Modular Sumset Labelling of Graphs

Graph labelling is an assignment of labels or weights to the vertices and/or edges of a graph. For a ground set X of integers, a sumset labelling of a graph is an injective map f : V G ð Þ ! P X ð Þ such that the induced function f ⊕ : E G ð Þ ! P X ð Þ is defined by f þ uv ð Þ ¼ f u ð Þ þ f v ð Þ , for all uv ∈ E G ð Þ , where f u ð Þ þ f v ð Þ is the sumset of the set-label, the vertices u and v . In this chapter, we discuss a special type of sumset labelling of a graph, called modular sumset labelling and its variations. We also discuss some interesting characteristics and structural properties of the graphs which admit these new types of graph labellings.


Basics of graph labelling
Labelling of a graph G can broadly be considered as an assignment of labels or weights to the elements (vertices and edges) of G subject to certain pre-defined conditions. The research on graph labelling has flourished in the second half of twentieth century after the introduction of the notion of β-valuations of graphs in [11]. The β-valuation of a graph G is an injective map f : V G ð Þ ! 1, 2, 3, … , jEj f g such that the induced function f * : E G ð Þ ! 1, 2, 3, … , jEj f g , defined by f * uv ð Þ ¼ | f u ð Þ À f v ð Þ| for all uv ∈ E G ð Þ, is also injective. Later, β-valuation of graphs was popularly known to be the graceful labelling of graphs (see [12]). Many variations of number valuations have been defined in the literature since then and most of those studies were based on the number theory and/or number theoretic properties of sets. For concepts and results in number theory, see [13][14][15][16].
Analogous to the number valuations of graphs, the notion of set-labelling of graphs has been introduced in [17] as follows: Given a non-empty ground set X, a set-labelling or a set-valuation of a graph G is an injective function f : V G ð Þ ! P X ð Þ, the power set of X, such that the induced function f ⊕ : V G ð Þ ! P X ð Þ defined by where ⨁ is the symmetric difference of two sets. A graph which admits a set-labelling is called a set-labelled graph or a set-valued graph. If the induced function f ⊕ is also injective, then the set-labelling f is called a set-indexer.
Subsequent to this study, many intensive investigations on set-labelling of graphs and its different types have been taken place. An overview of such studies on set-labelling of graphs can be seen in [17][18][19][20][21][22][23]. The study on set-labelling has been extended by replacing the binary operation ⨁ by some other binary operations of sets. For example, two different types of set-labellings-called disjunctive setlabelling and conjunctive set-labelling-of graphs have been studied in [24]. These set-labellings are defined respectively in terms of the union and the intersection of two sets instead of the symmetric difference of two sets.

Sumsets and integer additive set-labelled graphs
Integer additive set-labelling or sumset labelling of graphs has been a new addition to the theory of set-labelling of graphs recently. The notion of sumsets of two sets is explained as follows: Let A and B be two sets of numbers. The sumset of A and B is denoted by A þ B and is defined by f g (see [25]). Remember that a sumset of two sets can be determined if and only if both of them are number sets. If either A or B is countably infinite, then their sumset A þ B is also a countably infinite set and if any one of them is a null set, then the sumset is also a null set. If C is the sumset of two sets A and B, then both A and B are said to be the summands of C.
We note that A þ 0 f g ¼ A and hence A and 0 f g are called the trivial summands of the set A. Also, note that A þ B need not be a subset or a super set of A and/or B.
Furthermore, the sumset of two subsets of a set X need not be a subset of the ground set X. These observations are clear deviations from the other common binary operations of sets and thus the study of sumsets becomes more interesting. For the terms, concepts and results on sumsets, we refer to [25][26][27][28][29][30][31][32][33][34].
Note that if A and B are two non-empty finite sets of integers, then |A| þ |B| À 1 ≤ |A þ B| ≤ |A Â B| ¼ |A||B| (see [25]). The exact cardinality of the sumset A þ B always depends on the number as well as the pattern of elements in both the summands A and B. The counting procedure in this case is explained in [35] as follows: Two ordered pairs a, b ð Þ and c, d The cardinality of a compatibility class in A Â B lies between 1 and min jAj, jBj f g. Note that the sum of coordinates of all elements in a compatibility class is the same and this sum will be an element of the sumset A þ B. That is, the cardinality of the sumset of two sets is equal to the number of equivalence classes on the Cartesian product of the two sets generated by the compatibility relation defined on it.
Using the concepts of the sumsets of sets, the notion of integer additive setlabelling of graphs has been introduced in [36] as follows: Let X be a set of nonnegative integers and P 0 X ð Þ be the collection of the non-empty subsets of X. Then, an integer additive set-labelling or an integer additive set-valuation of a graph G is an Þ is the sumset of the set-label the vertices u and v (see [36,37]). A graph with an integer additive set-labelling is called an integer additive set-labelled graph. It can very easily be verified that every graph G admits an integer additive set-labelling, provided the ground set X is chosen judiciously.
Following to the above path-breaking study, the structural properties and characteristics of different types of integer additive set-labellings of graphs are studied intensively in accordance with the cardinality of the set-label, nature and pattern of elements in the set-label, nature of the collection of set-label, etc. Some interesting and significant studies in this area can be found in [35][36][37][38][39][40][41][42][43][44]. Later, the studies in this area have been extended by including the sets of integers (including negative integers also) for labelling the elements of a graph. Some extensive studies in this area, can be seen in [45,46].
As a specialisation of the sumset labelling of graphs, the notion of modular sumset labelling of graphs and corresponding results are discussed in the following section.

