Intuitionistic fuzzy graphs of n th type with applications

Abstract

In this article, a new concept of intuitionistic fuzzy graph of n^th type (IFGNT) is proposed as a generalization of intuitionistic fuzzy graphs (IFGs) and intuitionistic fuzzy graphs of second type (IFGST). Some light has also been shed upon the concepts of constant intuitionistic fuzzy graphs of second type (CIFGST) and constant intuitionistic fuzzy graphs of n^th type (CIFGNT). Moreover, some basic definitions of and results from IFGNT have been developed, supported with examples. Besides, the advantages of proposed new concept over the existing concepts have been highlighted and a comparative study of new and existing works made. Further, an application of IFGNT has been demonstrated in social networks context.

Keywords

Intuitionistic fuzzy graphs intuitionistic fuzzy graphs of second type intuitionistic fuzzy graphs of n-type constant intuitionistic fuzzy graphs of second type constant intuitionistic fuzzy graphs of n-type social networks

1 Introduction

Theory of graphs has been considered to play a vital role when it comes to its application in dealing with real life situations. The theory of fuzzy graphs (FGs) has its own significance as application of fuzzy set (FS) theory has no limits. FS theory has been applied to many situations like in intelligent systems by Yager and Zadeh [40], pattern recognition by Pedrycz [32], decision making by Maji et al. [24], traffic and transportation by Trabia et al. [37], clustering by Xie and Beni [38] and many other areas. It was Zadeh who initiated the concept of FSs [42] to deal with problems having some sort of uncertainty. For some recent developments in FS theory and other tools of uncertainty and their applications one is referred to [23 , 43–47]. Zadeh’s FS provided a solid ground for the theory of FGs which have been introduced by Kauffman [19] in 1973. After that, FGs have been comprehensively studied by Rosenfeld [34]. The study of FGs lead many scientists to contribute in this field such as Bhattacharya [6] discussed several graph theoretic results for FGs. Bhutani [7] worked on automorphisms of FGs and Mordeson and Chang-Shyh [26] developed operations on FGs. FGs have been applied to many practical situations like optimization problems by Kóczy [21], clustering by Yeh and Bang [41], shortest path problem by Klein [20]and social networks Nair and Sarasamma [27] etc. For some other work on FGs one may refer to [8 , 25] etc.

Atanassov and Gargov [3, 4] developed the concept of intuitionistic fuzzy set (IFS) as an extension of FS that deals with uncertain situations in a better way as its structure is not limited to membership grades only. The concept of IFSs 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 IFSs have been remarkably used in some areas so far. In [11] medical diagnosis is discussed based on IFSs by De et al and in [39], Xu defined some aggregation operators for IFSs which have been applied to multi-attribute decision making (MADM) by [22]. Some similarity measures for IFSs are discussed in [36] by Szmidt and Kacprzyk and applied to pattern recognition problems.

The concept of IFGs is introduced in [28] by Parvathi and Karunambigai. Like IFSs, some quality work on the theory of IFGs is also being done. Parvathi, Karunambigai and Atanassov [29] studied operations on IFGs, Gani and Begum [14] discussed the size, order and degree of IFGs, Akram and Davvaz [1] investigated strong IFGs, Parvathi and Thamizhendhi [30] discussed domination in IFGs, Akram and Dudek [2] proposed intuitionistic fuzzy hypergraphs, Karunambigai, Akram et al. [16] presented the concept of balanced IFGs, Karunambigai, Parvathi et al. [17] studied constant IFGs and Chountas, Shannon et al. [10] discussed intuitionistic fuzzy trees. For some other noteworthy work on IFGs one may refer to [18, 31] etc.

IFS is a basically a pair of two components namely membership and non-membership with a condition that their sum must belongs to [0,1]. This constraint is the only demerit in the structure of IFSs not allowing decision makers to choose values by their own consent. This was improved by Atanassov [3] in 1989 by defining IFS of second type. Still the condition for IFS of second type was limited so in [35] a new concept known as IFS of third type was introduced by Srinivasan and Begum which somehow enlarge the domain of previous concepts. Finally, to make IFS free of any restriction Atanassov, Szmidt et al. [5] proposed intuitionistic fuzzy pairs of n^th Type. This new concept has no restriction and it allows the decision makers to assign any value to membership and non-membership function in the interval [0,1].

