# Wiener Index of Intuitionistic Fuzzy Graphs with an Application to Transport Network Flow

Authors: Tulat Naeem; Muhammad Kamran Jamil; Khawaja Muhammad Fahd; Abdu alAmeri
Journal: Complexity
 (2022)

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## Abstract

The Wiener indexWI is one of the connectivity parameters used to know the biochemical and physicochemical properties of compounds depending upon their molecular structures. Intuitionistic fuzzy graphs IFGs are a convenient tool to represent the objects and relations between them with two types of information using truth membership degree and falsity membership degree. This research work presents the concept of WI under the structure IFGs, IF trees, and IF cycles. Some bounds on WI are investigated. The relationship between WI and connectivity index CI is also studied. In the end of this study, an application of the WI in transport network flow is proposed.


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## Body

## 1. Introduction

Zadeh [1] was the first person who presented his first publication on fuzzy sets (FSs) in 1965. He generalized crisp set by giving membership grades to every object of the set in the interval 0,1. Various concepts of ordinary sets were established for FSs. Further explorations in the work of Zadeh were made by Goguen [2] in 1967. Zadeh [3] discussed models including constraints and goals for the betterment and development of different sections of a country’s society in 1970. Interval-valued FSs were presented bySambuc [4] in 1975 under the name of H-fuzzy set, which is proposed by Zadeh [5] taken as a special case of type 2 FSs. In 1999, the authors in [6] studied a method for the calculation of correlation coefficient under fuzzy data. The value obtained by this formula provides the strength of correlation between the sets and positivity or negativity of this correlation. Ashraf et al. introduced spherical FSs and proposed their applications [7]. Mahmood et al. [8] introduced spherical and T-spherical FSs along with some operations on them. They discussed some problems using these sets. There have been a lot of publications on FSs and more to come. There are various areas in which FSs are applied, for example, pattern recognition [9], psychology [10], and decision-making [11].The graph is a convenient source by which objects and their relationships are presented by vertices and edges. When these objects and relationships contain vague information, it is necessary to present them by a FG. The first step in the field of FGs was taken by Rosenfeld [12] in 1975. In 1994, J.N. Mordeson [13] defined some operations on FGs. Bhattacharya [14] showed an association between FG and fuzzy group in 1987. The researchers completed a lot of work on FGs in the duration of 1981 to 1990. After that, the concept of domination in the environment of FGs was introduced by Somasundaram [15] in 1998. Akram [16] developed the concept of bipolar FGs in 2011, discussed different ways of their construction, and investigated some properties on them. By the same author [17], irregular bipolar FGs of different types were introduced in 2013. They also developed the concept of regular bipolar FGs [18]. FGs have applications in various fields such as controversial problems, decision-making, and social networks [19–21].FSs contain only membership information. Atanassov [22] was the first one who made the generalization of FSs by considering nonmembership information and invented a new set called intuitionistic fuzzy set (IFS). After this consideration, researchers started to generalize various concepts of FSs into the frame of IFSs. In 1991, Gerstenkorn and Manko [23] defined a function to measure the correlation of IFSs and introduced a coefficient of this correlation to analyze some of its properties. Angelov [24] extended the concept of fuzzy optimization and introduced IF optimization in 1997. The authors in [25] proposed the concept of distance between IFSs in 2000. Atanassov [26] introduced interval-valued IFs and discussed their properties. Li [27] investigated multiattribute decision-making under IFSs. From application point of view, IFSs have application in medicine [28], pattern classification [29], decision-making [30], etc.IF relations andIFGs are the foundations of Atanassov [31]. Akram et al. defined IFGs in a precise way and presented various concepts such as strong IFGs, balanced IFGs, IF hypergraphs, IF trees, and IF cycles [32–35]. Karunambigai et al. [36] introduced arcs in IFGs and classified them. The idea of nth type of IFGs was proposed by the authors in [37]. The concept of domination was developed by Parvathi [38]. Akram [39] introduced double domination for IFGs. Sun et al. [40] proposed the framework to increase the degree of consensus in the group of decision-making. The authors also made an analysis, and a numerical example is given to illustrate this work. Xing et al. [41] studied the concept of a new Choquet integral for the purpose to obtain the matrix of consensus of all individual experts. The authors also presented an illustration to show the effectiveness of the proposed work. IFGs have a variety of applications, for instance, decision support system, communication networks, and water supplier system [42–44].Mathematical chemistry deals with molecular structure in terms of mathematical techniques. In mathematical chemistry, molecular descriptors play a vital role. Chemical graph theory as a subject develops a connection among chemistry, graph theory, and mathematics. A graph representing atoms and bonds of a compound by vertices and edges is known as a molecular graph. Topological indices (TIs) are scalars connected with a molecular graph used to know the correlation of a chemical structure with many physical properties, biological activity, and chemical reactivity.WI [45] is the first TI proposed by Wiener in 1947. This TI is distance-based. There are many degree-based TIs discovered to know various properties of drugs and compounds, and some of them are Randic index, harmonic index, Zagreb indices, atomic bond connectivity index, geometric arithmetic index, and augmented Zagreb index. Binu et al. introduced CIs and WI for FGs with applications [46, 47]. The TIs based on vertex degrees such as Zagreb indices, Randic index, and harmonic index are defined in [48] for FGs with the discussion on two applications. Naeem et al. [49] proposed the concept of CIs under IFGs environment with applications in two types of networks. TIs are widely used in chemistry, mathematics, and pharmacy engineering [50–52]. Besides, TIs also have applications in human trafficking, Internet routing, and transport network flow [46–49].The setting of our research article is given as follows. Preliminary requirements are arranged in Section2. Throughout Section 3, the concept of WI with illustrations and some bounds for it is proved. The relationship between the WI and CI including supporting examples is presented in Section 4. An application of WI for IFGs is proposed in Section 5. In the last two sections, the conclusion and advantages of our proposed work are given.