Basics of modular sumsets
Recall that  n denotes the set of integers modulo n, where n is a positive integer. The modular sumset of two subsets A and B of  n is the set k : a ∈ A, b ∈ B, f a þ b k mod n ð Þ g. Unless mentioned otherwise, throughout this chapter, the notation A þ B denotes the modular sumset of the sets A and B. Unlike the ordinary sumsets, the modular sumset A þ B ⊆  n if and only if A, B ⊆  n . This fact will ease many restrictions imposed on the vertex set-label of a sumset graph G in order to ensure that the edge set-label are also subsets of the ground set.
If we assign the null set Ø to any vertex as the set-label, the set-label of every edge incident at that vertex will also be a null set. To avoid such an embarrassing situation, we do not consider the null set for labelling any vertex of graphs. Thus, the set of all non-empty subsets of a set X is denoted by P 0 X ð Þ. That is, P 0 X ð Þ ¼ P X ð Þn 0 f g.

Modular sumset graphs
In view of the facts stated above, the modular sumset labelling of a graph is defined as follows: Definition 2.1.
[47] A modular sumset labelling of a graph G is an injective is the modular sumset of the set labels of the vertices u and v. A graph which admits a modular sumset labelling is called a modular sumset graph. Definition 2.2.
[47] A modular sumset labelling of a graph G is said to be a uniform modular sumset labelling of G if the set-label of all its edges have the same cardinality. A modular sumset labelling f of G is said to be a k-uniform modular sumset labelling if f þ uv ð Þ ¼ k, ∀uv ∈ E G ð Þ. Proposition 2.1.
[47] Every graph G admits modular sumset labelling (for a suitable choice of n).
The proof of the above proposition is immediate from the fact that f u ð Þ þ f v ð Þ ⊆  n if and only if f u ð Þ, f v ð Þ ⊆  n . An immediate question that arises in this context is about the minimum size of the ground set  n (that is, the minimum value of n) required for the existence of a modular sumset labelling of G.
As in the case of sumsets, the cardinality of the modular sumsets also attracted the attention. Hence, we have the bounds for the cardinality of an edge set-label of a modular sumset graph G is as follows: Theorem 2.2.
[47] Let f : V G ð Þ ! P 0  n ð Þ be a modular sumset labelling of a given graph G. Then, for any edge uv ∈ E G ð Þ, we have The theorem follows immediately from the theorem on the cardinality of sumsets (see Theorem 2.7, p. 52, [25]).
In this context, it is quite interesting to investigate whether the bounds are sharp. It has also been proved in [25] that the lower bound is sharp when both f u ð Þ and f v ð Þ are arithmetic progressions (we call set an arithmetic progression if its elements are in arithmetic progression) with the same common difference. We shall discuss the different types of modular sumset graphs based on the set-labelling numbers of its vertices and edges, one by one in the coming discussions.