Keeping in view the development in the structure of IFSs the theory of IFGs was also improved accordingly and the concept of IFGST was introduced in [12] by Dhavudh and Srinivasan and some substantial work in this direction is carried out in [12 , 33] but less attention has been given to these advance structures such as IFSNT by the researchers. Therefore, in this manuscript, the concept of IFGNT are proposed in view of its significance.

This manuscript has five sections starting with introduction as section one. In section two, some basic concepts related to IFS, IFSST, IFSNT, IFG, IFGST are discussed. Section three is based on the theory of IFGNT and some related terms and notion are illustrated. In section four, the notion of CIFGST and CIFGNT are discussed. In section five, an application of IFGNT showing its significance over existing structures. The article ends with discussion and concluding remarks.

2 Preliminaries

In this section, some basic definitions like definition of IFS, IFSST and IFSNT are discussed and their spaces are illustrated graphically. Some basic graph theoretic concepts of IFG, complete IFG and IFGST are also discussed and illustrated with examples.

Definition 1. [4] An IFS in is defined as where represents the degree of membership function and represents the degree of non-membership function with the condition

Definition 2. [3] An IFSST in is defined as where represents the degree of membership function and represents the degree of non-membership function with the condition

Definition 3. [5]. An IFSNT in is defined as where represents the degree of membership function and represents the degree of non-membership function with the condition

A geometrical comparison of all structures is discussed in below Figs. (1 and 2).

All the above discussed theory and geometrical compression clearly shows the significance of IFSNT as it is the generalization of IFS and IFSST. The above figures show that every intuitionistic fuzzy number and intuitionistic fuzzy number of second type is an intuitionistic fuzzy number of n-type for n = 1 and n = 2 but converse is not true

Definition 4. [28] A pair is known as IFG if

is the set of vertices such that and represents the degree of membership and non-membership of the element respectively with a condition that for all .

where and represents the degree of membership and non-membership of the element such that and with a condition that for all .

Example 1. The following Fig. (3) is an example of IFG having four nodes and edges.

Definition 5. [28] A Pair is known as complete IFG if and for all .

Example 2. The graph in Fig. (4) is an example of complete IFG.

Definition 6. [12] A pair is known as IFGST if

is the set of vertices such that and represents the degree of membership and non-membership of the element respectively with a condition that for all .

where and represents the degree of membership and non-membership of the element such that and with a condition that for all .

Example 3. The graph in Fig. (5) is an IFGST where each node is an IFNST.

3 Intuitionistic fuzzy graphs of nth type

In this section, the notion of IFGNT is discussed with examples. The concepts of subgraph and complement of IFGNT are also discussed and exemplified. The degree of IFGNT is defined and illustrated with an example.

Definition 7. A Pair is known as IFGNT if

is the set of vertices such that and represents the degree of membership and non-membership of the element respectively with a condition that for all .

where and represents the degree of membership and non-membership of the element such that and with a condition that for all .

Example 4. Consider the following graphs where Fig. (6) represents an IFGNT while Fig. (7) is not an IFGNT because of edge . The vertices in the below Figs. (5 and 6) are purely intuitionistic fuzzy numbers (IFNs) of n-type for n = 4.

Remark 1. An IFG and IFGST are surely IFGNT but converse is not true in general.

Example 5. The graphs in Figs. (6 and 7) of an IFGNT for n = 4 clearly shows that none of them is either IFGST nor IFG. Because, for , we have also .

Definition 8. A pair is known as intuitionistic fuzzy subgraph of n-type of if and that is ; and for all i, j = 1, 2, … n.

Example 6. The graph depicted in Fig. (8) is an intuitionistic fuzzy subgraph of n-type of IFGNT portrayed in Fig. (6). Moreover, the vertices in the above Fig. (8) are purely IFNs of n-type for n = 4.

Definition 9. The set of all triples in an IFGNT where and for some i . j = 1, 2, … n is a subset of $\underset{˙}{V}$ for 0 ≤ t ≤ 1 and the set of all triples where and for some i . j = 1, 2, … n is a subset of $\dot{E}$ for 0 ≤ t ≤ 1. Here is a subgraph of $\dot{G}$ .

