# Connectivity Indices of Intuitionistic Fuzzy Graphs and Their Applications in Internet Routing and Transport Network Flow

Authors: Tulat Naeem; Abdu Gumaei; Muhammad Kamran Jamil; Ahmed Alsanad; Kifayat Ullah
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

Connectivity indexCI has a vital role in real-world problems especially in Internet routing and transport network flow. Intuitionistic fuzzy graphs IFGs allow to describe two aspects of information using membership and nonmembership degrees under uncertainties. Keeping in view the importance of CIs in real life problems and comprehension of IFGs, we aim to develop some CIs in the environment of IFGs. We introduce two types of CIs, namely, CI and average CI, in the frame of IFGs. In spite of that, certain kinds of nodes called IF connectivity enhancing node IFCEN, IF connectivity reducing node IFCRN, and IF neutral node are introduced for IFGs. We have introduced strongest strong cycles, θ-evaluation of vertices, cycle connectivity, and CI of strong cycle. Applications of the CIs in two different types of networks are done, Internet routing and transport network flow, followed by examples to show the applicability of the proposed work.


---

## Body

## 1. Introduction

Zadeh [1] presented the idea of fuzzy set FS by giving membership grades to the objects of a set ranging from zero to one. Many concepts of crisp set theory like inclusion, union, intersection, complement, etc., were established for FSs. FS theory opened the way to fuzzy logic and fuzzy control systems. In the beginning, probability theory was the only tool to handle problems of uncertainty, facing science, technology, and real life problems. FS theory has many applications in different areas such as inventory control model [2], decision-making problems [3], and intelligence science [4]. More applications can be found in [5, 6]. A recent research to treat COVID-19 disease with FS approach is given in [7].In 1975, Rosenfeld [8] studied fuzzy graphs FGs. After that, Yeh and Bang [9] presented the same concept independently during the same period. Rosenfeld defined some basic properties of fuzzy relations including fuzzy bridges and trees with their properties, while Yeh and Bang gave the concept of connectedness of FGs along with applications. Mordeson [10] proposed the work for fuzzy line graphs. Massa’deh [11, 12] introduced complete FGs, regular FGs, complement of FGs, and some other properties. Mathew and Sunitha introduced types of arcs such as α-strong, β-strong, and δ-arcs in FGs [13]. The authors gave various concepts like strong arcs in [14], fuzzy end nodes in [15], and geodesic in [16]. Recently, Akram presented concepts like bipolar FGs in [17] and energy of bipolar FGs in [18]. Jan et al. studied the concept of cubic bipolar FGs with an application in a social network [19]. FGs are useful in representing relationships under uncertainty. FGs are used in various areas like human trafficking, disaster management system, decision-making method, etc.The extension ofFS is intuitionistic fuzzy set IFS presented by Atanassov [20] in 1986. He added a new component in the definition of FS, which is known as the degree of nonmembership or falsity degree. IFS is the generalization of FS with the requirement that the sum of both degrees cannot exceed 1. Many researchers have applied IFSs in decision-making problems. Chen [21] proposed to measure the degree of similarity between vague sets. Similarity measures for discrete, as well as for continuous, sets are given in [22] and applied in pattern recognition problems. More work on IFSs can be found in [23–25]. Fields of applications of IFSs are Computer Science, Engineering, Medicine, Chemistry, Economics, etc.The generalization ofFG is intuitionistic fuzzy graph IFGs explained elaborately by Parvathi and Karunambigai [26]. They also gave the concepts of path, bridge, and cut vertices in IFGs. Dhavudh and Srinivasan [27] defined IFG of second kind, and IFG of nth type was developed by Davvaz et al. [28]. Karunambigai and Buvaneswari [29] introduced arcs in IFGs like strong arcs, weakest arcs, strong path, α-strong, β-strong, and δ-weak arcs. Karunambigai and kalaivani [30] presented IFGs as the matrix representation. Mishra and Pal [31, 32] discussed the product of two interval valued IFGs, their properties, and regular interval valued IFGs in 2013 and 2017. Fallatah et al. [33] and Alanser et al. [34] introduced new concepts as IF soft graphs and bipolar IFGs. Akram and Alshehri [35] introduced IF cycles and trees. Some misconceptions in the definitions of several generalizations of IFGs are corrected by Jan et al. [36]. IFGs are applied in different areas such as cellular network and decision support systems [37, 38].Connectivity is the most fundamental and normal parameter related to a network. The stability of a network depends on its connectivity. Binu et al. introduced two measures on connectivity, namely, cyclicCI and average cyclic CI of FGs [39]. Poulik and Ghorai brought the concept of CI, average CI, and types of connectivity nodes under bipolar fuzzy graph environment with applications [40]. Mathew and Mordeson [41] introduced CI and ACI, studied their properties, and investigated their applications. Binu et al. discussed the concept of Wiener index and relationship between Wiener index and connectivity index with an application to illegal immigration networks [42]. FGs describe only one type of opinion, that is, membership degree, while IFGs describe two types of opinions with the help of membership and nonmembership degrees.In our paper, we have consideredIFG and discussed certain concepts related to IFG. In Section 2, preliminary requirements are given for the work of this paper. In Section 3, we have extended concepts of CI and bounds of CI for IFG. Section 4 provides CI of vertex and edge deleted IF subgraphs. Section 5 presents concepts of the strongest strong cycles, θ-evaluation of vertices, cycle connectivity CC, and CI of strong cycle for IFG. Section 6 deals with ACI along with its properties. Applications of CIs are discussed in Section 7. Finally Section 8 concludes this study.


