Connectivity Concepts in Intuitionistic Fuzzy Incidence Graphs with Application.

In this research article, we presented the idea of intuitionistic fuzzy incidence graphs (IFIGs) along with connectivity concepts. IFIGs are the generalization of fuzzy incidence graphs (FIGs). Specific ideas analogous to intuitionistic fuzzy cut-vertices and intuitionistic fuzzy bridges in intuitionistic fuzzy graphs, intuitionistic incidence cut-vertices, and intuitionistic incidence bridges are explored. The notion of intuitionistic incidence gain and intuitionistic incidence loss for intuitionistic incidence paths and pairs of vertices is also initiated. In the case of FIGs, we have only membership value, and we do not have non-membership value (NMSV). Therefore, we use IFIGs because they are more reliable, valuable, and helpful than FIGs. Also, we can not apply graphs, fuzzy graphs, and FIGs to the application provided in Sect. 3 due to the non-availability of NMSV. An application in selecting the best paint company for investment among different companies by using IFIG is also obtained.

Introduction and Preliminaries

Graph theory has many applications in various fields of life, including computer science, engineering, and networking. Different ideas of paths and walks in graph theory are used in different problems of real-life like resource networking and database design. This guides the expansion of innovative algorithms that can be used in many applications. In this paper, we use simple and finite graphs. Zadeh [38] first paper on fuzzy sets extremely altered the face of science and technology. Zadeh [39] presented the idea of similarity, fuzzy preordering, fuzzy ordering, and partial fuzzy ordering. After his great paper on fuzzy sets, Roselfeld [28] initiated an innovative idea of FGs. Later, Yeh and Bang [37] discussed the basic terms and notations of FGs. They also explained how FGs could be applied to clustering analysis. Bhattacharya [6] introduced center, eccentricity in FGs and attached fuzzy groups with FGs. Rashmanlou and Jun [23] explored various products in complete interval-valued FGs. Rashmanlou and Pal [24, 25] examined balanced interval-valued and antipodal interval-valued FGs. Sunitha along with Vijayakumar [36] presented the notion of the complement, self-complementary and their properties in FGs. Mordeson and Nair [15] provided several applications of FGs, including clusters, cluster analysis, cohesiveness, and slicing. Later, Pramanik et al. [22] introduced interval-valued fuzzy planar graphs and their various properties. They also defined the interval-valued fuzzy dual graph, closely associated with the interval-valued fuzzy planar graph. Raut et al. [27] brought the idea of a fuzzy permutation graph. They defined two kinds of complements of a fuzzy permutation graph, including p-complement and f-complement. They also provided a real-life application of a fuzzy permutation graph. Zadeh [40] presented the concept that the position of centrality in the nontraditional view of fuzzy logic is that of precision. Samanta [33, 34] founded the fuzzy threshold graphs and fuzzy tolerance graphs. Atanassov [5] introduced the notion of intuitionistic fuzzy sets (IFSs) due to the non-availability of NMSV in fuzzy sets. Akram and Alshehri [1] investigated intuitionistic fuzzy bridges, cycles, and trees. Akram and Davvaz [2] introduced strong intuitionistic fuzzy graphs (IFGs) and intuitionistic fuzzy line graphs. Akram and Dudek [3] gave an application of intuitionistic fuzzy hypergraphs. Akram et.al [4] explained different metric aspects of IFGs. Sahoo and Pal [29-32] have done a comprehensive research on IFGs. Borzooei and Rashmanlou [7] discussed the ring sum of two product IFGs and explored some fascinating features of isomorphism on product IFGs. Gani and Begum [11] provided the new definition of complete IFG and regular FG. They also, explained various properties of IFG. Detail work on IFGs can be seen [8, 21, 35]. FGs are unable to speak about the effect of vertices on edges. For example, if vertices represent unlike universities and edges indicate roads connecting these universities, we can have a FG demonstrating the size of traffic from one university to another. The university has the maximum number of students and will have large number of ramps in university. So, if and are two different universities and is a road amalgamating them, then could reveal the ramp system from the road to the university . In the case of an unweighted graph, and both will have an effect of 1 on . In a directed graph, the result of on indicated by is 1 whereas is 0. We can generalized this concept by FIGs. Dinesh [9] proposed the idea of FIGs. After him, Malik et al. [13] apply FIGs in human trafficking. Mathew and Mordeson [14] discussed connectivity ideas and different properties in FIGs. Malik et al. [12] examined complementary FIGs. Fang et al. [10] presented the connectivity index and Wiener index in FIGs. They also defined different kinds of vertices. Later, Nazeer et al. [16, 17] presented the notion of order, size, domination, and strong pair domination in FIGs. They discussed two types of domination, including strong fuzzy incidence domination and weak fuzzy incidence domination. They also provided an application of fuzzy incidence domination to select the best medical lab among different labs for conducting tests for the coronavirus. The idea of cyclic connectivity, fuzzy incidence cycle, cyclic connectivity index, and average cyclic connectivity index was initiated by Nazeer et al. [18]. The number of operations including Cartesian product, composition, tensor product, and normal product in IFIG was presented by Nazeer et al. [19]. Later, Nazeer and Rashid [20] proposed the idea of picture fuzzy incidence graphs (PFIGs) as an extension of FIGs. They provided an application of PFIGs in controlling unlawful transit of people from India to America. The number of reasons and advantages to introducing the crucial idea of IFIGs. Firstly, in IFGs, a vertex c expresses the same impact on an edge cd and dc but in IFIGs, the impact of c on cd denoted by (c, cd) will be different from (d, cd). This motivated us to bring the concept of IFIGs. Another reason to use IFIGs is that graphs, FGs, and FIGs do not have NMSV, which becomes the fundamental reason to introduce IFIGs. We cannot use graphs, FGs, and FIGs in the application, provided in Sect. 3, due to the non-availability of NMSV. Therefore, we have used the approach of IFIGs to get the required results. IFIGs are more useful, helpful, and handy than graphs, FGs, and FIGs due to the availability of NMSV. Rashmanlou et al. [26] explored the idea of intuitionistic fuzzy cut-vertices, bridges, gain, and loss in IFGs. In this paper, we bring these ideas to IFIGs. The paper is organized as follows, Sect. 1 contains some fundamental definitions and results that are important to apprehend the remaining concepts of this paper. In Sect. 2, we introduce the notion of connectivity and its related properties in IFIGs. An application for selecting the best paint company to invest money in among various paint companies by using IFIG is provided. We provide some basic definitions and terminologies from [1, 5, 26].