Theorem 1. If is an IFGNT. Then is an intuitionistic fuzzy subgraph of for any such that .

Proof. Consider then so implies . Therefore, . Now, consider then so . Therefore, implies and hence . So, is an intuitionistic fuzzy subgraph of .

Theorem 2. Let be an intuitionistic fuzzy subgraph of an IFGNT . Then for any is an intuitionistic fuzzy subgraph of .

Proof. Given that and . To prove that is an intuitionistic fuzzy subgraph of For this we have to show that and . Assume that implies implies . Therefore, implies implies Now consider implies implies . Therefore, implies implies . Therefore is an intuitionistic fuzzy subgraph of .

Definition 10. The complement of an IFGNT is defined as

and for all i = 1, 2, … n.

and for every i . j = 1, 2, … n.

Example 6. Consider the graphs in Figs. (9 and 10) where graph in Fig. (10) represent the complement of graph depicted in Fig. (9). Moreover, the vertices in these graphs are purely IFNNT for n = 3.

Definition 11. Let be an IFGNT. Then the degree of vertex is defined by where and .

Example 8. Consider an IFGNT depicted in Fig. (11) where all the nodes are IFNNT for n = 4. The degrees of all vertices are determined below using Definition (11).

, .

4 Constant intuitionistic fuzzy graphs of n-type

This section is based on the novel concept of CIFGNT and CIFGST. These concepts are illustrated with the help of examples. The notion of total degree and constant function are also studied and supported with examples.

Definition 12. An IFGNT is said to be CIFGNT of degree or if and for all .

Example 9. The graph portrayed in Fig. (12) is an example of CIFGNT where all nodes are IFNNT for n = 4. The degree of all vertices is same i.e. (0.9,1.6).

Remark 2. Definition (11) reduces to definition of CIFGST if every element of and are purely IFNs of second type.

Remark 3. A complete IFGNT need not be CIFGNT.

This remark is demonstrated by the following example.

Example 10. Consider the following IFGNT depicted in Fig. (13) where all nodes are purely IFNNT for n = 3. Further, this IFGNT is complete but not constant.

Definition 13. For an IFGNT . The total degree of a vertex is defined as:

If total degree of each vertex is same, then is called IFGNT of total degree or -totally CIFGNT.

Example 11. Consider an IFGNT in Fig. (14) where all nodes are purely IFNNT for n = 2. Further, total degree of each vertex is (1.5,2.3).

Example 12. Consider an IFGNT in Fig. (15) where all nodes are purely IFNNT for n = 3. Further, $\dot{G}$ totally constant.

Theorem 3. Let be an IFGNT. Then is a constant function iff the following are equivalent.

is CIFGNT.

is totally CIFGNT.

Proof. Assume that is a constant function. Consider and for all where and are constants. Suppose that is a . Then and for all . So, , , for all . Therefore $\dot{G}$ is totally CIFGNT. Hence (1) ⇒ (2) is proved.

Now to prove (2) ⇒ (1). Suppose is totally CIFGNT to prove is CIFGNT. As is totally CIFGNT then , for all . , , . Likewise , . Therefor (1) and (2) are equivalent.

Conversely, suppose that (1) and (2) are equivalent i.e. is CIFGNT iff is totally CIFGNT. Assume that is not a constant function. Then , for at least . Let is totally CIFGNT. Then , . So, and . Likewise and . Therefore, , . We have , . So, is not totally CIFGNT which is contradiction to our supposition. Now, consider is totally CIFGNT. Then

Likewise, . So, is not constant which contradiction to our supposition. Therefore is a constant function.

Theorem 4. If an IFGNT is constant and totally constant. Then is constant function.

Proof. Consider is -constant and -totally CIFGNT. Then by definitions and for and , for , for all . implies that , for all . Hence is constant function. Similarly, , for all .

Remark 3. Converse of Theorem (4) does not hold. We show this by the following example.