## 2. Preliminaries

Throughout this section, definitions and examples are presented to recall concepts related to IFG, arcs in IFG, and IF-cycles relevant to the present work. Most of the definitions in preliminaries are taken from [29, 35, 43].The notion ofIFG was proposed by Akram and Davvaz [43] and given as follows:Definition 1 (see [43]).
AnIFG is a pair G=N,M such that(1)
V=u1,u2,u3,…,un with TN:V⟶0,1 and FN:V⟶0,1 representing the truth-membership degree and falsity-membership degree of the vertex ui∈V and 0≤TNui+FNui≤1 for each ui∈V0≤i≤n.(2)
E⊆V×V with TM:E⟶0,1 and FM:E⟶0,1 being as follows:(1)TMui,uj≤minTNui,TNuj,FMui,uj≥maxFNui,FNuj,and 0≤TMui,uj+FMui,uj≤1 for every edge ui,uj∈E.
The next definition is related to completeIFG given by Parvathi et al. [29].Definition 2 (see [29]).
AnIFGG is complete if TMui,uj=minTNui,TNuj and FMui,uj=maxFNui,FNujfor eachui,uj∈E.
The role of path is extremely famous and important inIFGs. The following definition gives us the concept of path in IFGs.Definition 3.
[29] A sequence u1,u2,u3,…,un of distinct vertices is a path P in an IFG, provided it has one of the conditions given below for someiandj.(1)
TMui,uj>0 and FMui,uj=0(2)
TMui,uj=0 and FMui,uj>0(3)
TMui,uj>0 and FMui,uj>0
The strength of paths plays a significant role inIFG settings. The following definition gives us component-wise and whole strength of paths in IFGs.Definition 4 (see [29]).
LetP=u1,u2,u3,…,un be a path in an IFG. Then, (1)
The T-strength ofP is denoted and defined by ST=minTMui,uj∀i,j(2)
The F-strength ofP is denoted and defined by SF=maxFMui,uj∀i,j(3)
SP=ST,SF is called the strength P if both ST and SF exist for the same edge
The highly connected nodes have significant role to a network. The next definition is about the strength of connectedness between the nodes.Definition 5 (see [29]).
The T-strength of connectedness between two verticesui and uj is defined by CONNTGui,uj=maxST and F-strength of connectedness between ui and uj is defined by CONNFGui,uj=minSF for all possible paths between ui and uj,
whereCONNTG−ui,ujui,uj, CONNFG−ui,ujvi,vj denote the T-strength of connectedness and F-strength of connectedness between ui and uj attained by removing the edge ui,uj from G, respectively.
The following definition gives us the idea of a bridge whose deletion from anIFG increase its number of connected components.Definition 6 (see [29]).
An edgevi,vj is called a bridge in G if either(2)CONNTG−ui,ujui,uj<CONNTGui,uj,CONNFG−ui,ujui,uj≥CONNFGui,ujorCONNTG−ui,ujui,uj≤CONNTGui,uj,CONNFG−ui,ujui,uj>CONNFGui,uj,for someui,uj∈VG.
In other words, deletion ofui,uj reduces the strength of connectedness between any pair of vertices.
The concept of strong and weakest edges is of much importance inIFGs as well as in our study. The next definition is related to the notions of strong and weakest edges.Definition 7 (see [29]).
An edgeui,uj in an IFG is(1)
Strong ifTMui,uj≥CONNTGui,uj and FMui,uj≤CONNFGui,uj for each ui,uj∈V(2)
Weakest ifTMui,uj<CONNTGui,uj and FMui,uj>CONNFGui,uj for each ui,uj∈V
The coming definition gives us the strongest paths between two vertices. This definition is relevant to our work.Definition 8 (see [29]).
A strongest path between two vertices in anIFGG is a path P having its strength equal to CONNTGui,uj and CONNFGui,uj lying in the same edge.
The following definition tells us the concept of strong path.Definition 9 (see [29]).
LetG=N,M be an IFG. A path P:ui−uj in G is called strong path if P consists of only strong edges.Example 1.
In Figure1, TMv1,v2=0.6=CONNTGv1,v2 and FMv1,v2=0.6=CONNFGv1,v2, which implies that v1,v2 is a strong arc. Similarly, v2,v3, v1,v5, v2,v4 are strong arcs and v3,v4, v4,v5 are weakest arcs. Here, P=v1v2v3 is a strong path. In fact, it is the strongest path.
The next definition provides us types of strong arcs inIFG.Figure 1
AnIFG with strong and weakest arcs.Definition 10 (see [29]).
An arcui,uj in an IFGG=N,M is(1)
Calledα-strong if TMui,uj>CONNTG−ui,ujui,uj and FMui,uj<CONNFG−ui,ujui,uj(2)
Calledβ-strong if TMui,uj=CONNTG−ui,ujui,uj and FMui,uj=CONNFG−ui,ujui,uj(3)
Calledδ-weak if TMu,uj<CONNTG−ui,ujui,uj and FMui,uj>CONNFG−ui,ujui,ujExample 2.
In Figure2, the arcs v1,v4,v2,v3,v4,v5,v1,v3 are α-strong, v3,v4 is β-strong, and v1,v2,v3,v5 are δ-weak.
The following definition gives us different types of strong paths inIFGs.Figure 2
AnIFG with α-strong, β-strong, and δ-weak arcs.Definition 11 (see [29]).
A path in anIFG containing only α-strong arcs is called α-strong and a path having only β-strong arcs is called β-strong.
The concept of a cycle has a vital role inIFGs. The following definition gives us the concept of a cycle in IFGs environment.Definition 12 (see [35]).
(1)
G=N,M is called a cycle if G∗=N∗,M∗ is cycle(2)
G is called an IF cycle if G∗=N∗,M∗ is cycle and ∄ unique x,y∈M∗ such that(3)TMx,y=minTMa,b:a,b∈M∗,FMx,y=maxFMa,b:a,b∈M∗.Example 3.
In this example, we takeTNu,FNu=0.3,0.2 for all u∈N∗. Here, minTMu,v=0.2 and maxFMu,v=0.3. Clearly, G=N,M is an IF cycle. The graph is shown in Figure 3.
The main goal of our study is to bring more accuracy and precision to the study of topological indices, especially in the context of connectivity indices.FGs have less information in comparison with IFGs. In particular situations like vagueness and uncertainty, FGs are described by only membership grades, but IFGs are characterized by the two grades known by membership and nonmembership. Due to the description of opinions using two membership grades, IFGs have less information loss as compared to FGs. So, that is why we aim to propose the concepts of several CIs for IFGs and study their applications.Figure 3
AnIF cycle.