A graph is an ordered pair , where V is the set of vertices of and E is the set of edges of . A graph without loops and multiple edges between any two vertices is named as a simple graph. A fuzzy subset (FS) on a set M is a map . A map is known as a fuzzy relation on if for each . A FG is a pair , where is a FS on a set V and is a fuzzy relation on . It is consider that V is finite and non empty, is reflexive and symmetric.

In this paper, minimum and maximum operators are expressed by or min and or max respectively.

An IFS A on universal set M is defined aswhere is named as degree of membership of w in A, is named as degree of non-membership of w in A, and , satisfies for every . An intuitionistic fuzzy relation (IFR) on universal set is an IFS of the form , where , and fulfills for every . An IFR R on a universal set is reflexive if for every , symmetric if for any . Let be a crisp graph. An IFG on is a pair in which is an IFS on V and is an IFR on E of the type , and for all . Also, S is a symmetric IFR on R. A partial intuitionistic fuzzy subgraph (PIFS) of an IFG, is an IFG such that and for each , and for every . An intuitionistic fuzzy subgraph of an IFG, is an IFG such that and for each of H, and for every of H. An IFG is strong if and for every . An IFG is complete if and for every .

The definitions given below are taken from [10, 14].

Assume is a graph with non empty vertex set V. Then, an incidence graph (IG) of is indicated by where . The members of I are called pairs or incidence pairs (IPs). Then a FIG of is expressed by where , and are FS of vertices, edges and IPs respectively, such that for all . If there exist such a path v, (v, vw), vw, (w, vw), w between v and w then vertices v and w are connected. v and vw are connected if there is a path from v to vw such that v, (v, vw), vw.

