On m-Polar Interval-valued Fuzzy Graph and its Application

In this paper, the concept of the -polar fuzzy graph ( -PFG) and interval-valued fuzzy graph (IVFG) is integrated and introduced an unprecedented kind of fuzzy graph designated as -polar interval-valued fuzzy graph ( -PIVFG). Complement of the -PIVFG is defined and the failure of this definition in some cases are highlighted. Various examples are cited and then redefined the notation of complement such that it applies to all -PIVFGs. The other algebraic properties such as isomorphism, weak isomorphism, co-weak isomorphism of the -PIVFG are investigated. Moreover, some basic results on the isomorphic property of -PIVFG are proved. Finally, an application of -PIVFG is explored. Abbreviations: The following abbreviations are employed in this study: FS: Fuzzy set; FG: Fuzzy graph; IVFS: Interval-valued fuzzy sets; IVFG: Interval-valued fuzzy graph; -PFS: -polar fuzzy sets; -PFG: -polar fuzzy graph; -PIVFS: -polar interval-valued fuzzy sets; -PIVFG: -polar interval-valued fuzzy graph.


Introduction
A graph is a mathematical structure used to represent pairwise relations between objects. It is defined as an ordered pair G = (V, E) consisting of a set of vertices, designated as V and a set of edges, denoted by E. When there is a vagueness either in vertices or in edges or in both then a fuzzy model is needed to describe a fuzzy graph. With the Konigsberg bridge problem, the graph theory was started in 1735. The concept was first introduced by Swiss Mathematician Euler in 1736. Then, Euler studied and incorporated a structure that solves the Konigsberg bridge problem which is also known as a Eulerian graph. Thereafter, the complete and bipartite graphs were proposed by Mobius in 1840. Recently, applications of graph theory are mostly promoted to the areas of computer networks, electrical networks, coding theory, operational research, architecture, data mining, etc.
Observing the vast application of graph theory motivated to explain fuzzy graph which is a non-empty set V together with a fuzzy set and a fuzzy relation. In 1973, Kauffman [1] defined fuzzy graph depending on the idea of fuzzy set introduced by Zadeh [2]. In 1975, Rosenfeld [3] first proposed another definition of the Fuzzy graph which is a generalization of Euler's Graph theory. He also elaborated definition of fuzzy vertex, fuzzy edges and several fuzzy concepts such as cycles, paths, connectedness, etc. The idea of isomorphism, weak isomorphism, co-weak isomorphism between fuzzy graphs was introduced by Bhutani [4] in 1989. The extension of the concept of fuzzy set and the idea of bipolar fuzzy sets were given in 1994 by Zhang [5,6]. Several properties of fuzzy graphs and hypergraphs were discussed by Mordeson and Nair [7][8][9] in 2000.
IVFG was defined by Hongmei and Lianhua [10] in 2009 and some operations on this were studied by Akram and Dudek [11] in 2011. Complete fuzzy graph was defined by Hawary [12]. He also studied three new operations on it. Nagoorgani and Malarvizhu [13,14] studied isomorphic properties on fuzzy graphs and also defined the self-complementary fuzzy graphs. The extension of bipolar fuzzy set and the idea of m-polar fuzzy sets (m-PFS) were introduced by Chen et al. [15] in 2014. Samanta and Pal [16][17][18][19] investigated on fuzzy tolerance graph, fuzzy threshold graph, fuzzy k-competition graphs, p-competition fuzzy graphs and also fuzzy planar graphs. Some properties of isomorphism and complement on IVFG were studied by Talebi and Rashmanlou [20]. Later, Ghorai and Pal [21,22] described various properties on m-PFGs. They examined isomorphic properties on m-PFG. Different types of research on generalized fuzzy graphs were discussed on [23][24][25][26][27][28][29][30][31]. The main contribution of this study is as follows: Some propositions and theorems related to this property are also discussed. Section 6 provides the application of an m-PIVFG in decision-making problems. Section 7 is based on a summary of this article.