## 3. Connectivity Index for Intuitionistic Fuzzy Graphs

When we talk about the network like Internet or transport network, naturally, we think about the connectivity of this network. The connectivity means how stable and dynamic this network is! So, we can say that this measure of connectivity is the most fundamental and natural. The measure of connectivity is already available inFGs. But IFG is the generalization of FG, and it gives better results in situations where FGs are not preferable. So, because of this reason, we have proposed this concept of connectivity from FGs to IFGs. We have made some results of connectivity of FGs to IFGs. We define CI formally as follows.Definition 13.
TheCI of an IFGG=N,M is defined as(4)CIG=∑u,v∈VGTNu,FNuTNv,FNv×CONNGu,v=∑u,v∈VGTNu,FNuTNv,FNvCONNTGu,v,CONNFGu,v=∑u,v∈VGTNuTNvCONNTGu,v+FNuFNvCONNFGu,v=∑u,v∈VGTNuTNvCONNTGu,v+∑u,v∈VGFNuFNvCONNFGu,v=TCIG+FCIG,where TCIG is T-connectivity index of G, FCIG is F-connectivity index of G, and CONNTGu,v and CONNFGu,v are T-strength of connectedness and F-strength of connectedness between u and vExample 4.
Refer to Figure1,(5)TCIG=∑u,v∈VGTNuTNvCONNTGu,v=0.70.60.6+0.70.50.4+0.70.40.4+0.70.60.6+0.70.50.4+0.70.40.4+0.70.60.6+0.50.40.4+0.50.60.4+0.40.60.4=0.252+0.14+0.112+0.252+0.12+0.096+0.216+0.08+0.12+0.096=1.484,FCIG=∑u,v∈VGFNuFNvCONNFGu,v=0.30.30.3+0.30.40.4+0.30.60.6+0.30.40.4+0.30.40.4+0.30.60.6+0.30.40.4+0.40.60.6+0.40.40.6+0.60.40.6=0.027+0.048+0.108+0.048+0.048+0.108+0.048+0.144+0.096+0.144=0.819.
So,CIG=TCIG+FCIG=1.484+0.819=2.303.
It may be observed thatTCIG>FCIG, which shows that the level of FCIG is lower than the level of TCIG in this problem. This comparison is interesting and useful in applications of connectivity index.Theorem 1.
LetG=N,M be a complete IFG with N∗=v1,v2,v3,…,vn such that t1≤t2≤⋯≤tn and s1≥s2≥s3≥⋯≥sn, where ti=TNvi and si=FNvi. Then,(6)CIG=∑i=1n−1ti2∑j=i+1ntj+∑i=1n−1si2∑j=i+1nsj.Proof.
Letv1 be the vertex having least truth-membership value t1. For a complete IFG, CONNTGu,v=TMu,v for each u,v∈N∗. So, TMv1,vi=t1 ; 2≤i≤n and hence, TNv1TNviCONNTGv1,vi=t1⋅ti⋅t1=t12ti ; 2≤i≤n. Taking summation over i, we have(7)∑i=2nTNv1TNviCONNTGv1,vi=∑i=2nt12ti.
Similarly, for vertexv2, we obtain(8)∑i=3nTNv2TNviCONNTGv2,vi=∑i=3nt22ti,and for vertex v3(9)∑i=4nTNv3TNviCONNTGv3,vi=∑i=4nt32ti,and so on, for vertex vn−1(10)∑i=nnTNvn−1TNviCONNTGvn−1,vi=∑i=nntn−12ti.
By adding all the above equations, we get(11)TCIG=∑i=2nt12ti+∑i=3nt22ti+∑i=4nt32ti+⋯+∑i=nntn−12ti=∑i=1n−1ti2∑j=i+1ntj.
Now, letv1 be the vertex with the largest falsity-membership value s1. For a complete IFG, CONNFGu,v=FMu,v for each u,v∈N∗. Thus, FMv1,vi=s1 ; 2≤i≤n and hence, FNv1FNviCONNFGv1,vi=s1⋅si⋅s1=s12si ; 2≤i≤n. Summing over i, we have(12)∑i=2nFNv1FNviCONNFGv1,vi=∑i=2ns12si.
Similarly, for vertexv2, we have(13)∑i=3nFNv2FNviCONNFGv2,vi=∑i=3ns22si,and for vertex v3(14)∑i=4nFNv3FNviCONNFGv3,vi=∑i=4ns32si,and so on, for vertex vn−1(15)∑i=nnFNvn−1FNviCONNFGvn−1,vi=∑i=nnsn−12si.
By adding all the above equations, we get(16)FCIG=∑i=2ns12si+∑i=3ns22si+∑i=4ns32si+⋯+∑i=nnsn−12si=∑i=1n−1si2∑j=i+1nsj.
Hence, by the definition of connectivity, we see(17)CIG=TCIG+FCIG=∑i=1n−1ti2∑j=i+1ntj+∑i=1n−1si2∑j=i+1nsj.Example 5.
In Figure4, it can be easily seen that k3 is a complete IFG. So,(18)TCIG=∑i=13TNviTNvjCONNTGvi,vj=0.40.50.4+0.50.50.5+0.40.50.4=0.285,FCIG=∑i=13FNviFNvjCONNFGvi,vj=0.50.40.5+0.40.30.4+0.50.30.5=0.223.
Therefore,CIG=TCIG+FCIG=0.285+0.223=0.508.
Now, we use above theorem(19)∑i=1n−1ti2∑j=i+1ntj=∑i=12ti2∑j=i+13tj=t12t2+t3+t22t3=0.420.5+0.5+0.520.5=0.16+0.125=0.285,∑i=1n−1si2∑j=i+1nsj=∑i=12si2∑j=i+13sj=s12s2+s3+s22s3=0.520.4+0.3+0.420.3=0.175+0.048=0.223.
Adding these two summations, we get(20)∑i=1n−1ti2∑j=i+1ntj+∑i=1n−1si2∑j=i+1nsj=0.285+0.223=0.508.
Hence, it is verified that(21)CIG=∑i=1n−1ti2∑j=i+1ntj+∑i=1n−1si2∑j=i+1nsj.Figure 4
A completeIFG with CI=0.508.