Intuitionistic Fuzzy Incidence Graphs

IFIGs with different kinds of examples are elaborated in this section. In the whole article we will denote an IG by and a IFIG by .

Definition 2.1

Let be an IG. An IFIG, is an ordered-triplet in which

is an IFS on V.

is an IFS on .

is an IFS on such that

,

,

and .

An example of IFIG is provided here.

Example 2.2

Here we include a daily life example of three different cities. As an illustrative case, consider a network of (IFIG) of three vertices representing the three different cities. The membership value (MSV) of the vertices indicates the percentage of people who can speak English very well, and the non-membership value (NMSV) of the vertices represents the percentage of people who can not speak English fluently. The MSV of the edges expresses the percentage of those people contacting the people of other cities to enhance their English language, and the NMSV of the edges shows the percentage of those who are taking an interest in improving their English. The MSV of the pairs represents the percentage of those people who have improved their English, and the NMSV of the pairs shows the percentage of those people who fail to improve their English.

Consider an IG, such that , and as shown in Fig. 1. Let be an IFIG associated with provided in Fig. 2 where,

Fig. 1

Incidence graph

Fig. 2

IFIG

,

,

,

Definition 2.3

The support of an IFIG is defined as where

support of

support of

support of

Definition 2.4

A PIFS is called a partial intuitionistic fuzzy incidence subgraph (PIFIS) of an IFIG if and

for all .

Definition 2.5

An intuitionistic fuzzy subgraph is called an intuitionistic fuzzy incidence subgraph (IFIS) of an IFIG if and

for all is the set of incidence pair of H. Figs. 3 and 4 is an example of PIFIS and IFIS of IFIG given in Fig. 2.

Fig. 3

PIFIS of IFIG provided in Figure 2

Fig. 4

IFIS of IFIG given in Figure 2

Definition 2.6

A strong IFG, is called strong IFIG if and for each and in .

Definition 2.7

A complete IFG, is called complete IFIG if and for each .

Connectivity ideas are significant in whole graph theory. But in classical problems these ideas only tackle the disconnection of the networks. The reduction in flow is more frequent than the disconnection. Here, we are going to talk about intuitionistic incidence walk (IIW), intuitionistic incidence path (IIP), intuitionistic incidence gain path (IIGP), intuitionistic incidence loss path (IILP) intuitionistic incidence balanced path (IIBP), intuitionistic incidence optimal path (IIOP), intuitionistic incidence cut-vertices and intuitionistic incidence bridges . These ideas will help us to deal with the reduction in the strength of connectedness between distinct pairs of vertices in an IFIGs.

Definition 2.8

If then vw is called an edge of the IFIG and if then (v, vw) and (w, vw) are IPs of .

Definition 2.9

A sequence

in is called IIW. If then an IIW is closed.

Definition 2.10

An IIP

.

is a sequence of different vertices such that either one of the given below condition is satisfied:

and for some i, j,

and for some i, j,

and for some i, j.

Example 2.11

Let be an IFIG provided in Fig. 5 where,

Fig. 5

IFIG

,

,

.

An IIW in Fig. 5 is closed because its beginning and final vertex is identical but it is not an IIP because all vertices are not different.

The idea of loss and gain is crucial in various problems in economics, networks and computer science. We are going to connect these notions to an IFIG.

Example 2.12

Consider is an IFIG provided in Fig. 5 where,

,

,

,

.

Here we include a real life example of four different university friends. As an illustrative case, consider a network (IFIG) of four different vertices representing mobile phones. The MSV of the vertices indicating the percentage of useful data in a mobile phone and the NMSV is representing the percentage of useless data. The MSV of the edges is the percentage of helpful data which they are sharing with each other and the NMSV of the edges is the percentage of harmful data which they are sharing with each other. The MSV of the pair is the percentage of effective data which one friend is sending to other friend every week and the NMSV of the pair is the percentage of worthless data which one friend is transferring to other friend each week.

Definition 2.13

Let be an IFIG. For any IIP

in ,

we define as the intuitionistic incidence gain (IIG) of P. IIG of P is expressed by and

as the intuitionistic incidence loss (IIL) of P. IIL of P is denoted by . In Fig. 5, for an IIP, , and

Definition 2.14

An IIP, P is called an IIGP if and IILP if .