Preliminaries
In this part, some definitions related to m-PFG are defined and demonstrated with the help of examples. The basic definition of IVFG is also discussed in this part, followed by an example for demonstration.
A fuzzy set is a set whose elements have degrees of membership. Fuzzy sets were introduced by Zadeh [2] in 1965 as an extension of the classical notion of the set. A fuzzy set A is a pair (S, m) where S is a set and m : S → [0, 1] is a membership function. Throughout this article, G * is a crisp graph, and G is a fuzzy graph.        Tables 3 and 4, respectively: In the following section the m-PIVFG, a combination of IVFG and m-PFG is defined.

m-polar Interval-valued Fuzzy Graph (m-PIVFG)
Herein   of the edge xy in G respectively satisfying   Similarly, we get the edges vw and wu. Hence, the graph G is complete, since for all the pair of vertices x, y ∈ V the conditions But, for the graph of Figure 4 is not complete. Here, } for all the edges xy ∈ E and for each i = 1, 2, . . . , m.
Above described example is an example of a strong m-PIVFG. Already, we have discussed that Figure 4 is not complete.   Example 5 The following Figure 5 is an example of an m-PIVFG while Figure 6 represents its complement. Let us consider a 3-PIVFG G = (V, A,

The complementḠ of G is
Similarly for others, Thus, we get . Figure 7),   The complementḠ ( Figure 8

Construction of complements we just stated by the above definition fails for some m-PIVFG. For further illustration, we consider the examples as follows. Example 6 Let us consider a 3-PIVFG
, which is not an interval. So, we can't construct this type of m-PIVFG.
Keeping in mind the limitations of definition 9 as demonstrated by example 5, we propose a new definition of the complement of m-PIVFG which is well defined given below.

DEFINITION 4.2: Let G = (V, A, B) be an m-PIVFG. Also let A and B represent min{p
for each i = 1, 2, . . . , m and for every x, y ∈ V.    Proof Let us consider an m-PIVFG is classic. Then,

DEFINITION 4.3: An m-PIVFG G = (V, A, B) of a crisp graph G * (V, E) is called classic m-PIVFG if all its m-pole of all its edge satisfy the condition min{p
, for each i = 1, 2, . . . , m and for every x, y ∈ V, i.e. for each edge this condition satisfies. Hence all of the edges are perfect. The proof of the converse part is straight forward.
In the next section, we study various types of isomorphic property of m-PIVFG with proper examples. Thereafter, we describe some propositions and theorems of m-PIVFG with the proofs.
Example 8 Here for any two 3-PIVFG, Consider a mapping φ : Since all the conditions of homomorphism are hold therefore, there exists a homomorphism φ : G 1 → G 2 (See Figures 10 and 11).
The following m-PIVFG depicted in Figures 12 and 13 show that there exists an isomorphism between them by the help of Definition 14.  We consider a homomorphism (See Figures 12 and 13) φ : G 1 → G 2 where the mapping φ : V 1 → V 2 satisfies the following criteria,
Again from the definition of complement for the complete graph, 1 and for each i = 1, 2, . . . , m. This implies that G 1 is isomorphic to G 2 . The proof of the converse part is the same as above. Here, we see G is isomorphic toḠ. Hence, G is self-complementary. A, B) is a complete m-PIVFG that is then G is self complementary (Figures 16 and 17).
Similarly, we can prove that p i • μ u B (xy) = p i • μ ū B (xy) for any xy ∈ E. Therefore, G is selfcomplementary.
Note 1 Let G = (V, A, B) of G * = (V, E) be a strong m-PIVFG. ThenḠ is a strong m-PIVFG if (xy). Hence the proof.

m, then the identity mapping I : V → V is an isomorphism from G toḠ . Clearly I satisfies the condition of vertices for isomorphism, that is, p i
And by the Theorem 2, ∀xy ∈ E and i = 1, 2, . . . , m, That imply I satisfies also the condition of edges for isomorphism. Therefore, G ∼ =Ḡ. That is G is self-complementary.
Conversely, let G 1 ∼ = G 2 , then there exists a bijective mapping φ : Case I: If xy ∈ E 1 and for each i = 1, 2, . . . , m, for each x ∈ V 1 and for each i = 1, 2, . . . , m, i.e. the weight of the nodes of the intervals are preserved but the weight of the edges are not necessarily preserved. We define a mapping φ :