## 4. Edge Deleted and Vertex Deleted IFGs with Connectivity Index

TheCI is affected or not by deleting a vertex or an edge. It is based on the nature of vertex and edge to be removed.Example 6.
Taking theCI=2.017G=N,M shown in Figure 5. Here, v1,v4,v2,v3,v4,v5 are α-strong arcs, v1,v3,v3,v4 are β-strong arcs, and v1,v2,v3,v5 are δ-arcs. Then,(22)TCIG=∑i=110TNviTNvjCONNTGvi,vj=0.50.30.3+0.50.40.3+0.50.60.4+0.50.10.1+0.30.40.3+0.30.60.3+0.30.10.1+0.40.60.3+0.40.10.1+0.60.10.1=0.405,FCIG=∑i=110FNviFNvjCONNFGvi,vj=0.40.50.5+0.40.40.5+0.40.40.4+0.40.80.8+0.50.40.5+0.50.40.5+0.50.80.8+0.40.40.5+0.40.80.8+0.40.80.8=1.612.
So, we haveCIG=TCIG+FCIG=0.405+1.612=2.017.
So, we haveCIG−v1,v4=TCIG−v1,v4+FCIG−v1,v4=0.375+1.628=2.003. Thus, CIG−v1,v4<CIG, which means that CI of G has been reduced by deleting α-strong edge v1,v4. The IFG, G−v1,v2 is shown in Figure 6(a). If we delete the β-strong edge v1,v3, then the strength of connectedness between every pair of vertices is invariant, i.e., CONNTGvi,vj=CONNTG−V1,V3vi,vj and CONNFGvi,vj=CONNFG−V1,V3vi,vj, so CIG−v1,v3=CIG. The graph of G−v1,v3 is shown in Figure 6(b). Similarly, when we delete the δ-arcv1,v2, then the strength of connectedness between every pair of vertices does not change, and so is the CI. The graph of G−v1,v2 is shown in Figure 6(c).Figure 5
An IFG withCI=2.017.Figure 6
G−v1,v4, G−v1,v3 and G−v1,v2. (a) G−v1,v4. (b) G−v1,v3. (c) G−v1,v2.
(a)(b)(c)(23)TCIG−v1,v4=∑i=110TNviTNvjCONNTG−v1,v4vi,vj=0.50.30.3+0.50.40.3+0.50.60.3+0.50.10.1+0.30.40.3+0.30.60.3+0.30.10.1+0.40.60.3+0.40.10.1+0.60.10.1=0.375,FCIG−v1,v4=∑i=110FNviFNvjCONNFG−v1,v4vi,vj=0.40.50.5+0.40.40.5+0.40.40.5+0.40.80.8+0.50.40.5+0.50.40.5+0.50.80.8+0.40.40.5+0.40.80.8+0.40.80.8=1.628.Theorem 2.
LetH be the IF subgraph of an IFG G=N,M formed by removing an edge uv∈MG from G. Then, CIG>CIH or CIG<CIH iff uv is a bridge.Proof.
Takeuv as a bridge. According to the definition, there exit u and v such that their strength of connectedness will be decreased. So, we conclude that CIG>CIH or CIG<CIH.
Conversely, suppose thatCIG>CIH or CIG<CIH and consider the possibilities given below.Case 1: suppose thatuv is a δ-arc. Then, CONNTG−uvu,v=CONNTGu,v and CONNFG−uvu,v=CONNFGu,v. So, we have TCIG=TCIH and FCIG=FCIH and therefore, CIG=CIH.Case 2: takeuv as β-strong edge. Then, TMu,v=CONNTG−uvu,v , FMu,v=CONNFG−uvu,v. This implies that there is another u−v path different from uv edge. Therefore, the removal of the arc uv will have no effect on the strength of connectedness between u and v. So, CIG=CIH.Case 3: now, takeuv as α-strong edge. Then, TMu,v>CONNTG−uvu,v, FMu,v<CONNFG−uvu,v. So, the only strongest path is uv edge having strength equal to TMu,v,FMu,v. Then, clearly CIG>CIH, or CIG<CIHTCIG−TCIH>FCIG−FCIH or TCIG−TCIH<FCIG−FCIH since α-strong edges are IF bridges. This implies that uv is a bridge.Corollary 1.
LetH be the IF subgraph of an IFG G=N,M formed by removing an edge uv∈MG from G. Then, CIG=CIH iff uv is either δ edge or β-strong.Corollary 2.
Suppose thatuv an edge of a complete IFG G=N,M. Then, CIG≠CIG−uv iff uv is a unique IF bridge of G.Proof.
LetG=N,M be a complete IFG. Suppose that CIG≠CIG−uv. Then, uv is the only IF bridge of G.
Conversely, letuv be a unique IF bridge of G. Hence, by Theorem 2, it follows that CIG≠CIG−uv.Theorem 3.
LetG1=N1,M1 and G2=N1,M2 be the two isomorphic IFGs. Then, CIG1=CIG2.Proof.
Suppose thatG1=N1,M1 and G2=N2,M2 are isomorphic IFGs. Then, ∃ is a mapping h:N1⟶N2 such that h is bijective and TN1ui=TN2hui and FN1ui=FN2hui for all ui∈N∗ as well as TM1ui,uj=TM2hui,huj and FM1ui,uj=FM2hui,huj for ui,uj∈M∗. As G1 and G2 are isomorphic, then the strength of any strongest path between ui and uj is equal to that between hui and huj in G2. Thus, for u,v∈N∗(24)CONNTG1ui,uj=CONNTG2hui,huj,CONNFG1ui,vuj=CONNFG2hui,huj.
So, we have(25)TCIG1=∑ui,uj∈VG1TN1uiTN1ujCONNTG1ui,uj=∑hui,huj∈VG2TN2huiTN2hujCONNTG2hui,huj=TCIG2,FCIG1=∑ui,uj∈VG1FN1uiFN1ujCONNFG1ui,uj=∑hui,huj∈VG2FN2huiFN2hujCONNFG2hui,huj=FCIG2.
Thus,TCIG1+FCIG2=TCIG2+FCIG2. This implies that CIG1=CIG2.