Example 13. The following IFGNT depicted in Fig. (16) where each node is purely IFNNT for n = 3 support Remark (3) i.e. is constant function, but neither CIFGNT nor totally CIFGNT.

5 Application

It is discussed that the framework of IFGNT is diverse in nature than that of IFS and IFSST. It allows the membership and non-membership values to be chosen from anywhere in the interval [0,1] regardless of any condition. So, this type of structure can be applied to many real-life problems with no limitations.

Social networks:

We discussed the application of IFGNT in social networks where the relationship between different countries based on different matters has been studied. We used IFGNT to determine the level of relationship. For this purpose, we consider different matters that are important in the relationship of different countries with each other. These includes culture, area, religion, behavior of peoples, budget for defense expenditure, visa policy, trade, political, border management and media.

Consider Fig. (17) where a list of countries is taken into account and their relations are studied keeping in mind the above discussed matters in the environment of IFGNT. The countries are India, Saudi-Arab, Iran, Pakistan, and China. The following graph gives us a brief information about the relationship of these countries with each other.

The edge between two countries represent their relation. We list the edge values of each pair of countries in Table (1). These edges are in the form of IFNNT having a membership and non-membership grade which mean that if the membership degree is greater compared to the non-membership degree then the relationship is considered as strong otherwise weak. The degree of each vertex will give us the strength of relationship of a country with all other countries.

Table 1

Relation of every pair of countries in the form of IFNNT

Country	Country	Relationship
Pakistan	India	(0.4, 0.8)
Iran	India	(0.6, 0.8)
China	Saudi-Arab	(0.7, 0.8)
Pakistan	China	(0.5, 0.7)
India	Saudi-Arab	(0.5, 0.8)
Pakistan	Iran	(0.6, 0.8)
Pakistan	Saudi-Arab	(0.5, 0.7)
China	Iran	(0.6, 0.7)

Now the degree of relation of each country is calculated based on Definition (11). The high degree of membership shows the good relation of it with other countries and vice versa. The degree of relation of each country is listed in Table (2).

Table 2

Degree of relation of each country de-termined from Fig. (17)

Country	Degree
Iran	(1.8, 2.3)
Pakistan	(2.0, 3.0)
Saudi-Arab	(1.1, 1.6)
India	(1.5, 2.4)
China	(1.8, 2.1)

For simplicity, the strength of each vertex is measured by using the effective degree which is defined as the difference of membership degree and non-membership degree. The effective degree of each vertex is calculated in Table 3.

Table 3

Effective degree of relation of each country

Country	Score
India	–0.9
Saudi-Arab	–0.5
Pakistan	–1
China	–0.3
Iran	–0.5

From the calculation in Table (3) it is clear that on different matters, china has better relationship with other countries. The effective degree of Saudi-Arab and Iran shows that their relationship is not as much stronger compared to china but better than Pakistan and India.

6 Advantages

The proposed framework IFGNT generalizes both IFG and IFGST. The main advantage of proposed framework is that the space of IFGNT is much larger than that of IFS and IFSST and is free of any barriers i.e. it allows the decision makers to assign membership and non-membership values from anywhere in the interval [0, 1]. All the works done so far in IFG and IFGST can be done in the proposed structure of IFGNT. On the other hand, the work information in the form of IFNNT could not be processed using the notions of IFSs or IFSSTs because of their limited structures. For example, if we look at Fig. (16) of social networks, all the information is in the form of IFNNT, hence IFGs and IFGST are failed to describe it.

7 Conclusion

In this article, the theory of IFGNT, CIFGST and CIFGNT have been proposed. Some basic graph theoretic concepts are defined for IFGNT and their properties are investigated. The structure of IFS, IFSST and IFSNT are compared and it is proved that IFSNT generalizes IFS and IFSST which also prove the generalization of IFGNT over IFG and IFGST. A real-life application of proposed IFGNT is discussed showing is worth. In future some further contribution to this theory could be made such as the concepts of cycles, tree could be defined in this frame work. The minimum spinning tree problems could be discussed along with some other real-life problems. Further this study could be extended to the directions of soft set theory and rough set theory to deal with some MADM problems as demonstrated in [23 , 43–47].

Compliance with Ethical Standards

Conflict of interest

We declare that there are no conflicts of interest regarding the publication of this paper.