Example 11 Let us consider any two 3-PIVFGs
Since all the conditions satisfied, thus, G 1 is weak-isomorphic to G 2 .
THEOREM 6: Let us consider a weak isomorphism φ : G →Ḡ, then for xy ∈ E, and for Taking summation both sides, x =y Similarly we can prove, Hence, the result.
. . , m, then G has a weak isomorphism φ from G to it's complementḠ. Let G = (V, A, B) be an m-PIVFG, satisfying
Example 12 Let us consider any two 3-PIVFGs Here, we define a mapping φ : Figures 18 and 19).
Hence, the result.

Application
Fuzzy graphs have many applications for problems concerning group structures, solving fuzzy intersection equations, etc. An m-PFG has applications in decision-making problems including co-operative games, medical diagnosis, signal processing, pattern recognition, robotics, database theory, expert systems and so on. Also, m-PIVFG is used in many decisionmaking problems. This happens when a democratic country elects its leader, a group of people decide which movie to watch when a company decides which product design to manufacturing, when a candidate for the winning trophy depending on their qualities that are voice tone, smoothness, confidence, facial expression, presentation. Suppose Judge 'a' is an expert of 'Sufi music', judge 'b' an expert of 'Ghazal music', judge 'c' an expert of 'folk music' and judge 'd' an expert of 'Indian filmy music'. By default, all the Judges have sufficient knowledge in 'Classical music'. For each candidate a judge from J can give marks in the form of interval value in [0,1] to x ∈ V; such as, Assuming Table 5 is constructed by the four Judges. The first column represents the performance marks of Aman given by four Judges. Similar to other columns. On the other hand first row represents the marks to all participants given by First Judge. From this table, one can construct a 5-PIVFG shown in Figure  The judges give marks to the singers by the following rule: Table 5. Marks given to each candidate.  Table 7. Rank given to each candidate. Marks = {(upper limit of the interval + lower limit of the interval)÷2}×100. Marks of each candidate (v i ) given by the judges are listed in following table. Then each pair of judges give rank (R i ) to all the candidates (v i ) according to their marks (Tables 6-8).
Depending on the performance of the competitions, each pair of judges prepared a panel for the candidates. Again, to find the combined rank of each candidate based on the rank of all judges we consider weights for a different rank. Suppose w i be the weights for the rank i. Obviously w i > w j for i < j. Thus the combined rank or say a score of a candidate is given by the formula s j = i × w i . Using this formula the score (s j ) of all five candidates are calculated below: Hence according to the final score, Bibhu get the first position, Karan gets the second position, Piu gets the third position, Survi gets the fourth position and Aman gets the fifth position. The determination of which singer to win the trophy is called the decision-making problem. Moreover, m-PIVFG has applications in different areas of computer science, neural intelligence, astronomy, autonomous system and industrial field and so on.

Conclusion and Future Research Direction
We have been seen that IVFG being viewed as a generalization of fuzzy graph and m-PFG also viewed as an extension of bi-polar fuzzy graph. In this study, we have been introduced the m-PIVFG, a generalization of IVFG and m-PFG, and its complements with examples. The definition of complement has been failed in some cases. Therefore, we have been modified the definition with examples. The definitions of homomorphism, isomorphism, weak isomorphism, co-weak isomorphism of m-PIVFG have been defined with proper given examples. Furthermore, we have been stated the complete m-PIVFG and strong m-PIVFG. In fact, some properties related to complements of complete m-PIVFG and strong m-PIVFG have been described. Thereafter, we also have been discussed few properties regarding self-complementary of m-PIVFG.
We should feature that regarding this investigation, there are distinctive developing regions that we need not demonstrate here as they are outside of our feasible region. In any case, there can be interesting points for future research; for example, one may examine the m-PIVFG with various kinds of environments [39], e.g. domination, Pythagorean, fuzzy soft graph [40][41][42][43][44], etc. In the future, we shall investigate other results of m-PIVFG and extend them to solve various problems of decision-making problems under different fuzzy environments.

Disclosure Statement
No potential conflict of interest was reported by the author(s).