## 5. Strongest Strong Cycles,θ-Evaluation of Vertices, Cycle Connectivity, and CI of Strong Cycle

This section contains some concepts aboutIF cycles. IF strongest strong cycles, θ-evaluation of IF vertices, IF cycle connectivity, and CI of IF strong cycles are defined in the current section. Also, some properties related to these concepts are studied.Definition 14.
The truth and falsity values of the weakest edge in a cycleC are defined to be the strength of C in an IFGG.Definition 15.
LetC denote a cycle in an IFGG. Then, C is called IF strongest strong cycle (IFSSC) if it is the union of two strongest strong u−v paths for each of u and v in C with the exception when uv in C is an IF bridge of G.Remark 1.
We observe that whenuv is an IF bridge of G and it lies in C, the condition for C to be the union of two strongest strong u to v paths can be omitted for u and v. Also, CONNTGa,b=CONNTCa,b and CONNFGa,b=CONNFCa,b∀a,b∈G.Definition 16.
IfC is a cycle in an IFGG, then C is said to be strong, provided each of its edges is strong.Example 7.
In Figure7, we take TNu,FNu=0.6,0.3 for all u∈N∗. The edges v1,v2,v3,v5 are bridges in G. C1=v1v2v3v5v1 and C2=v5v3v4v5 are strongest strong cycles as C3=v1v2v3v4v5v1 is not the union of two strongest strong v3−v5 paths. So, it is not a strong cycle. Also, CONNTGv3,v5=0.6 and CONNFGv3,v5=0.3. But C3 has no strongest v3−v5 path. Moreover, we see that v3,v5 is a fuzzy bridge of G that is outside of C3.Figure 7
Strongest strong cycles.Definition 17.
LetG be an IFG. Then, Tθ-evaluation of two vertices ui and uj in G is the set θTui,uj defined by(26)θTui,uj=α:α∈0,1,where α represents T-strength of a SC passing through both ui and uj. Similarly, Fθ-evaluation of ui and uj is the set θFui,uj defined by(27)θFu,v=β:β∈0,1,where β represents F-strength of the same SC passing through both ui and uj.Note 1.
If cycles throughu and v do not exist, then θTui,uj=ϕ and θFui,vj=ϕ. With this θ-evaluation, we define another connectivity measure in IFGs called cyclic connectivity (CC).Definition 18.
LetG be an IFG. Then, cycle T-connectivity between ui and uj in G is denoted and defined by(28)TCui,ujG=∨α:α∈θTui,uj;ui,uj∈N∗.
Similarly, cycle F-connectivity betweenui and uj in G is denoted and defined by(29)ℱCui,ujG=∧β:β∈θFui,uj;ui,uj∈N∗.Note 2.
IfθTui,uj=ϕ and θFui,uj=ϕ for any two vertices ui and uj, then we define cycle T-connectivity and cycle F-connectivity to be zero, i.e., TCu,vG=0 and ℱCu,vG=0.Example 8.
From Figure8, we have θTv1,v3=0.7,0.6 and hence, TCv1,v3G=0.7. Similarly, θFv1,v3=0.3,0.4 and therefore, ℱCv1,v3G=0.3.Figure 8
The cycle connectivity.Theorem 4.
LetG be an IFG and for any ui,uj∈N∗, both ui and uj lie on a common SC. Then,(30)CIG=∑ui,uj∈N∗TNuiTNujTCui,ujG+∑ui,uj∈N∗FNuiFNujℱCui,ujG.Proof.
Suppose thatui,uj∈N∗ such that both of them lie on a common IFSSC. Then, TCui,ujG=maxα:α∈θTui,uj;ui,uj∈N∗, where θTui,uj=α∈0,1:αisT−strength of a strong cycle throughuandv. Therefore, CONNTGui,uj=TCui,ujG and hence,(31)TCIG=∑ui,uj∈N∗TNuiTNujTCui,ujG.
Similarly,ℱCui,ujG=minβ:β∈θFui,uj;ui,uj∈N∗, where θFui,uj=β∈0,1:βisF−strength of a strong cycle throughuandv. Thus, we obtain CONNFGui,uj=ℱCui,ujG and hence,(32)FCIG=∑ui,uj∈N∗FNuiFNujℱCui,ujG.
So, we obtain(33)CIG=∑ui,uj∈N∗TNuiTNujTCui,ujG+∑ui,uj∈N∗FNuiFNujℱCui,ujG.


