Some isomorphic properties of m-polar fuzzy graphs with applications.

The theory of graphs are very useful tool in solving the combinatorial problems in different areas of computer science and computational intelligence systems. In this paper, we present a frame work to handle m-polar fuzzy information by combining the theory of m-polar fuzzy sets with graphs. We introduce the notion of weak self complement m-polar fuzzy graphs and establish a necessary condition for m-polar fuzzy graph to be weak self complement. Some properties of self complement and weak self complement m-polar fuzzy graphs are discussed. The order, size, busy vertices and free vertices of an m-polar fuzzy graphs are also defined and proved that isomorphic m-polar fuzzy graphs have same order, size and degree. Also, we have presented some results of busy vertices in isomorphic and weak isomorphic m-polar fuzzy graphs. Finally, a relative study of complement and operations on m-polar fuzzy graphs have been made. Applications of m-polar fuzzy graph are also given at the end.

Background

After the introduction of fuzzy sets by Zadeh (1965), fuzzy set theory have been included in many research fields. Since then, the theory of fuzzy sets has become a vigorous area of research in different disciplines including medical and life sciences, management sciences, social sciences engineering, statistic, graph theory, artificial intelligence, signal processing, multi agent systems, decision making and automata theory. In a fuzzy set, each element is associated with a membership value selected from the interval [0, 1]. Zhang (1994, 1998) introduced the concept of bipolar fuzzy sets. Instead of using particular membership value as in fuzzy sets, m-polar fuzzy set can be used to represent uncertainty of a set more perfectly. Chen et al. (2014) introduced the notion of m-polar fuzzy set as a generalization of fuzzy set theory. The membership value in m-polar fuzzy set is more expressive in capturing uncertainty of data.

An m-polar fuzzy set on a non-void set X is a mapping . The idea behind this is that “multipolar information” exists because data of real world problems are sometimes come from multiple agents. m-polar fuzzy sets allow more graphical representation of vague data, which facilitates significantly better analysis in data relationships, incompleteness, and similarity measures. Graph theory besides being a well developed branch of Mathematics, it is an important tool for mathematical modeling. Realizing the importance, Rosenfeld (1975) introduced the concept of fuzzy graphs, Mordeson and Nair (2000) discussed about the properties of fuzzy graphs and hypergraphs. After that, the operation of union, join, Cartesian product and composition on two fuzzy graphs was defined by Mordeson and Peng (1994). Sunitha and Vijayakumar (2002) further studied the other properties of fuzzy graphs. The concept of weak isomorphism, co-weak isomorphism and isomorphism between fuzzy graphs was introduced by Bhutani (1989). Later many researchers have worked on fuzzy graphs like in Bhutani et al. (2004); Al-Hawary (2011); Koczy (1992); Lee-kwang and Lee (1995); Nagoorgani and Radha (2008), Samanta and Pal (2011a, b, 2013, 2014, 2015). Akram (2011, 2013) introduced and defined different operations on bipolar fuzzy graphs. Again, Rashmanlou et al. (2015a, 2015b, 2016) studied bipolar fuzzy graphs with categorical properties, product of bipolar fuzzy graphs and their degrees, etc. Using these concepts many research is going on till date on bipolar fuzzy graphs such as Ghorai and Pal (2015b), Samanta and Pal (2012a, b, 2014), Yang et al. (2013). Chen et al. (2014) first introduced the concept of m-polar fuzzy graphs. Then Ghorai and Pal (2016a) presented properties of generalized m-polar fuzzy graphs, defined many operations and density of m-polar fuzzy graphs (2015a), introduced the concept of m-polar fuzzy planar graphs (2016b) and defined faces and dual of m-polar fuzzy planar graphs (2016c). Akram and Younas (2015), Akram et al. (2016) introduced irregular m-polar fuzzy graphs and metrics in m-polar fuzzy graphs. In this paper, weak self complement m-polar fuzzy graphs is defined and a necessary condition is mentioned for an m-polar fuzzy graph to be weak self complement. Some properties of self complement and weak self complement m-polar fuzzy graphs are discussed. The order, size, busy vertices and free vertices of an m-polar fuzzy graphs are also defined and proved that isomorphic m-polar fuzzy graphs have same order, size and degree. Also, we have proved some results of busy vertices in isomorphic and weak isomorphic m-polar fuzzy graphs. Finally, a relative study of complement and operations on m-polar fuzzy graphs have been made.