Arithmetic modular sumset graphs
As mentioned above, the lower bound of the inequality (1) is sharp if both summand set-label are arithmetic progressions with the same common difference. If the context is clear, the common difference of the set-label (if exists) of an element may be called the common difference of that element. The deterministic ratio of an edge of G is the ratio, k ≥ 1 between the common differences of its end vertices. In view of this terminology we have the following definition. The difference set of a non-empty set A, denoted by D A , is the set defined by Note that if A is an arithmetic progression, then its difference set D A is also an arithmetic progression and vice versa. Analogous to the corresponding result of the edge-arithmetic sumset labelling of graphs (see [44,46]), the following result is a necessary and sufficient condition for a graph G to be edge-arithmetic modular sumset graph in terms of the difference sets the setlabel of vertices of G. Theorem 3.

Let f be a modular sumset labelling defined on a graph G. If the setlabel of an edge of G is an arithmetic progression if and only if the sumset of the difference sets of set-label of its end vertices is an arithmetic progression.
Proof. Let f : V G ð Þ ! P ℕ 0 ð Þ be a modular sumset labelling defined on G. Let a i , a j be two arbitrary elements in f u ð Þ and let b r , b s be two elements in f v ð Þ. Then, Then, we have D A is an arithmetic progression. Since the difference set D A is an arithmetic progression, then by the above remark, we have is also an arithmetic progression. Hence, the edge e ¼ uv has an arithmetic progression as its set-label.
□ In view of the notions mentioned above, we note that there are some graphs, all whose elements have arithmetic progressions as their set-label and there are some graphs, the set-label of whose edges are not arithmetic progressions. Keeping this in mind, we define the following notion.
Definition 3.2. An arithmetic sumset labelling of a graph G is a modular sumset labelling f of G, with respect to which the set-label of all vertices and edges of G are arithmetic progressions. A graph that admits an arithmetic modular sumset labelling is called an arithmetic modular sumset graph.
Analogous to the condition for an arithmetic sumset graphs (see [44]), a necessary and sufficient condition for a graph to admit an arithmetic modular sumset labelling is discussed in the following theorem.
Theorem 3.2. A graph G admits an arithmetic modular sumset labelling f if and only if for any two adjacent vertices in G, the deterministic ratio of every edge of G is a positive integer, which is less than or equal to the set-labelling number of its end vertex having smaller common difference.
Proof. Here, we need to consider the following two cases: Case 1: First note that if the set-label of two adjacent vertices are arithmetic progressions with the same common difference, say d, then the set-label of the corresponding edge is also an arithmetic progression with the same common difference d. Then, it is clear that a vertex arithmetic modular sumset graph is an arithmetic modular sumset graph if the common differences between any two adjacent vertices of G are the same.
Case 2: Assume that u, v be any two adjacent vertices in G with common differ- Þth row in such a way that equal terms of different rows come in the same column of this arrangement. Without loss of generality, assume that That is, a few final elements of each row of the above arrangement occur as the initial elements of the succeeding row (or rows) and the difference between two successive elements in each row is d u itself. If k ¼ m, then the difference between the final element of each row and the first element of the next row is d u and the difference between two consecutive elements in each row is d u . Hence, if k ≤ m, then f þ uv ð Þ is an arithmetic progression with common difference d u .
In both cases, note that if the given conditions are true, then f is an arithmetic modular sumset labelling of G.
We prove the converse part by contradiction method. For this, assume that f is an arithmetic modular sumset labelling of G. Let us proceed by considering the following two cases.
Case-1: Assume that d j is not a multiple of d i (or d i is not a multiple of d j ). Without loss generality, let d i < d j . Then, by division algorithm, Then, the differences between any two consecutive terms in f þ v i v j À Á are not equal. Hence, in this case also, f is not an arithmetic modular sumset labelling, contradiction to the hypothesis. Therefore, d i |d j .
Then, the difference between two successive elements in each row is d i , but the difference between the final element of each row and the first element of the next row is td i , where t ¼ k À m þ 1 6 ¼ 1. Hence, f is not an arithmetic modular sumset labelling, a contradiction to the hypothesis. Hence, Therefore, from the above cases it can be noted that if a vertex arithmetic modular sumset labelling of G is an arithmetic modular sumset labelling of G, then the deterministic ratio of every edge of G is a positive integer, which is greater than or equal to the set-labelling number of its end vertex having smaller common difference. This completes the proof. □ In the following theorem, we establish a relation between the common differences of the elements of an arithmetic modular sumset graph G. Theorem 3.3. If G is an arithmetic modular sumset graph, the greatest common divisor of the common differences of vertices of G and the greatest common divisor of the common differences of the edges of G are equal to the smallest among the common differences of the vertices of G.
Proof. Let f be an arithmetic modular sumset labelling of G. Then, by Theorem 3.2, for any two adjacent vertices v i and v j of G with common differences d i and d j respectively, either If the common differences of the elements of G are the same, the result is obvious. Hence, assume that for any two adjacent vertices v i and v j of G, where d i is the smallest among the common differences of the vertices of G. If v r is another vertex that is adjacent to v j , then it has the common difference d r which is equal to either d i or d j or ld j . In all the three cases, d r is a multiple of d i . Hence, the greatest common divisor of d i , d j , d r is d i . Proceeding like this, we have the greatest common divisor of the common differences of the vertices of G is d i .
Also, by Theorem 3.2, the edge u i v j has the common difference d i . The edge v j v k has the common difference d i , if d k ¼ d i , or d j in the other two cases. Proceeding like this, we observe that the greatest common divisor of the common differences of the edges of G is also d i . This completes the proof.
□ The study on the set-labelling number of edges of an arithmetic modular sumset graphs arouses much interest. Analogous to the result on set-labelling number of the edges of an arithmetic sumset graph (see [43]), The set-labelling number of an edge of an arithmetic modular sumset graph G, in terms of the set-labelling numbers of its end vertices, is determined in the following theorem.
Theorem 3.4. Let G be a graph which admits an arithmetic modular sumset labelling, say f and let d i and d j be the common differences of two adjacent vertices v i and v j in Proof. Let f be an arithmetic modular sumset labelling defined on G. For any two Let d i and d j be the common differences of the vertices v i and v j respectively, such that d i < d j . Since f is an arithmetic modular sumset labelling on G, by Theorem 3.2, there exists a positive integer k such that d j ¼ k: That is, the set-labelling number of the edge v i v j is m þ k n À 1 ð Þ. □