Footnotes

Acknowledgments

The authors are highly thankful to the honorable editor and reviewers for their valuable suggestions.

References

Akram

, Davvaz

, Strong intuitionistic fuzzy graphs, Filomat 26(1) (2012), 177–196.

Akram

, Dudek

W.A.

, Intuitionistic fuzzy hypergraphs with applications, Information Sciences 218 (2013), 182–193.

Atanassov

, Geometrical interpretation of the elements of the intuitionistic fuzzy objects, Preprint IM-MFAIS-1-89, Sofia, 1989.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy sets and Systems 20(1) (1986), 87–96.

Atanassov

, Szmidt

, Kacprzyk

, Vassilev

, On intuitionistic fuzzy pairs of n-th type, Issues Intuit Fuzzy Sets Gen Nets 13 (2017), 136–142.

Bhattacharya

, Some remarks on fuzzy graphs, Pattern Recognition Letters 6(5) (1987), 297–302.

Bhutani

K.R.

, On automorphisms of fuzzy graphs, Pattern Recognition Letters 9(3) (1989), 159–162.

Bhutani

K.R.

, Battou

, On M-strong fuzzy graphs, Information Sciences 155(1-2) (2003), 103–109.

Bhutani

K.R.

, Rosenfeld

, Strong arcs in fuzzy graphs, Information Sciences 152 (2003), 319–322.

10.

Chountas

, Shannon

, Rangasamy

, Atanassov

, On intuitionistic fuzzy trees and their index matrix interpretation, Notes on Intuitionistic Fuzzy Sets 15(4) (2009), 52–56.

11.

S.K.

, Biswas

, Roy

A.R.

, An application of intuitionistic fuzzy sets in medical diagnosis, Fuzzy Sets and Systems 117(2) (2001), 209–213.

12.

Dhavudh

S.S.

, Srinivasan

, Intuitionistic fuzzy graphs of second type, Advances in Fuzzy Mathematics 12(2) (2017), 197–204.

13.

Dhavudh

S.S.

, Srinivasan

, A study on intuitionistic fuzzy graphs of second type, International Journal of Mathematical Archive EISSN 8(8) (2017), 2229–5046.

14.

Gani

A.N.

, Begum

S.S.

, Degree, order and size in intuitionistic fuzzy graphs, International journal of algorithms, Computing and Mathematics 3(3) (2010), 11–16.

15.

Gani

A.N.

, Latha

S.R.

, On irregular fuzzy graphs, Applied Mathematical Sciences 6(11) (2012), 517–523.

16.

Karunambigai

, Akram

, Sivasankar

, Palanivel

, Balanced intuitionistic fuzzy graphs, Applied Mathematical Sciences 7(51) (2013), 2501–2514.

17.

Karunambigai

, Parvathi

, Buvaneswari

, Constant intuitionistic fuzzy graphs, NIFS 17(1) (2011), 37–47.

18.

Karunambigai

, Sivasankar

, Palanivel

, Some properties of a regular intuitionistic fuzzy graph, International Journal of Mathematics & Computation™ 26(4) (2015), 138–146.

19.

Kaufmann

, Introduction à la théorie des sous-ensembles flous à l’usage des ingénieurs: Éléments théoriques de base, Masson, 1973.

20.

Klein

C.M.

, Fuzzy shortest paths, Fuzzy Sets and Systems 39(1) (1991), 27–41.

21.

Kóczy

, Fuzzy graphs in the evaluation and optimization of networks, Fuzzy Sets and Systems 46(3) (1992), 307–319.

22.

D.-F.

, Multiattribute decision making method based on generalized OWA operators with intuitionistic fuzzy sets, Expert Systems with Applications 37(12) (2010), 8673–8678.

23.

, Zhan

, Ali

M.I.

, Mehmood

, A survey of decision making methods based on two classes of hybrid soft set models, Artificial Intelligence Review 49(4) (2018), 511–529.

24.

Maji

, Roy

A.R.

, Biswas

, An application of soft sets in a decision making problem, Computers & Mathematics with Applications 44(8-9) (2002), 1077–1083.

25.