In Fig. 5P : psr is an IILP because .

Definition 2.15

An IFIG is named as connected if there exists an IIP between each pair of vertices.

Definition 2.16

Let l and m be vertices in a connected IFIG . Among all IIP in , an IIP whose IIG is greater than or equal to of any other IIP in is called a maximum IIGP. It is shown by . In the same way, a IIP whose IIL is smaller than or equal to that of any other IIP in is known as minimum IILP. It is expressed by . That is an IIP, P is if and is a if , where is any IIP in .

Example 2.17

Table 1 shows the intuitionistic incidence max gain and intuitionistic incidence min loss between each pair of vertices of an IFIG given in Fig. 5.

Definition 2.18

In an IFIG a IIP, P is named as IIBP if and P is called IIOP if P is a and .

Example 2.19

Figure 5 does not have any IIBP because for each IIP but is an IIOP.

Definition 2.20

Assume is an IFIG and let l and m be any two vertices of . The IIG of l and m is defined as IIG of . It is expressed by . Similarly, the IIL of l and m is defined as IIL of . It is expressed by .

Example 2.21

In Fig. 5, , , and .

Definition 2.22

Let H ba an IFIS of , l and m be any two vertices of H then an IIG of l and m in H is the IIG of strictly belongs to H and it is shown by . In a similar manner, IIL of l and m in H is the IIL of strictly belongs to H and it is denoted by . If there does not exist and in H then and .

Example 2.23

Let be an IFIG provided in Fig. 6 which is a IFIS of IFIG provided in Fig. 5 where,

Fig. 6

IFIS of Figure 5

,

,

.

It can be seen that and .

Proposition 2.24

If H is an IFIS of an IFIG, then and for each pairs of vertices l and m.

Proof

Let H be an IFIS of having same number of vertices, edges and IPs with equal membership degree and non-membership degree of vertices, edges and IPs this implies that and for every pair of vertices l and m in H. Now, if H has less number of vertices, edges and IPs then membership degree and non-membership degree of vertices,edges, and IPs will less this implies and . Hence, and .

Next, we have proposed an idea of intuitionistic incidence gain loss matrix (IIGLM) in an IFIGs.

Definition 2.25

Let be an IFIG with n vertices, . The IIGLM is defined as where and for and , for .

Example 2.26

IIGLM of shown in Fig. 5 is given below.It is clear from the matrix that IIGLM of an IFIG is a symmetric matrix.

In crisp graphs, a cut-vertex is one whose deletion results the disconnection of a graph and a bridge is an edge whose deletion results the disconnection of graph. But in IFIGs these definitions are different.

Definition 2.27

Assume is an IFIG. A vertex is named as an if there exists two vertices of the type such that and . A vertex k in an IFIG is said to be an intuitionistic incidence gain cut-vertex if is satisfied and an an intuitionistic incidence loss cut-vertex if is satisfied. In Fig. 7, p is an because and r is an because .

Fig. 7

IFIG having p as and r as

Theorem 2.28

A vertex k in an IFIG is an if and only if (iff) k is a vertex in each and in each , for some l and m in .

Let be an IFIG. Suppose that k is an . According to the definition of , there exists vertices l and m in such that and and . From (i) it is clear that the deleting of k from deletes all and from (ii) it can be seen that the deletion of k deletes every . Thus, k is in every and . Conversely, assume that k is in each and in each . Then the deletion of k form results in the deletion of all and . This implies the will lessen and will enhance between l and m. So, and . Hence k is an .

Definition 2.29

Let be an IFIG. An edge in is said to be an if and for some . If at least one of or is distinct from l and m then is called intuitionistic incidence bond and an intuitionistic incidence cutbond if both and are dissimilar from l and m.

Example 2.30

Let be an IFIG provided in Fig. 8 where,

Fig. 8

IFIG having

,

,

.

In Fig. 8, edges qr and qs are .

Theorem 2.31

An edge of an IFIG is an iff it is in each and in every for some l and m .