Strongly modular sumset graphs
The next type of a modular sumset labelling we are going to discuss is the one with the upper bound in Inequality (1) is sharp (that is, |A þ B| ¼ |AkB|). Thus, we have the following definition.
[47] A modular sumset labelling f : V G ð Þ ! P 0  n ð Þ defined on a given graph G is said to be a strongly modular sumset labelling if for the associated A graph which admits a strongly modular sumset labelling is called a strongly modular sumset graph.
Invoking the notion difference set of a set, a necessary and sufficient condition of a modular sumset labelling of a graph G to be a strongly modular sumset labelling is given below:

Þ of a given graph G is a strongly modular sumset labelling of G if and only if D f u
ð Þ ! P 0  n ð Þ be a modular sumset labelling on a given graph G. For any vertex u ∈ V G ð Þ, define D f u ð Þ ¼ a i À a j : a i , a j ∈ f u ð Þ È É . Let uv be an arbitrary edge in E G ð Þ. Assume that f is a strong modular sumset labelling of G. Then, by definition | f þ uv Therefore, the difference sets of the set-label of any two adjacent vertices are disjoint.
Conversely, assume that the difference Hence, f is a strongly modular sumset labelling of G.
Also, note that the maximum possible cardinality in the set-label of any element of G is n, the product | f u ð Þ|| f v ð Þ| cannot exceed the number n. This completes the proof.
□ A necessary and sufficient condition for a modular sumset labelling of a graph G to be a strongly k-uniform modular sumset labelling is given below: Theorem 4.2.
[47] For a positive integer k ≤ n, a modular sumset labelling f :