Preliminaries

First of all we give the definitions of m-polar fuzzy sets, m-polar fuzzy graphs and other related definitions from the references (Al-Harary 1972; Lee 2000).

Throughout the paper, (m-power of [0, 1]) is considered to be a poset with point-wise order , where m is a natural number. is defined by for each , where and is the ith projection mapping.

As a generalization of bipolar fuzzy sets, Chen et al. (2014) defined the m-polar fuzzy sets in 2014.

Definition 1

(Chen et al. 2014) Let X be a non-void set. An m-polar fuzzy set on X is defined as a mapping .

The m-polar fuzzy relation is defined below.

Definition 2

(Ghorai and Pal 2016a) Let A be an m-polar fuzzy set on a set X. An m-polar fuzzy relation on A is an m-polar fuzzy set B of such that for all , . B is called symmetric if for all .

We define an equivalence relation on as follows:

We say if and only if either or and . Then we obtain an quotient set denoted by . The equivalence class containing the element (x, y) will be denoted as xy or yx.

We assume that is a crisp graph and is an m-polar fuzzy graph of throughout this paper.

Chen et al. (2014) first introduced m-polar fuzzy graph. We have modified their definition and introduce generalized m-polar fuzzy graph as follows.

Definition 3

(Chen et al. 2014; Ghorai and Pal 2016a) An m-polar fuzzy graph (or generalized m-polar fuzzy graph) of is a pair where is an m-polar fuzzy set in V and is an m-polar fuzzy set in such that for all , and for all , is the smallest element in . We call A as the m-polar fuzzy vertex set of G and B as the m-polar fuzzy edge set of G.

Example 4

Let be a crisp graph where and . Then, be a 3-polar fuzzy graph of where and .

Ghorai and Pal (2016a) introduced many operations on m-polar fuzzy graphs such as Cartesian product, composition, union and join which are given below.

Definition 5

(Ghorai and Pal 2016a) The Cartesian product of two m-polar fuzzy graphs and of the graphs and respectively is denoted as a pair such that for

for all .

for all , .

for all , .

for all .

Definition 6

(Ghorai and Pal 2016a) The composition of two m-polar fuzzy graphs and of the graphs and respectively is denoted as a pair such that for

for all .

for all , .

for all , .

for all .

for all .

Definition 7

(Ghorai and Pal 2016a) The union of the m-polar fuzzy graphs and of and respectively is defined as follows: for

if .

Definition 8

(Ghorai and Pal 2016a) The join of the m-polar fuzzy graphs and of and respectively is defined as a pair such that for

if .

if .

if , where denotes the set of all edges joining the vertices of and .

if .

Remark 9

Later on, Akram et al. (2016) applied the concept of m-polar fuzzy sets on graph structure and also defined the above operations on them.

Different types of morphism are defined on m-polar fuzzy graphs by Ghorai and Pal (2016a).

Definition 10

(Ghorai and Pal 2016a) Let and be two m-polar fuzzy graphs of the graphs and respectively. A homomorphism between and is a mapping such that for each is said to be an isomorphism if it is a bijective mapping and for In this case, we write .

for all ,

for all .

for all ,

for all .

Definition 11

(Ghorai and Pal 2016a) A weak isomorphism between and is a bijective mapping such that

is a homomorphism,

for all , for each .

Definition 12

(Ghorai and Pal 2016a) is called strong if for all , .

A strong m-polar fuzzy graph G is called self complementary if .

Degree of a vertex in an m-polar fuzzy graph is defined as below.

Definition 13

(Akram and Younas 2015) The neighborhood degree of a vertex v in the m-polar fuzzy graph G is denoted as where , .

Remark 14

If and are two m-polar fuzzy graphs. Then the canonical projection maps and are indeed homomorphisms from to and to respectively. This can be seen as follows:

for all and for all and . In a similar way we can check the other conditions also. This shows that the canonical projection maps is a homomorphism from to .

Weak self complement m-polar fuzzy graphs

Self complement m-polar fuzzy graphs have many important significant in the theory of m-polar fuzzy graphs. If an m-polar fuzzy graph is not self complement then also we can say that it is self complement in some weaker sense. Simultaneously we can establish some results with this graph. This motivates to define weak self complement m-polar fuzzy graphs.

