Complement of max product of intuitionistic fuzzy graphs

In this paper, the complement of max product of two intuitionistic fuzzy graphs is defined. The degree of a vertex in the complement of max product of intuitionistic fuzzy graph is studied. Some results on complement of max product of two regular intuitionistic fuzzy graphs are stated and proved. Finally, we provide an application of intuitionistic fuzzy graphs in school determination using normalized Hamming distance.


Introduction
Graph theory has been considered to play an important role when it comes to its application in dealing with real-life situations. The fuzzy graph theory has its own significance as application of fuzzy set theory has no limits. In 1975, Rosenfeld [9] introduced the concept of fuzzy graphs. Yeh and Bang [22] also introduced fuzzy graphs independently. Fuzzy graphs are useful to represent relationships which deal with uncertainty and it differs greatly from classical graphs. It has numerous applications to problems in computer science, electrical engineering, system analysis, operations research, economics, networking routing, transportation, etc. After Rosenfeld [9], the fuzzy graph theory increases with its various types of branches, such as fuzzy tolerance graph [13], fuzzy threshold graph [12], bipolar fuzzy graphs [7,8], balanced interval-valued fuzzy graphs [4,6], fuzzy planar graphs [11], etc. Also, several works have been done on fuzzy graphs by Samanta and Pal [14].
Atanassov [1,2] developed the concept of intuitionistic fuzzy set (IFS) as an extension of fuzzy set that [23] deals with uncertain situations in a better way as its structure is not limited to membership grades only. The concept of intuitionistic fuzzy sets is a better tool to use due to its diverse structure describing membership as well as non-membership grades of an element. The theory of intuitionistic fuzzy sets has been remarkably used in some areas so far. Shannon and Atanassov [15] introduced the concept of intuitionistic fuzzy graphs in 1994. Parvathi and Karunambigai [5] gave a definition for intuitionistic fuzzy graph as a special case of intuitionistic fuzzy graphs defined by Shannon and Atanassov [16]. Sankar Sahoo and Madhumangal Pal [10] defined three types of products, namely direct product, semistrong product, and strong product. Yaqoob et al. [21] discussed the four basic operations, namely Cartesian product, composition, union, and join of complex intuitionistic fuzzy graphs. Yahya Mohamed and Mohamed Ali [19,20] defined modular and max product on intuitionistic fuzzy graph. In this paper, the complement of max product of two intuitionistic fuzzy graphs and the degree of a vertex in the complement of this product are studied under regularity conditions. The max product of intuitionistic fuzzy graphs is applied to solve the decision-making problem in school determination. The research paper is organized as follows: "Introduction" presents the literature review of fuzzy graphs and intuitionistic fuzzy graphs. In "Preliminaries", we have provided some basic concepts of intuitionistic fuzzy graphs. The definition of complement of max product on two intuitionistic fuzzy graphs and its degree are given in "Complement of max product of intuitionistic fuzzy graphs". In "Applications of max product of intuitionistic fuzzy graphs in school determination", we studied an application of intuitionistic fuzzy graphs in school determination using normalized hamming distance. This distance function was used to measure the distance between each student and each school. The schools in which each of the students has been enrolled were determined using normalized hamming distance function based on examination that is performed for transition to high school education. The result is determined by calculating the smallest distance between each student and each school. In "Conclusion", we conclude present studies and recommendations for future studies.

Preliminaries
Throughout this paper, assume that G * = (V, E) is a crisp graph and G is an intuitionistic fuzzy graph, where V is a non-empty vertex set and E is an edge set.
Definition 2.1 [5] An intuitionistic fuzzy graph is of the form G = 1 , 2 , 1 , 2 on G * = (V, E) and denote the degree of membership and non-membership of the element x i ∈ V respectively, such that 0 where   1 x i , x j and 2 x i , x j denote the degree of membership and degree of non-membership values of the edge x i , x j , respectively, such that 1 x i , x j ≤ 1 x i ∧ 1 x j and For notational convenience, instead of representing an edge as (x, y) , we denote this simply by xy. Definition 2.2 [5] An intuitionistic fuzzy graph G = 1 , 2 , 1 , 2 is called strong intuitionistic fuzzy graph if