Þ of a given connected graph G is a strongly k-uniform modular sumset labelling of G if and only if either k is a perfect square or G is bipartite.
Proof. If k is a perfect square, say k ¼ l 2 , then we can label all the vertices of a graph by distinct l-element sets in such a way that the difference sets of the set-label of every pair of adjacent vertices are disjoint. Hence, assume that k is not a perfect square.
Let G be a bipartite graph with bipartition X, Y ð Þ. Let r, s be two divisors of k. Label all vertices of X by distinct r-element sets all of whose difference sets are the same, say D X . Similarly, label all vertices of Y by distinct s-element sets all of whose difference sets the same, say D Y , such that D X ∩ D Y ¼ Ø. Then, all the edges of G have the set-labelling number k ¼ rs. Therefore, G is a strongly k-uniform modular sumset graph.
Conversely, assume that G admits a strongly k-uniform modular sumset labelling, say f.
Since, f is a strong modular sumset labelling, the set-labelling number of every vertex of G is a divisor of the set-labelling numbers of the edges incident on that vertex. Let v be a vertex of G with the setlabelling number r, where r is a divisor of k, but r 2 6 ¼ k. Since f is k-uniform, all the vertices in N v ð Þ, must have the set-labelling number s, where rs ¼ k. Again, all vertices, which are adjacent to the vertices of N v ð Þ, must have the set-labelling number r. Since G is a connected graph, all vertices of G have the set-labelling number r or s. Let X be the set of all vertices of G having the set-labelling number r and Y be the set of all vertices of G having the set-labelling number s. Since r 2 6 ¼ k, no two elements in X (and no elements in Y also) can be adjacent to each other. Therefore, G is bipartite. □ The following result is an immediate consequence of the above theorem. Theorem 4.3.
[47] For a positive non-square integer k ≤ n, a modular sumset

Þ of an arbitrary graph G is a strongly k-uniform modular sumset labelling of G if and only if either G is bipartite or a disjoint union of bipartite components.
For a positive integer k ≤ n, the maximum number of components in a strongly kuniform modular sumset graph is as follows.
[47] Let f be a strongly k-uniform modular sumset labelling of a graph G with respect to the ground set  n . Then, the maximum number of components in G is the number of distinct pairs of divisors r and s of k such that rs ¼ k.
The following theorem discusses the condition for an arithmetic modular sumset labelling of a graph G to be a strongly modular sumset labelling of the graph.

Theorem 4.5. Let G be a graph which admits an arithmetic modular sumset labelling, say f . Then, f is a strongly modular sumset labelling of G if and only if the deterministic ratio of every edge of G is equal to the set-labelling number of its end vertex having smaller common difference.
Proof. Let f be an arithmetic modular sumset labelling of G. Let v i and v j are two adjacent vertices in G and d i and d j be their common differences under f . Without loss of generality, let d i < d j . Then, by Theorem 3.4, the set-labelling number of the Conversely, assume that the common differences d i and d j of two adjacent vertices v i and v j respectively in G, Þ-th row in such a way that equal terms of different rows come in the same column of this arrangement. Then, the common difference between consecutive elements in each row is d i . Since k ¼ | f v i ð Þ|, the difference between the final element of any row (other than the last row) and first element of its succeeding row is also d i . That is, no column in this arrangement contains more than one element. Hence, all elements in this arrangement are distinct. Therefore, total number of elements in Hence, f is a strongly modular sumset labelling. □

Supreme modular sumset labelling of G
In both types of modular sumset labelling discussed above, it is observed that the cardinality of the edge set-label cannot exceed the value n. This fact creates much interest in investigating the case where all the edge set-label have the cardinality n.
[47] A modular sumset labelling f : V G ð Þ ! P 0  n ð Þ of a given graph G is said to be a supreme modular sumset labelling or maximal modular sumset labelling of G if and only if f þ E G ð Þ ð Þ¼  n f g. Put in a different way, a modular sumset labelling f : V G ð Þ ! P  n ð Þ of a given graph G is a supreme modular sumset labelling of G if the set-label of every edge of G is the ground set  n itself.
A necessary and sufficient condition for a modular sumset labelling of a graph G to be its supreme modular sumset labelling is discussed in the theorem given below: Theorem 5.1.
[47] The modular sumset labelling f : V G ð Þ ! P  n ð Þ of a given graph G is a supreme modular sumset labelling of G if and only if for every pair of adjacent vertices u and v of G some or all of the following conditions hold.
The strict inequality hold when D f u ð Þ and D f v ð Þ are arithmetic progressions containing the same elements. ii Proof. For two adjacent vertices u and v in G, let D f u ð Þ ¼ D f v ð Þ ¼ d f g are arithmetic progressions containing the same elements. Then, the elements in f u ð Þ and f v ð Þ are also in arithmetic progression, with the same common difference d.
A necessary and sufficient condition for a strong modular sumset labelling of a graph G to be a maximal modular sumset labelling of G.
[47] Let f be a strong sumset-labelling of a given graph G. Then, f is a maximal sumset-labelling of G if and only if n is a perfect square or G is bipartite or a disjoint union of bipartite components.
Proof. The proof is an immediate consequence of Theorem 4.2, when k ¼ n. □