Definition 15

Let be an m-polar fuzzy graph of the crisp graph . The complement of G is an m-polar fuzzy graph of such that and is defined by for , .

Example 16

Let be a 3-polar fuzzy graph of the graph where , , , . Then by Definition 15, we have constructed the complement of G which is shown in Fig. 1.

Fig. 1

G and it’s complement

Remark 17

Let be the complement of where andHence, .

Definition 18

The m-polar fuzzy graph is said to be weak self complement if there is a weak isomorphism from G onto . In other words, there exist a bijective homomorphism such that for

for all ,

for all .

Example 19

Let be a 3-polar fuzzy graph of the graph where , , , . Then is also a 3-polar fuzzy graph where and . We can easily verify that, the identity map is an weak isomorphism from G onto (see Fig. 2). Hence G is weak self complement.

Fig. 2

Weak self complement 3-polar fuzzy graphs

In Ghorai and Pal (2015a), Ghorai and Pal proved that if G is a self complementary strong m-polar fuzzy graph then for all and The converse of the above result does not hold always.

Example 20

For example, let us consider a 3-polar fuzzy graph of where , , , . Then we have the followingSo,Similarly,andHence for we have,But G is not self complementary as there exists no isomorphism from G onto (see Fig. 3).

Fig. 3

Example of 3-polar fuzzy graph G which is not self complement

Now suppose an m-polar fuzzy graph is a weak self complement. Then the following inequality holds.

Theorem 21

Let be a weak self complement m-polar fuzzy graph of . Then for

Proof

Since G is weak self complement, therefore there exists a weak isomorphism such that for all and for all , .

Using the above we have,Therefore, for all , i.e.,i.e.,

Remark 22

The converse of the above theorem is not true in general. For example, consider the 3-polar fuzzy graph of Fig. 3. We see that for the 3-polar fuzzy graph G, the condition of Theorem 21 is satisfied. But, G is not weak self complementary as there is no weak isomorphism from G onto .

Theorem 23

If for all , then G is a weak self complement m-polar fuzzy graph.

Let be the complement of G where for all and for , .

Let us now consider the identity map . Then for all andSo, for and . Hence, is a weak isomorphism.

Example 24

Consider the 3-polar fuzzy graph of where , , , . We see that for each and , .

Also, consider the complement of G of Fig. 4. Let us now consider the identity mapping such that for all . Then, I is the required weak isomorphism from G onto . Hence, G is weak self complementary.

Fig. 4

Example of 3-polar fuzzy graph G which is weak self complement

Order, size and busy value of vertices of m-polar fuzzy graphs

In this section, the order, size, busy value of vertices of an m-polar fuzzy graph is defined.

Definition 25

The order of the m-polar fuzzy graph is denoted by |V| (or O(G)) whereThe size of G is denoted by |E| (or S(G)) where

Theorem 26

Two isomorphic m-polar fuzzy graphs and of the graphs and have same order and size.

Let be an isomorphism from onto . Then for all and for , .

Now,and

Definition 27

The busy value of a vertex u of an m-polar fuzzy graph G is denoted as where ; are the neighbors of u. The busy value of G is denoted as D(G) where , .

Example 28

Consider the 3-polar fuzzy graph of where , , and . Then we have from Fig. 5,So, .

Fig. 5

3-Polar fuzzy graph G and busy value of its vertices

Definition 29

If for , then the vertex u of G is called a busy vertex. Otherwise it is a free vertex.

Definition 30

If , for , then it is called an effective edge of G.

Definition 31

Let be a vertex of the m-polar fuzzy graph .

u is called a partial free vertex if it is a free vertex of G and .

u is called a fully free vertex if it is a free vertex of G and it is a busy vertex of .

u is called a partial busy vertex if it is a busy vertex of G and .

u is called a fully busy vertex if it is a busy vertex in G and it is a free vertex of .

Theorem 32

Let be an isomorphism from onto . Then for all .

Since is an isomorphism between and , we have for all and for all , .

Hence, for , . So, for all .

Theorem 33

If is an isomorphism from onto and u is a busy vertex of , then is a busy vertex of .

Since is an isomorphism between we have, and for , .

If u is a busy vertex of , then for . Then by the above and Theorem 32, for . Hence, is a busy vertex in .

