Regularity and duality of intuitionistic fuzzy k -partite hypergraphs

: A graph in which the edge can connect more than two vertices is called a Hypergraph. A k -partite hypergraph is a hypergraph whose vertices can be split into k different independent sets. In this paper, regular, totally regular, totally irregular, totally neighborly irregular Intuitionistic Fuzzy k -Partite Hypergraphs (IF k -PHGs) are defined. Also order and size along with the properties of regular and totally regular IF k -PHGs are discussed. It has been proved that the size S ( H ) of a r -regular IF k -PHG is tr 2 where t = | V | . The dual IF k -PHG has also been defined with example.


Introduction
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.The history of graph theory may be specifically traced

Preliminaries
Basic definitions relating to intuitionistic fuzzy set, intuitionistic fuzzy hypergraph and intuitionistic fuzzy k-partite hypergraph are discussed in this section.

Definition 2.1 ( [1]
).Let a set E be fixed.An intuitionistic fuzzy set (IFS) V in E is an object of the form V = {⟨v i , µ i (v i ), ν i (v i )⟩ |v i ∈ E}, where the function µ i : E → [0, 1] and ν i : E → [0, 1] determine the degree of membership and the degree of non-membership of the element v i ∈ E, respectively and for every v i ∈ E, 0 ≤ µ i (v i ) + ν i (v i ) ≤ 1.

Definition 2.2 ( [8]
).An intuitionistic fuzzy hypergraph (IFHG) is an ordered pair H = ⟨V, E⟩ where (i) V = {v 1 , v 2 , . . ., v n }, is a finite set of intuitionistic fuzzy vertices (ii Here, the hyperedges E j are crisp sets of intuitionistic fuzzy vertices, µ j (v i ) and ν j (v i ) denote the degrees of membership and non-membership of vertex v i to edge E j .Thus, the elements of the incidence matrix of IFHG are of the form ⟨v ij , µ j (v i ), ν j (v j )⟩.The sets (V, E) are crisp sets.Definition 2.3 ([5]).The IFk-PHG is an ordered triple H where Definition 2.5 ([6]).The order of an IFk-PHG, Definition 2.6 ([6]).The size of an IFk-PHG, H = (V, E, ψ) is defined to be Definition 2.7 ([6]).The degree of a vertex v in an IFk-PHG, H is denoted by d H (v) and defined by d Note: The degree of each vertex in a k-partite hyperedge is nothing but the membership and non-membership values of the corresponding k-partite hyperedge.
Example 1.For an IFk-PHG H, let Then Example 2. For an intuitionistic fuzzy k-partite hypergraph H, define Example 3. From the above example, it is known that the vertex v 3 has the closed neighborhood degree of ⟨1.6, 0.8⟩.Definition 3.3.The total degree of a vertex v in an intuitionistic fuzzy k-partite hypergraph H, denoted by td H (v) is defined as td

Example 4.
From Example 1, it is clear that the total degree of each of the vertex is ) be an IFk-PHG.If all the vertices in V have the same degree r, then H is said to be an r-regular intuitionistic fuzzy k-partite hypergraph.
Definition 3.5.Let H = (V, E, ψ) be an IFk-PHG.If all the vertices in V have the same total degree s, then H is said to be an s-totally regular intuitionistic fuzzy k-partite hypergraph.
Definition 3.6.If H = (V, E, ψ) is both r-regular and s-totally regular IFk-PHG, then it is called perfectly regular IFk-PHG.
Example 5. Consider an IFk-PHG, Proof.Let H be a regular IFk-PHG and µ k i , ν k i be a constant function, then Hence, H is a totally regular IFk-PHG.

Theorem 3.2. Let H be an IFk-PHG. If H is a totally regular IFk-PHG and µ k
Proof.Let H be a totally regular IFk-PHG and ⟨µ k i , ν k i ⟩ be a constant function, then Remark 1.The converse of the above theorem is also true.Consider Figure 1, in which Hence, H is both regular and totally regular IFk-PHG.Definition 3.7.An IFk-PHG, H is said to be totally irregular if there exists a vertex which is adjacent to the vertices with distinct td H (v), i.e., total degree.Definition 3.8.If the total degree, i.e., td H (v) of every pair of adjacent vertices of H are distinct then H = (V, E, ψ) is said to be totally neighborly irregular.Example 6. Assume an IFk-PHG, H as the one shown below.
Theorem 3.4.The sufficient condition for a regular and totally regular IFk-PHG to be perfectly regular IFk-PHG is that each vertex of ψ i , i = 1, 2, . . ., k is linked through an hyperedge.
Proof.Assume that H is both regular and totally regular IFk-PHG and each vertex of As each vertex of ψ i , i = 1, 2, . . ., k is linked through the hyperedge, which means that all vertices of ψ i , i = 1, 2, . . ., k are adjacent.
From the above results we have d Hence, the degree and total degree of all the vertices of ψ i are the same.So, H is perfectly regular IFk-PHG.
Theorem 3.5.The size S(H) of a r-regular IFk-PHG is tr Proof.Since H is an s-totally regular IFk-PHG, This implies that from equation (3.1) and from Definition 2.5, ts = 2S(H) + O(H) which completes the proof of the theorem.Definition 3.9.An IFk-PHG H is defined to be (l, m)-uniform if all the k-partite hyperedges have same cardinality, i.e., |supp(µ k i j , ν k i j )| = (l, m).The IFk-PHG can be represented by the following incidence matrix: The dual IFk-PHG H * is shown in Figure 5.

3 Regular and irregular intuitionistic fuzzy k-partite hypergraphs Definition 3 . 1 .
The open neighborhood degree d

Fig. 4 .
Fig. 4. IFk-PHG 2, . . ., k and no two vertices in the same set are adjacent such that E k =

Theorem 3.1. Let
8, 0.6⟩.Hence, H is a ⟨0.8, 0.6⟩-totally regular intuitionistic fuzzy k-partite hypergraph.The above IFk-PHG, H is both regular and totally regular, hence it is called perfectly regular IFk-PHG.H be an intuitionistic fuzzy k-partite hypergraph.If H is a regular IFk-PHG and µ k Hence H is a regular IFk-PHG.If H is both regular and totally regular IFk-PHG, then ⟨µ k i , ν k i ⟩ is a constant function.Proof.Let H be both regular and totally regular IFk-PHG, then d µ