Weakly modular sumset graphs
Another interesting question we address in the beginning of this section is whether the lower bound and the upper bound of the sumset can be equal. Suppose that A and B be two non-empty subsets of  n such that the bounds of their sumset are equal. Then, we have which is possible only when |A| ¼ 1 or |B| ¼ 1 (or both). Also, note that in this case the cardinality of the sumset is equal to equal to that of one of the summands. This interesting phenomenon leads us to a new type of a modular sumset labelling called weakly modular sumset labelling. This type of labelling is investigated in the following section.

Weakly modular sumset labelling of graphs
Definition 6.1. A modular sumset labelling f of a graph G is said to be a weakly modular sumset labelling of G if the cardinality of the set-label of every edge of G is equal to the cardinality of the set-label of at least one of its end vertices. A graph which admits a weakly modular sumset labelling is called a weakly modular sumset graph.
From the above definition, it can be observed that for any edge uv in weakly modular sumset graph G, Putting it in a different way, the set-labelling number of at least one end vertex of every edge of a weakly modular sumset graph is a singleton. An element (a vertex or an edge) of modular sumset graph G with set-labelling number 1 is called a sparing element or a monocardinal elements of G. Hence, analogous to the condition for a sumset graph to be a weak sumset graph (see [39]), we have. Theorem 6.

A graph G admits a weak modular sumset labelling if and only if G is bipartite or contains sparing edges.
Proof. Note the fact that at least one end vertex of every edge of G is a sparing vertex. Also, we note that no two vertices with non-singleton set-label in weakly modular sumset graph can be adjacent to each other. Thus, if every of edge of G has exactly one end vertex with singleton set-label, then we can partition the vertex set of G into two subsets X with all sparing vertices and Y with all non-sparing vertices. Here, no two vertices in the same partition are adjacent and hence G is a bipartite graph. If G is not a bipartite graph, then obviously G should have at least one sparing edge, completing the proof.
Invoking Theorem 6.1, the following two results are immediate. Corollary 6.2. Every graph G admits a weakly modular sumset labelling.

Corollary 6.3. A graph G admits a weakly uniform modular sumset labelling if and only if G is bipartite.
The above results are similar to the corresponding result of integer additive setlabelled graphs (see [39]) and hence the notion of sparing number of graphs defined and studied in [48][49][50][51][52][53][54][55][56][57] can be extended to our current discussion also. The notion of the sparing number of graphs is defined as follows: Definition 6.2. Let G be a weakly modular sumset labelled graph. Then, the sparing number of G is the number of sparing edges in G.
A set of vertices X of a graph G is said to have maximal incidence if the maximum number of edges of incidence at the elements of X. Then, analogous to the corresponding result of integer additive set-valued graphs (see [40]), we have. Theorem 6.4. Let G be a weakly modular sumset labelled graph and I be I be the largest independent set of G with maximum incidence. Then, the sparing number of G is Recall that the degree of a vertex v, denoted by d v ð Þ, is equal to the number of edges incident on a vertex. Note that any vertex v i ∈ I can have a nonsingleton set-label which gives non-singleton set labels to d v i ð Þ edges incident on it. Since I is an independent set, the edges incident at the vertices in I assumes nonsingleton set-label. Therefore, the number of edges having non-singleton set-label incident at the vertices in I is Since I is a maximal independent set of that kind, the above expression counts the maximal non-sparing edges in G. Hence, the number of sparing edges in G is |E G ð Þ| À