## 6. Average Connectivity Index of an IFG

The concept of averageconnectivity index is present in the literature of FGs. So, we introduced this concept for IFGs. The stability of a network is guaranteed by its average flow.Example 9.
LetG=N,M be the IFG in Figure 9, with TNv,FNv=0.9,0.1 for all v∈N∗. Then, by routine calculations, we have TCIG=1.944,FCIG=0.006 and CIG=1.95.
From Example10, CIG=1.95 and the number of pairs in G is 42=6. By averaging the TCIG,FCIG and CIG, we get ATCIG=1/6TCIG=1/61.944=0.324, AFCIG=1/6FCIG=1/60.006=0.001 and ACIG=0.324+0.001=0.325. Now, consider G−V4, and we have TCIG−v4=1.701,FCIG−v4=0.003 and CIG−v4=1.701+0.003=1.704. On averaging them, we have ATCIG−v4=1.701/3=0.567,AFCIG−v4=0.003/3=0.001 and ACIG−v4=0.567+0.001=0.568. The overall connectivity of G is increased by deleting vertex v4 from G. The following definitions and results are led by this example.Figure 9
An IFGG and G−v4. (a) An IFG with CIG=1.95. (b) G−v4.
(a)(b)Definition 19.
LetG=N,M be an IFG. Then, the average T-connectivity index of G is denoted and defined as(34)ATCIG=1n2∑u,v∈N∗TNuTNvCONNTGu,v,and the average F-connectivity index of G is denoted and defined as(35)AFCIG=1n2∑u,v∈N∗FNuFNvCONNFGu,v,where CONNTGu,v is the T- strength of connectedness, and CONNFGu,v is the F- strength of connectedness between the nodes u and v.Definition 20.
LetG=N,M be an IFG. The average connectivity index of G is defined to be the sum of average T- connectivity index and average F- connectivity index of G, i.e.,(36)ACIG=1n2∑u,v∈N∗TNuTNvCONNTGu,v+1n2∑u,v∈N∗FNuFNvCONNFGu,v=ATCIG+AFCIG,where CONNTGu,v is the T-strength of connectedness, and CONNFGu,v is the F-strength of connectedness between the nodes u and v.Note 3.
Obviously,CIG will not be enhanced by the removal of an edge, and therefore, ACIG also. For an IFG, 0≤ACIG≤1.Definition 21.
LetG=N,M be an IFG and u∈N∗. Then, u is said to be an IF connectivity reducing node IFCRN of G if ACIG−u<ACIG. u is said to be an IF connectivity enhancing node IFCEN of G if ACIG−u>ACIG. u is said to be an IF neutral node of G if ACIG−u=ACIG.Example 10.
Consider theIFGG as shown in Figure 10. We have taken here TNu,FNu=0.5,0.5 for all u∈N∗. ACIG=0.16667,ACIG−v1=0.1583,ACIG−v2=0.16667,ACIG−v3=0.1583, ACIG−v4=0.1833.ACIG−v1,ACIG−v3<ACIG,ACIG−v2=ACIG and ACIG−v4>ACIG.Thus v1,v3 are IFCRNs, v2 is an IF neutral node, and v4 is a IFCEN. We characterize these nodes using CI in the following theorem.Figure 10
IFG withIFCRN, IFCEN, and IF neutral nodes.Theorem 5.
LetG=N,M be an IFG and u∈N∗ with N∗≥3. Let r=CIG/CIG−u. u is a IFCEN iff r<n/n−2. u is a IFCRN iff r>n/n−2. u is an IF neutral node iff r=n/n−2.Proof.
Suppose thatu is an IF neutral node. Then, by definition, ACIG−u=ACIG. By definition of average connectivity index, we obtain(37)1n2CIG=1n−12CIG−u.
From here, we get(38)CIGCIG−u=n2n−12=nn−1/2n−1n−2/2=nn−2.
The converse part can be proved by reversing the arguments. Similarly, we can prove other cases.Note 4.
An isolated vertexw in an IFG, and G=N,M is an IFCEN. Further, if G¯=N¯,M¯ denote complement of IFGG=N,M, then 0≤CIG¯≤n2 and therefore,(39)CIG+CIG¯≤n2+n2=2n2=2nn−12=nn−1.Theorem 6.
LetG=N,M be an IFG with N∗≥3. If w∈N∗ is an end vertex of G, let l=∑u∈N∗−wCONNTGu,w+∑u∈N∗−wCONNFGu,w. Then,(1)
w is an IFCEN if l<2/n−2CIG−w(2)
w is an IFCRN if l>2/n−2CIG−w(3)
w is an IF neutral node if l=2/n−2CIG−wProof.
Letw be an IF neutral node. Then, by definition, ACIG−w=ACIG. We see that(40)CIG=CIG−w+∑u∈N∗−wCONNTGu,w+∑u∈N∗−wCONNFGu,wCIG=CIG−w+l,l=CIG−CIG−w,1n2l=1n2CIG−1n2CIG−w=1n−12CIG−w−1n2CIG−w=CIG−w1n−12−1n2,l=CIG−wn2n−12−1=2n−2CIG−w.
The converse part can be proved by reversing these steps. Similarly, we can prove other parts.
At this stage, we can classify anIFG depending on the nature of vertices in it.Definition 22.
AnIFGG containing at least one IFCEN is called an IF connectivity enhancing graph. If there is no IFCEN node in G and at least one IFCRN, then G is said to be an IF connectivity reducing graph. If G has all vertices as IF neutral nodes, then it is said to be an IF neutral graph.


## 7. Applications

In this section, two applications are given. One is on Internet routing, and the other is on transport network flow.


### 7.1. Internet Routing

The strength of connectedness between points in a network has much importance in various areas, for example, shortest path problem, routing problem, network flow problem, and maximum band width problem. Consider a networkG that connects routers in a part of a network. For convenience in calculations, we have taken TNv,FNv=0.9,0.1 for all v. Here, the edge values represent maximum bandwidth between the corresponding routers. Membership value of the edge represents correct information and nonmembership value for incorrect information. Also, if P is a path connecting two routers in the network, then STP denotes the truth bandwidth of P and similarly, SFP is for falsity bandwidth of P. Hence, CONNTGu,v and CONNFGu,v denote the maximum possible truth bandwidth and the minimum possible falsity bandwidth between u and v. Consider the fuzzified network as shown in Figure 11.Figure 11
Fuzzified Network of routers.After computations, we haveTCIG=20.3715,ATCIG=20.3715/55=0.3704, FCIG=0.055,AFCIG=0.055/55=0.001. By removal routers v11 and v2, we obtain TCIG−v11=15.50745,ATCIG−v11=15.50745/45=0.34461,FCIG−v11=0.045, TCIG−v11=15.50745,ATCIG−v11=15.50745/45=0.34461,FCIG−v11=0.045, ATCIG−v2=18.3465/45=0.4077,FCIG−v2=0.045,AFCIG−v2=0.001. Hence, CIG=TCIG+FCIG=20.3715+0.055=20.4265,ACIG=0.3714, ACIG−v11=0.34561,ACIG−v2=0.4087. We see that ACIG−v2>ACIG and ACIG−v11<ACIG. This implies that v2 is IFCEN, and v11 is a IFCRN.This problem has less incorrect information becauseTCIG>FCIG. Also, the average bandwidth of the network is increased by the removal of router v2 and removal of v11 causes reduction in average bandwidth.