Theorem 34

Let the two m-polar fuzzy graphs and be weak isomorphic. If is a busy vertex of , then the image of u under the weak isomorphism is also busy in .

Let be a weak isomorphism between and .

Then, for all and for all , .

Let be a busy vertex. Then, for , .

Now by the above for Hence, is a busy vertex in .

Complement and isomorphism in m-polar fuzzy graphs

In this section some important properties of isomorphism, weak isomorphism, co weak isomorphism related with complement are discussed.

Theorem 35

Let and be two m-polar fuzzy graphs of the graphs and . If then .

Let . Then there exists an isomorphism such that for all and , for each and .

Now, for all .

Also, for and we have,Hence, is an isomorphism between and i.e., .

Remark 36

Suppose there is a weak isomorphism between two m-polar fuzzy graphs and . Then there may not be a weak isomorphism between and .

For example, consider two 3-polar fuzzy graphs and of Fig. 6. Let us now define a mapping such that , , . Then is a weak isomorphism from onto . But, there is no weak isomorphism from onto (see Fig. 7) because , and .

Fig. 6

Weak isomorphic 3-polar fuzzy graphs and

Fig. 7

Example of weak isomorphic graphs whose complement is not weak isomorphic

Remark 37

In a similar way, we can construct example to show that if there is a co-weak isomorphism between two m-polar fuzzy graphs and then there may not be a co-weak isomorphism between and .

Theorem 38

Let and be two m-polar fuzzy graphs of the graphs and such that . Then .

To show that , we need to show that there exists an isomorphism between and .

We will show that the identity map is the required isomorphism between them. For this, we will show the following:

for all , ,

and for and .

Let .

ThenNow for each and we have,

Theorem 39

Let and be two m-polar fuzzy graphs of the graphs and such that . Then .

Consider the identity map . We will show that I is the required isomorphism between and .

For this, we will show the following:

for all , ,

and for and .

Let .

Thenand for , we have,This completes the proof.

Theorem 40

Let and be two strong m-polar fuzzy graphs of the graphs and respectively. Then .

Let be an m-polar fuzzy graph of the graph where and .

We show that the identity map I is the required isomorphism between the graphs and . Let us consider the identity map .

In order to show that I is the required isomorphism, we show that for each and for all , . Several cases may arise.

Case (i): Let where , . Then .

Since is strong m-polar fuzzy graph, we have for each (since is strong and , therefore for each , ).

Case (ii): Let where , . Then .

So for each , andAgain, since , therefore for each ,

Case (iii): Let where , .

Then . So for each , as in Case (i).

Also, since , therefore for each , .

Case (iv): Let where , . Then .

Hence for each , ,

Case (v): Let where , . Then . So we have for each , as in Case (i).

Also, since , we have for each , .

Case (vi): Let where , . Then and hence for each , ,and since ,

Case (vii): Finally, let where , . Then and hence for each , ,Now, and if , then we have the Case (iv).

Again, if and if , then we have Case (vi).

Thus combining all the cases we have, for each , and ,

Remark 41

If and are not strong, then always. For example, consider the two 3-polar fuzzy graphs and which are not strong (see Fig. 8). From Figs. 8 and 9, we see that, .

Fig. 8

, , and

Fig. 9

Example of 3-polar fuzzy graphs and where

Applications

Now a days, fuzzy graphs and bipolar fuzzy graphs are most familiar graphs to us and they can also be thought of as 1-polar and 2-polar fuzzy graphs respectively. These graphs have many important application in social networks, medical diagnosis, computer networks, database theory, expert system, neural networks, artificial intelligence, signal processing, pattern recognition, engineering science, cluster analysis, etc. The concepts of bipolar fuzzy graphs can be generalized to m-polar fuzzy graphs. For example, consider the sorting of mangoes and guavas. Now the different characteristics of a given fruit can change the decision in sorting process more towards the decision mango or vice versa. There are two poles present in this case. One is sure mango and the other is sure guava. This shows that the situation is bipolar. This situation can be generalized further by adding a new fruit, for example sweet lemon into the sorting process.