Let be an IFIG. Suppose that is an . According to the definition of , there exists vertices and in such that , and . From (i) it can be observe that the deleting of e from deletes every and from (ii) it can be seen that the deletion of e deletes every . Thus, e is in every and . Conversely, assume that e is in each and in each . Then the deletion of e form results in the deletion of every and . This implies the will lessen and will enhance between and . So, and . This proves that is an .

Here, we are presenting a simple theorem to examine whether a specific edge is an or not.

Theorem 2.32

An edge lm is an iff and for some .

Let be an IFIG and is an edge in of the type and . Since and , we have , . This implies that lm is an . Conversely, consider that lm is an then by Theorem 2.28. there will be vertices a and b in R such that lm is present on every and each . Assume . Then, . This implies there will be a in say, J which is dissimilar from lm. Assume K is a in . Replace lm in K by J to get an IIW. This IIW carries an IIP. The IIG of this IIP is larger than or equal to which is impossible. Therefore, . Now, consider that . Then, . This follows that there is a in call which is distinct from lm. Let be a . Replace lm by to get IIW. This IIW contains IIP. The IIL of this IIP is smaller than or equal to which is impossible. Therefore, .

Selection of Company or Companies to Investment by Using IFIG

Eight different paint companies , , , , , , and are doing their business with each other on certain and . Investors are interested in investing their money in these companies to earn more. Still, before the investment, they are a bit confused about which company or companies they should invest their money in because they are unaware of the company or companies with a maximum and minimum of . A mathematical model of this scenario is discussed here. Let be an IFIG as shown in Fig. 9.

Fig. 9

A mathematical model to select best company for investment

showing the different paint companies with their and before starting their business with each other.

representing the contract requirements of and among all companies at which they will do their business with each other.

,

,

,

,

,

,

,

,

,

shows which company or companies are accomplishing and which are not achieving the said and contract requirements. After computing the and of all companies, the maximum and minimum is of company and . So, investors can invest their money either in or in .

Conclusion

It is well known that graphs are among the omnipresent models for various kinds of structures. Graphs have the bulk of applications in different areas of life. However, they have failed to discover the impact of vertices on edges. This thing makes a way to introduce the idea of FIGs. FIGs can provide the effect of a vertex on edge. For example, if vertices show alike hotels and edges show the roads connecting these hotels, we can have a FG expressing the the area of traffic from one hotel to another hotel. The hotel has a large number of customers and will have maximum ramps in a hotel. So, if and are two hotels and roads connecting them then represent the ramp system from the road to the hotel . In unweighted graph, and both will have an effect of 1 on . In the case of directed graph,, the impact of on indicated by is 1 whereas is 0. This concept is generalized by FIGs. FIGs are beneficial, but there is a lack of FIGs because they do not have a NMSV. Therefore, we have introduced IFIGs because IFIGs have MSV and NMSV. Also, we can not apply FIGs to the application provided in Sect. 3 due to the non-availability of NMSV. This motivates us to initiate the vital idea of IFIGs. IFIGs are more beneficial than FIGs due to the availability of MSV. Another reason to introduce IFIGs is that FIGs cannot tackle the concept of and due to the unavailability of NMSV. Still, IFIGs can handle the idea of and due to the availability of NMSV. IFIG is an extension of a FIG. In this paper, different connectivity ideas related to IFIGs are discussed. The notion of and for IIP and different pairs of vertices are examined. and are introduced with various examples. A comprehensive study on the effectiveness of IFIGs on different communication systems by using and is the main target of our forthcoming research papers. Another significant aspect of our upcoming research work is to make an application of IIGLM of IFIG in transportation problems.

Table 1

Intuitionistic incidence Max-gain and intuitionistic incidence Min-loss of vertices

Vertices	Max-IIG	Max-IIGP	Min-IIL	Min-IILP
p-q	0.2	pq	0.8	pq,prq
p-r	0.2	pqr	0.8	pr,pqr
p-s	0.1	Any path	0.9	Any path
q-r	0.2	qr	0.5	qr
q-s	0.1	Any path	0.9	Any path
r-s	0.1	Any path	0.5	Any path