### 7.2. Transport Network Flow

Consider a directed networkG of traffic flow as shown in Figure 12, which is fuzzified. Conveniently, we have taken TNv,FNv=0.8,02 for all v∈VG. The connectivity of directed IFG and undirected IFG is similar. So, we can extend these concepts for directed IFG. The vertices are junctions containing correct and incorrect values for vehicles. The edges represent roads connecting two junctions, and their weights indicate number of vehicles consisting of correct and incorrect information. Now, we discuss some connectivity properties of the network flow. Firstly, we find the associated T-connectivity matrix TMG← of the directed IFG.(41)TMG←=00.40.40.40.60.800.20.40.60.80.800.40.60.60.60.600.60.40.40.40.40.Figure 12
Transport flow network.As the graph is directed, the above matrix is not symmetric. So, we need to sum up all the elements of the matrix. Thus,TCIG←=6.656,ATCIG←=6.656/10=0.6656. Now, the associated F- connectivity matrix FMG← is given as follows.(42)FMG←=00.40.40.40.40.200.40.40.40.20.200.40.40.40.40.400.40.40.40.40.40.By adding all the entries ofFMG←, we obtain FCIG←=0.296,AFCIG←=0.296/10=0.0296. Thus, ACIG←=ATCIG←+AFCIG←=0.6656+0.0296=0.6952.ConsiderG←−v5. It is also directed IFG. The matrices TMG←−v5 and TMG←−v5 are given by(43)TMG←−v5=00.20.20.20.800.20.20.80.800.20.60.60.60,FMG←−v5=00.60.60.60.200.60.60.20.200.60.40.40.40.By calculations, we haveTCIG←−v5=3.456,ATCIG←−v5=3.456/6=0.576 and FCIG←−v5=0.216,AFCIG←−v5=0.216/6=0.036. Thus, ACIG←−v5=ATCIG←−v5+AFCIG←−v5=0.576+0.036=0.612. As ACIG←>ACIG←−v5, which implies that v5 is IFCRN.Next, we considerG←−v1. The matrices TMG←−v1 and TMG←−v1 are given by(44)TMG←−v1=00.20.20.20.800.20.20.60.600.20.40.40.40,FMG←−v1=00.60.60.60.200.60.60.40.400.60.40.40.40.We have after calculations,TCIG←−v1=2.816,ATCIG←−v1=2.816/6=0.4693 and FCIG←−v1=6.232,AFCIG←−v1=0.232/6=0.0387. Thus, ACIG←−v1=ATCIG←−v1+AFCIG←−v1=0.4693+0.0387=0.508. As ACIG←>ACIG←−v1, which implies that v1 is also IFCRN.ConsiderG←−v2. The matrices TMG←−v2 and TMG←−v2 are given by(45)TMG←−v2=00.40.40.6000000.60000.40.40,FMG←−v2=00.60.40.4000000.40000.40.40.We calculate using above matricesTCIG←−v2=1.792,ATCIG←−v2=1.792/6=0.2987 and FCIG←−v2=1.664,AFCIG←−v2=1.664/6=0.2773. Thus, ACIG←−v2=ATCIG←−v2+AFCIG←−v2=0.2987+0.2773=0.576. As ACIG←>ACIG←−v2, which implies that v2 is also IFCRN. Now, we consider G←−v3. The matrices TMG←−v3 and TMG←−v3 are given by(46)TMG←−v3=000.40.600.40.60000000.40,FMG←−v3=000.40.40.200.40.40000000.40.From the above matrices, we haveTCIG←−v3=1.8,ATCIG←−v3=1.8/6=0.3 and FCIG←−v3=0.1375,AFCIG←−v3=0.1375/6=0.0229. Thus, ACIG←−v3=ATCIG←−v3+AFCIG←−v3=0.3+0.0229=0.3229. As ACIG←>ACIG←−v3, which implies that v3 is also IFCRN. Finally, we consider G←−v4. The matrices TMG←−v4 and TMG←−v4 are given by(47)TMG←−v4=00.40.40.60.800.40.60.80.800.60.40.40.40,FMG←−v4=00.60.60.40.200.60.40.20.200.40.60.60.60.From the above matrices, we obtainTCIG←−v4=4.224,ATCIG←−v4=4.224/6=0.704 and FCIG←−v4=0.216,AFCIG←−v4=0.216/6=0.036. Thus, ACIG←−v4=ATCIG←−v4+AFCIG←−v4=0.704+0.036=0.74. As ACIG←<ACIG←−v4, which implies that v4 is IFCEN. So, the removal of junction v4 increases the average connectivity amongst the other junctions. Table 1 shows that there is a small difference between ACIG← and ACIG←−v4. So, the removal of v4 has no too much effect on the network.Table 1
Difference in connectivity indices after removing a vertex.
G−viACIG−viACIG−ACIG−viG−v10.5080.1872G−v20.5760.1192G−v30.32290.3723G−v40.740.0448We also see that the difference betweenACIG← and ACIG←−v3 is highest than other differences, so the removal of v3 has maximum negative effects on the connectivity.


## 7.1. Internet Routing

The strength of connectedness between points in a network has much importance in various areas, for example, shortest path problem, routing problem, network flow problem, and maximum band width problem. Consider a networkG that connects routers in a part of a network. For convenience in calculations, we have taken TNv,FNv=0.9,0.1 for all v. Here, the edge values represent maximum bandwidth between the corresponding routers. Membership value of the edge represents correct information and nonmembership value for incorrect information. Also, if P is a path connecting two routers in the network, then STP denotes the truth bandwidth of P and similarly, SFP is for falsity bandwidth of P. Hence, CONNTGu,v and CONNFGu,v denote the maximum possible truth bandwidth and the minimum possible falsity bandwidth between u and v. Consider the fuzzified network as shown in Figure 11.Figure 11
Fuzzified Network of routers.After computations, we haveTCIG=20.3715,ATCIG=20.3715/55=0.3704, FCIG=0.055,AFCIG=0.055/55=0.001. By removal routers v11 and v2, we obtain TCIG−v11=15.50745,ATCIG−v11=15.50745/45=0.34461,FCIG−v11=0.045, TCIG−v11=15.50745,ATCIG−v11=15.50745/45=0.34461,FCIG−v11=0.045, ATCIG−v2=18.3465/45=0.4077,FCIG−v2=0.045,AFCIG−v2=0.001. Hence, CIG=TCIG+FCIG=20.3715+0.055=20.4265,ACIG=0.3714, ACIG−v11=0.34561,ACIG−v2=0.4087. We see that ACIG−v2>ACIG and ACIG−v11<ACIG. This implies that v2 is IFCEN, and v11 is a IFCRN.This problem has less incorrect information becauseTCIG>FCIG. Also, the average bandwidth of the network is increased by the removal of router v2 and removal of v11 causes reduction in average bandwidth.