Graphical representation of tug of war

Consider the another example of tug of war where two people pull the rope in opposite directions. Here, who uses the bigger force, the center of the rope will move in the respective direction of their pulling. The situation is symmetric in this case. We present an example where m people pull a special rope in m different directions. We use this example to represent it as an m-polar fuzzy graph. We assume that O is the origin and there are m straight paths leading from O. We also assume that there is a wall in between these paths. In this setting, we have the special rope with one node at O and m endings going out from this nodes—one end corresponding to each of the paths. Suppose on every path there is a man standing and pulling the rope in the direction of the path on which he is standing. This situation can be represented as an m-polar fuzzy graph by considering the nodes as m-polar fuzzy set and edges between them as m-polar fuzzy relations, which is shown in Fig. 10. In this context, one can ask the question what is the strength require in order to pull the node O from the center into one of the paths (assuming no friction)? The answer to this is that if the corresponding forces which are pulling the rope are , , then the node O will move to the path if .

Fig. 10

Graphical representation of tug of war

Evaluation graph corresponding to the teacher’s evaluation by the students

In this section we present the model of m-polar fuzzy graph which is used in evaluating the teachers by the students of 4th semester of a department in an university during the session 2015–2016. Here the nodes represent the teachers of the corresponding department and edges represent the relationship between two teachers. Suppose the department has six teachers denoted as . The membership value of each node represents the corresponding teachers feedback response of the students depending on the following: {regularity of classes, style of presentation, quality of lectures, generation of interest and encouraging future reading among students, updated information}. Since all the above characteristics of a teacher according to the different students are uncertain in real life, therefore we consider 5-polar fuzzy subset of the vertex set T (Fig. 11).

Fig. 11

5-Polar fuzzy evaluation graph corresponding to the teacher’s evaluation by students

In the Table 1, the membership values of the teacher’s are given which is according to the evaluation of the students.

Table 1

5-Polar fuzzy set A of T

	t_1	t_2	t_3	t_4	t_5	t_6
p_1∘A	0.6	0.7	0.8	0.8	0.8	0.7
p_2∘A	0.7	0.6	0.9	0.7	0.9	0.8
p_3∘A	0.8	0.7	0.7	0.8	0.7	0.9
p_4∘A	0.9	0.8	0.8	0.9	0.7	0.7
p_5∘A	0.9	0.8	0.9	0.8	0.8	0.8

Edge membership values which represent the relationship between the teachers can be calculated by using the relation for all , . These values are given in the Table 2.

Table 2

5-Polar fuzzy relation B on A

	t_1t_2	t_1t_5	t_1t_6	t_2t_3	t_2t_4	t_2t_5	t_3t_4	t_3t_5	t_4t_5	t_4t_6	t_5t_6
p_1∘A	0.6	0.5	0.6	0.6	0.8	0.7	0.8	0.7	0.8	0.6	0.7
p_2∘A	0.6	0.7	0.7	0.6	0.7	0.5	0.7	0.5	0.7	0.7	0.8
p_3∘A	0.7	0.7	0.8	0.6	0.6	0.6	0.7	0.7	0.7	0.7	0.6
p_4∘A	0.8	0.6	0.7	0.7	0.6	0.7	0.8	0.7	0.7	0.7	0.6
p_5∘A	0.7	0.8	0.8	0.8	0.8	0.6	0.8	0.8	0.8	0.7	0.7

We rank the teacher’s performance according the following:From the Table 3, we see that the performance of the teachers are very good whereas the performance of the teachers and are excellent. Among these teachers, teacher is the best teacher according the response score of the students of the department during the session 2015–2016.

Table 3

Average response score of the teachers

	Teachers
	t_1	t_2	t_3	t_4	t_5	t_6
Scores	0.78	0.72	0.82	0.8	0.78	0.78

Teacher’s average response score <60%, teacher’s performance according to the students is .

Teacher’s average response score ≥60% and <70%, teacher’s performance according to the students is .

Teacher’s average response score ≥70% and <80%, teacher’s performance according to the students is .

Teacher’s average response score is ≥80%, teacher’s performance according to the students is .

Conclusions

The theory of fuzzy graphs play an important role in many fields including decision makings, computer networking and management sciences. An m-polar fuzzy graph can be used to represent real world problems which involve multi-agent, multi-attribute, multi-object, multi-index, multi-polar information and uncertainty. In this research paper, we have studied the isomorphic properties of m-polar fuzzy graphs with some applications. We are extending our research work on m-polar fuzzy intersection graphs, m-polar fuzzy interval graphs, properties of m-polar fuzzy hypergraphs, degrees of vertices of m-polar fuzzy graphs and its application in decision making, etc.