labeling λ as follows.

It is clear that the maximum value of the labeling is , so is a The edge incident with the vertex is only the edge , hence the weight of the vertex is

and the set of modular weight in modulo 4n is:

The edges incident with the vertex are the edge and , therefore the weight of the vertex is

Hence, the set of modular weight in modulo 4n is:

Lastly, the edges that incident with the vertex are the edges , the edge , and the edges that incident with in , hence the weight of the vertex is

Hence, the set of modular weight in modulo 4n is:

So, we have that the set of modular weights of all the vertices of is

Therefore, is a modular irregular labeling of , and we can conclude that

To minimize the greatest label used, the greater weights must be in the vertices with the greatest degree. Because has 1 as its minimum degree, the weight 0 must be obtained at least by 4n. Suppose that the weights of up to 4n are located in the vertices of because there are n vertices with . Then, the weight of the vertices from must consist of 1 to 3n. Let ms . Since we want to minimize the weight to 3n must be located in the vertices of degree 2, so we get

Hence, we get the lower-bound

By (10) and (11), we can conclude that

Theorem 2.4. Let be a regular graph of order n containing perfect matching, and be a path graph of order 5. Then

Proof. Let be a d-regular graph of order n that contains a perfect matching . Let be the vertices of and be the vertices of an i-th copy of , adjacent to an

We define an edge labeling as follows.

Observe that the maximum value of the labeling is . Hence, we obtain that is a -labeling.

The weight of vertex can be calculated as follows

For

For .

For

Hence, we get

and the set of modular weight of vertex , is

The edges that are incident with the vertex in are the edges that are incident with in and the edges .. Therefore, the modular weight of the vertex under the labeling is:

Hence, we get

The set of modular weights of the vertex is

We have the set of all the modular weights of

Therefore, is a modular irregular labeling of , and we can conclude that

Using Theorem 1.1 and Theorem 1.3 gives us

Since ms must be an integer, we have ms

Suppose that there exists an -modular irregular labeling of The weight of vertices of degree 2 or 3 is in the range or , respectively. Observe that only weights are available for these vertices, but the number of vertices of degree 2 or 3 is 5n. So, there must be some vertices that have the same weight. Therefore, ms n + 1 \end{document} ]]>

To increase the lower bound, we use a diferent approach. We can safely assume that the highest weight must be in the vertices to the greatest extent to minimize the highest label used Because has as its minimum degree, the weights 0 and 1 must be obtained at least by 6n and , respectively. Suppose that the weights of up to 6n+1 are located in the vertices of , because there are n vertices with . Then, the vertices of must ”cover” the weights from 2 to 5n. Let ms . Since we want to minimize we get that

Hence, we get the new lower-bound

From (12) and (13) we can conclude that

□

In this research, we determined the modular irregularity strength for that is, ms , which we proved in Theorem 2.1 by constructing the modular irregular labeling for and show the lower bound for its modular irregularity strength. We completed the previous result on the modular irregularity strength for the corona product , where is a d-regular graph containing a perfect matching G and H is a graph of order 3, as follows.

1.\end{array}\right.\end{array} \end{document} ]]>

Lastly, we also determined the modular irregularity strength for the corona product of a regular graph containing a 1-factor with a path graph of order ms. Further research can be conducted to determine the modular irregularity strength for the corona product of the circulant graph and the complement of complete graph for other parameters of the circulant graph, e. g. or the modular irregularity strength for the corona product of the graph with path for m or m .