Definition 2.3 [5] An intuitionistic fuzzy graph
Definition 2.4 [3] Let G = 1 , 2 , 1 , 2 be an intuitionistic fuzzy graph, and then, the order of G is defined to Definition 2.5 [3] Let G = 1 , 2 , 1 , 2 be an intuitionistic fuzzy graph, then the size of G is defined to be Definition 2.6 [5] The complement of an intuitionistic fuzzy graph G = (V, E) is an intuitionistic fuzzy graph Ḡ = 1 , 2 , 1 , 2 , where 1 , 2 = 1 , 2 and Example 3.2 Consider the two intuitionistic fuzzy graphs as shown in Figs. 1 and 2 and their corresponding max product G 1 × m G 2 shown in Fig. 3. Then, the complement of max product of G 1 and G 2 is shown in Fig. 4. Theorem 3.1 Let G 1 and G 2 be two regular intuitionistic fuzzy graphs. If underlying crisp graphs G * 1 and G * 2 are complete graphs and Then, complement of max product of two intuitionistic fuzzy graphs G 1 and G 2 is regular intuitionistic fuzzy graph.
Proof: Let G 1 and G 2 be two regular intuitionistic fuzzy graphs. The underlying crisp graphs G * 1 and G * 2 are complete graphs of degrees d 1 and d 2 for every vertices of V 1 and V 2 .
Given that By theorem, max product of two regular intuitionistic fuzzy graphs is regular intuitionistic fuzzy graph.
By Eq. (6) Since by the definition of max product of two intuitionistic fuzzy graphs

Similarly
Since G 1 and G 2 are two regular intuitionistic fuzzy graphs & G * 1 and G * 2 are complete graphs, then are constants say e 1 , e 2 and e 3 , e 4 .

Similarly
Hence, complement of max product of two regular intuitionistic fuzzy graphs is regular. Theorem 3.2 Let G 1 and G 2 be two regular intuitionistic fuzzy graphs of the underlying crisp graphs; G * 1 and G * 2 are regular graphs with the vertex sets; and edge sets of G 1 and G 2 are different constants which satisfies Then, complement of the max product of two regular intuitionistic fuzzy graphs G 1 and G 2 is regular intuitionistic fuzzy graph.
Proof: Let G 1 and G 2 be two regular intuitionistic fuzzy graphs. The underlying crisp graphs G * 1 and G * 2 are regular graphs of degrees g 1 and g 2 for every vertices in V 1 and V 2 .
Given that G 1 , G 2 , G 1 and G 2 are constants, say

Case (i) : If
Applying the fact that shortest distance between two intuitionistic fuzzy sets shows more similarity between them, therefore, it can be described that for the student q i is to be enroll in the school corresponding to min The relation between school points and subject papers and the relation between students and their corresponding average entrance marks are given in Tables 3 and 4, respectively.
In the following table, we have used intuitionistic fuzzy sets as a tool, since it incorporates the membership grades (the average marks of the questions that have been correctly answered by the student) and the non-membership grades (the average marks of the questions that have been incorrectly answered by the student). The max product of G 1 and G 2 is shown in Fig. 7.
The following decision matrix has been obtained by finding distance between each student (Table 3) and each school ( From the above decision matrix, less distance between the student and school implies more possibility to get enrollment in the corresponding school. Therefore, the student Ali is to enroll in the school S 1 , the student Mathew is to enroll in the school S 4 , the student Sunil is to enroll in the school S 1 , the student Yusuf is to enroll in the school S 3 , and the student Joseph is to enroll in the school S 1 .

Conclusion
Graph theory has numerous applications in solving various networking problems encountered in different fields such as signal processing, transportation, and error codes. In particularly, the shortest path problem is a well-known combinatorial optimization problem in graph theory. Intuitionistic fuzzy graph models are more practical and useful than fuzzy graph models as it provides membership and nonmembership grades for representing imprecise information which occur in real-life situations. This paper has introduced the complement of max product of two intuitionistic fuzzy graphs. Using the max product, the different types of structural models can be combined to produce a better one. The special attention on the regularity in the complement of two intuitionistic fuzzy graphs has been given as it can be applied widely in designing reliable communication and network systems. Finally, an application of intuitionistic fuzzy graphs in decision-making concern the school determination for the students based on their entrance score has been presented. In future, we are going to extend our work to: (1) Pythagorean fuzzy graphs; (2) Interval-valued Pythagorean fuzzy graphs, and (3) Spherical fuzzy graphs.