Mordeson

J.N.

, Fuzzy line graphs, Pattern Recognition Letters 14(5) (1993), 381–384.

26.

Mordeson

J.N.

, Chang-Shyh

, Operations on fuzzy graphs, Information Sciences 79(3-4) (1994), 159–170.

27.

Nair

P.S.

, Sarasamma

S.T.

, Data mining through fuzzy social network analysis, Fuzzy Information Processing Society, NAFIPS’07 Annual Meeting of the North American, IEEE, 2007.

28.

Parvathi

, Karunambigai

, Intuitionistic fuzzy graphs, Computational Intelligence, Theory and Applications, Springer (2006), 139–150.

29.

Parvathi

, Karunambigai

M.G.

, Atanassov

K.T.

, Operations on intuitionistic fuzzy graphs, In Fuzzy Systems, 2009 FUZZ-IEEE 2009 IEEE International Conference on, IEEE, 2009, pp. 1396–1401.

30.

Parvathi

, Thamizhendhi

, Domination in intuitionistic fuzzy graphs, Notes on Intuitionistic Fuzzy Sets 16(2) (2010), 39–49.

31.

Parvathi

, Thilagavathi

, Thamizhendhi

, Karunambigai

, Index matrix representation of intuitionistic fuzzy graphs, Notes on Intuitionistic Fuzzy Sets 20(2) (2014), 100–108.

32.

Pedrycz

, Fuzzy sets in pattern recognition: Methodology and methods, Pattern Recognition 23(1-2) (1990), 121–146.

33.

Rashmanlou

, Jouybari

M.N.

, Firouzian

, Kumar

P.K.

, A study on irregular intuitionistic fuzzy graphs, International Journal of Mathematics & Computation™ 29(2) (2018), 17–28.

34.

Rosenfeld

, Fuzzy graphs. Fuzzy sets and their applications to cognitive and decision processes, Elsevier, 1975, pp. 77–95.

35.

Srinivasan

, Begum

S.S.

, Some properties of intuitionistic fuzzy sets of third type, International Journal of Science and Humanities 1(1) (2015), 53–58.

36.

Szmidt

, Kacprzyk

, A similarity measure for intuitionistic fuzzy sets and its application in supporting medical diagnostic reasoning, International Conference on Artificial Intelligence and Soft Computing, Springer, 2004.

37.

Trabia

M.B.

, Kaseko

M.S.

, Ande

, A two-stage fuzzy logic controller for traffic signals, Transportation Research Part C: Emerging Technologies 7(6) (1999), 353–367.

38.

Xie

X.L.

, Beni

, A validity measure for fuzzy clustering, IEEE Transactions on Pattern Analysis and Machine Intelligence 13(8) (1991), 841–847.

39.

, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems 15(6) (2007), 1179–1187.

40.

Yager

R.R.

, Zadeh

L.A.

, An introduction to fuzzy logic applications in intelligent systems, Springer Science & Business Media, 2012.

41.

Yeh

R.T.

, Bang

, Fuzzy relations, fuzzy graphs, and their applications to clustering analysis, Fuzzy Sets and Their Applications to Cognitive and Decision Processes, Elsevier, 1975, pp. 125–149.

42.

Zadeh

L.A.

, Information and control, Fuzzy Sets 8(3) (1965), 338–353.

43.

Zhan

, Alcantud

J.C.R.

, A novel type of soft rough covering and its application to multicriteria group decision making, Artificial Intelligence Review (2018), 1–30.

44.

Zhan

, Wang

, Certain types of soft coverings based rough sets with applications, International Journal of Machine Learning and Cybernetics (2018), 1–12.

45.

Zhan

, Xu

, Two types of coverings based multigranulation rough fuzzy sets and applications to decision making, Artificial Intelligence Review (2018), 1–32.

46.

Zhang

, Zhan

, Fuzzy soft β-covering based fuzzy rough sets and corresponding decision-making applications, International Journal of Machine Learning and Cybernetics, pp. 1–16.

47.

Zhang

, Zhan

, Novel classes of fuzzy soft β-coverings-based fuzzy rough sets with applications to multi-criteria fuzzy group decision making, Soft Computing (2018).