## 7.2. Transport Network Flow

Consider a directed networkG of traffic flow as shown in Figure 12, which is fuzzified. Conveniently, we have taken TNv,FNv=0.8,02 for all v∈VG. The connectivity of directed IFG and undirected IFG is similar. So, we can extend these concepts for directed IFG. The vertices are junctions containing correct and incorrect values for vehicles. The edges represent roads connecting two junctions, and their weights indicate number of vehicles consisting of correct and incorrect information. Now, we discuss some connectivity properties of the network flow. Firstly, we find the associated T-connectivity matrix TMG← of the directed IFG.(41)TMG←=00.40.40.40.60.800.20.40.60.80.800.40.60.60.60.600.60.40.40.40.40.Figure 12
Transport flow network.As the graph is directed, the above matrix is not symmetric. So, we need to sum up all the elements of the matrix. Thus,TCIG←=6.656,ATCIG←=6.656/10=0.6656. Now, the associated F- connectivity matrix FMG← is given as follows.(42)FMG←=00.40.40.40.40.200.40.40.40.20.200.40.40.40.40.400.40.40.40.40.40.By adding all the entries ofFMG←, we obtain FCIG←=0.296,AFCIG←=0.296/10=0.0296. Thus, ACIG←=ATCIG←+AFCIG←=0.6656+0.0296=0.6952.ConsiderG←−v5. It is also directed IFG. The matrices TMG←−v5 and TMG←−v5 are given by(43)TMG←−v5=00.20.20.20.800.20.20.80.800.20.60.60.60,FMG←−v5=00.60.60.60.200.60.60.20.200.60.40.40.40.By calculations, we haveTCIG←−v5=3.456,ATCIG←−v5=3.456/6=0.576 and FCIG←−v5=0.216,AFCIG←−v5=0.216/6=0.036. Thus, ACIG←−v5=ATCIG←−v5+AFCIG←−v5=0.576+0.036=0.612. As ACIG←>ACIG←−v5, which implies that v5 is IFCRN.Next, we considerG←−v1. The matrices TMG←−v1 and TMG←−v1 are given by(44)TMG←−v1=00.20.20.20.800.20.20.60.600.20.40.40.40,FMG←−v1=00.60.60.60.200.60.60.40.400.60.40.40.40.We have after calculations,TCIG←−v1=2.816,ATCIG←−v1=2.816/6=0.4693 and FCIG←−v1=6.232,AFCIG←−v1=0.232/6=0.0387. Thus, ACIG←−v1=ATCIG←−v1+AFCIG←−v1=0.4693+0.0387=0.508. As ACIG←>ACIG←−v1, which implies that v1 is also IFCRN.ConsiderG←−v2. The matrices TMG←−v2 and TMG←−v2 are given by(45)TMG←−v2=00.40.40.6000000.60000.40.40,FMG←−v2=00.60.40.4000000.40000.40.40.We calculate using above matricesTCIG←−v2=1.792,ATCIG←−v2=1.792/6=0.2987 and FCIG←−v2=1.664,AFCIG←−v2=1.664/6=0.2773. Thus, ACIG←−v2=ATCIG←−v2+AFCIG←−v2=0.2987+0.2773=0.576. As ACIG←>ACIG←−v2, which implies that v2 is also IFCRN. Now, we consider G←−v3. The matrices TMG←−v3 and TMG←−v3 are given by(46)TMG←−v3=000.40.600.40.60000000.40,FMG←−v3=000.40.40.200.40.40000000.40.From the above matrices, we haveTCIG←−v3=1.8,ATCIG←−v3=1.8/6=0.3 and FCIG←−v3=0.1375,AFCIG←−v3=0.1375/6=0.0229. Thus, ACIG←−v3=ATCIG←−v3+AFCIG←−v3=0.3+0.0229=0.3229. As ACIG←>ACIG←−v3, which implies that v3 is also IFCRN. Finally, we consider G←−v4. The matrices TMG←−v4 and TMG←−v4 are given by(47)TMG←−v4=00.40.40.60.800.40.60.80.800.60.40.40.40,FMG←−v4=00.60.60.40.200.60.40.20.200.40.60.60.60.From the above matrices, we obtainTCIG←−v4=4.224,ATCIG←−v4=4.224/6=0.704 and FCIG←−v4=0.216,AFCIG←−v4=0.216/6=0.036. Thus, ACIG←−v4=ATCIG←−v4+AFCIG←−v4=0.704+0.036=0.74. As ACIG←<ACIG←−v4, which implies that v4 is IFCEN. So, the removal of junction v4 increases the average connectivity amongst the other junctions. Table 1 shows that there is a small difference between ACIG← and ACIG←−v4. So, the removal of v4 has no too much effect on the network.Table 1
Difference in connectivity indices after removing a vertex.
G−viACIG−viACIG−ACIG−viG−v10.5080.1872G−v20.5760.1192G−v30.32290.3723G−v40.740.0448We also see that the difference betweenACIG← and ACIG←−v3 is highest than other differences, so the removal of v3 has maximum negative effects on the connectivity.


## 8. Conclusion

We have developed someCIs in the IFGs framework due to the reason that IFGs cover uncertainty and vagueness with the help of two membership grades. Some key results of our study are as follows:(i)
We introduced the notion ofCI for IFGs and developed results on CI. Examples are also given to support results of CI.(ii)
CI of edge and vertex deleted IFGs with an example is also given.(iii)
We developedSSC, θ-evaluation of vertices, CC, and CI of SC and related results.(iv)
ACI of IFG is defined.(v)
Types of connectivity nodes, namely,CEN,CRN, and neutral node, and results on them are introduced.(vi)
Applications in two types of networks, namely, Internet routing and transport flow network.In the future, we aim to extend our work to the environment of picture fuzzy graphs [44] and T-spherical fuzzy graphs [45]. We also aim to introduce some other connectivity indices in IFGs and investigate their applications.


## 9. Advantages

The main advantages and characteristics of our study are as follows:(i)
The main feature of our study is to develop the concept ofCIs under IFGs environment due to the fact that IFGs handle uncertain information with two membership grades.(ii)
IFGs are described by two types of components that is membership and nonmembership, while FGs are characterized by only one component.(iii)
Our findings are the generalization of the results ofCIs in FGs. For example, if we neglect the second component, then our results are converted into the results of FGs. Thus, the results of CIs in FGs become the special case of our results in IFGs.(iv)
Comparatively, our study proves thatIFGs would have less loss of information as compared to FGs.



---

*Source: 4156879-2021-08-14.xml*