Weakly modular sumset number of graphs
As a special case of the modular sumset number, the notion of weakly modular sumset number is introduced in [47] as follows: Definition 6.3. The weakly modular sumset number of a graph G, denoted by σ w is defined to be the minimum value of n such that a modular sumset labelling f : V G ð Þ ! P 0  n ð Þ is a weakly modular sumset labelling of G. The following theorem discussed the weak sumset number of an arbitrary graph G in terms of its covering and independence numbers. Theorem 6.5.
[47] Let G be a modular sumset graph and α and β be the covering number and independence number of G respectively. Then, the weak modular sumset number of G is max α, r f g, where r is the smallest positive integer such that 2 r À r À 1 ≥ β. [4]). Since G is a modular sumset graph, no two adjacent vertices can have non-singleton set-label simultaneously. Therefore, the maximum number of vertices that have non-singleton set-label is β. Let V 0 be the set of these independent vertices in G. Therefore, the minimum number of sparing vertices is |V G ð Þ| À β ¼ α. Since all these vertices in V À V 0 must have distinct singleton set-label, the ground set must have at least α elements.
Also, the number non-empty, non-singleton subsets of the ground set must be greater than or equal to α. Otherwise, all the vertices in V 0 cannot be labelled by non-singleton subsets of this ground set. We know that the number of non-empty, non-singleton subsets of a set A is 2 |A| À |A| À 1, where A ⊆  n is the ground set used for labelling. Therefore, the weak modular sumset number G is α if 2 α À α À 1 ≥ β. Otherwise, the ground set must have at least r elements such that 2 r À r À 1 ≥ β. Therefore, in this case, the weak modular sumset number of G is r, where r is the smallest positive integer such that 2 r À r À 1 ≥ β. Hence, σ * G ð Þ ¼ max α, r f g. This completes the proof.
□ The weakly modular sumset number some fundamental graph classes are given in Table 1.
The following theorem discusses the minimum cardinality of the ground set when the given graph G admits a weakly uniform modular sumset labelling.
[47] Let G be a weakly k-uniform modular sumset graph with covering number α and independence number β, where k < α. Then, the minimum cardinality of the ground set  n is max α, r f g, where r is the smallest positive integer such that r k ≥ β .
Proof. Let a weakly k-uniform modular sumset labelling be defined on a graph G over the ground set A ⊂  n . Then, by Corollary 6.3, G is bipartite. Let X, Y be the bipartition of the vertex set V G ð Þ. Without loss of generality, let |X| ≤ |Y|. Then, α ¼ |X| and β ¼ |Y|. Then, distinct elements of X must have distinct singleton setlabel. Therefore, n ≥ α.
On the other hand, since f is k-uniform, all the elements in Y must have distinct k-element set-label. The number of k-element subsets of a set A (obviously, with more than k elements) is |A| k . The ground set A has α elements only if α k ≥ β .
Otherwise, the ground set A must contain at least r elements, where r > α is the smallest positive integer such that r k ≥ β . Therefore, n ¼ max α, r f g. □ In view of the above theorem, the following result is immediate. Corollary 6.7. Let G be a weakly k-uniform modular sumset graph, where k ≥ α, where α is the covering number of G. Then, the minimum cardinality of the ground set  n is the smallest positive integer n such that n k ≥ β , where β is the independence number of G.
The following result explains a necessary and sufficient condition for a weak modular sumset labelling of a given graph G to be a maximal modular sumset labelling of G. Proposition 6.8.
[47] A weakly modular sumset labelling of a graph G is a supreme modular sumset labelling of G if and only if G is a star graph.
Proof. Let f be a weak modular sumset labelling of given graph G. First, assume that f is a maximal modular sumset labelling of G. Then, the set-labelling number of one end vertex of every edge of G is 1 and the set-labelling number of the other end vertex is n. Therefore,  n be the set-label of one end vertex of every edge of G, which is possible only if G is a star graph with the central vertex has the set-label  n and the pendant vertices of G have distinct singleton set-label.
Conversely, assume that G is a star graph. Label the central vertex of G by the ground set  n and label other vertices of G by distinct singleton subsets of  n . Then, all the edges of G has the set-labelling number n. That is, this labelling is a supreme modular sumset labelling of G. □

Conclusion
In this chapter, we have discussed certain types of modular sumset graphs and their structural properties and characterisations. These studies are based on the cardinality of the set-label of the elements of the graphs concerned and the patterns of the elements in these set-label. It is to be noted that several other possibilities can be investigated in this regard. For example, analogous to the topological setvaluations of graphs, the case when the collection of set-label of vertices and/or edges of a graph G forms a topology of the ground set  n can be studied in detail. Another possibility for future investigation is to extend the graceful and sequential concepts of set-labelling of graphs to modular sumset labelling also. All these points highlight the wide scope for further studies in this area.