## 2. Preliminaries

To understand this research paper in a better way, we give some definitions along with some results throughout the preliminary section.Definition 1.
(see [32]). A pair χ=N,M under V and E known as the sets of vertices and edges is an IFG satisfying the following:(1)
TN,FN:V⟶0,1 are functions that stand for the truth membership degree, and FNwi indicates the falsity membership degree of every wi∈V under the condition that TNwi+FNwi∈0,1.(2)
TM,FM:E⟶0,1 are functions, TMwi,wj represents the truth membership degree, and FMwi,wj denotes falsity membership degree of wi,wj∈E with the following:(1)TMwi,wj≤minTNwi,TNwj,FMwi,wj≥maxFNwi,FNwj,such that TMwi,wj+FMwi,wj∈0,1.Definition 2.
(see [53]). A partial subgraph of an IFG is also an IFGH=N′,M′ such that(1)
TN′wi≤TNwi and FN′wi≥FNwi(2)
TM′wi,wj≤TMwi,wj and FM′wi,wj≥FMwi,wjDefinition 3.
(see [36]). An IFG is known as complete if TMwi,wj=∧TNwi,TNwj and FMwi,wj=∨FNwi,FNwj.Definition 4.
(see [36]). A vertex sequence v1,v2,v3,…,vn with vi≠vj for all i≠j is known as an IF path if one of the following holds for at least one i and j.(1)
TMwi,wj>0 and FMwi,wj=0(2)
TMwi,wj=0 and FMwi,wj>0(3)
TMwi,wj>0 and FMwi,wj>0Definition 5.
(see [36]). The T strength and F strength of a path P=v1v2v3…vn are defined by ST=minTMvi,vj and SF=maxFMvi,vj for every i and j. The pair containing both the strengths is known as the strength of P provided both of them lie on the common arc.Definition 6.
(see [36]). The T strength and F strength of connectedness between two vertices vi and vj are denoted and defined by CONNTχvi,vj=maxST and CONNFχvi,vj=minSF for all paths P:vi−vj.Definition 7.
(see [36]). An arc wi,wj is known as a bridge in χ if it has one of the following:(1)
CONNTχ−wi,wjwi,wj<CONNTχwi,wj and CONNFχ−wi,wjwi,wj≥CONNFχwi,wj(2)
CONNTχ−wi,wjwi,wj≤CONNTχwi,wj and CONNFχ−wi,wjwi,wj>CONNFχwi,wjDefinition 8.
(see [36]). An edge wi,wj is strong if TMwi,wj≥CONNTχwi,wj and FMwi,wj≤CONNFχwi,wj, and wi,wj is weakest if TMwi,wj<CONNTχwi,wj and FMwi,wj>CONNFχwi,wj.Definition 9.
(see [36]). If P is a path between any two nodes vi and vj such that STP=CONNTχvi,vj and SFP=CONNFχvi,vj, then it is called strongest path between vi and vj.Definition 10.
(see [36]). A path P from vertex vi to vertex vj in an IFG is known as strong provided it has just strong arcs.Example 1.
In Figure1, the arcs v1,v2,v1,v3,v1,v4, v2,v3, and v2,v4 are all strong arcs, while v3,v4 is the only weak arc. The path P:v3v2v1v4 is a strong path, as well as strongest path between the vertices v3 and v4.Figure 1
An intuitionistic graphχ.Definition 11.
(see [36]). An arc vi,vj in an IFG is such that(1)
TMvi,vj>CONNTχ−vi,vjvi,vj and FMvi,vj<CONNFχ−vi,vjvi,vj are known as α-strong(2)
TMvi,vj=CONNTχ−vi,vjvi,vj and FMvi,vj=CONNFχ−vi,vjvi,vj are known as β-strong(3)
TMvi,vj<CONNTχ−vi,vjvi,vj and FMvi,vj>CONNFχ−vi,vjvi,vj are known as δ-weakDefinition 12.
(see [35]). A connected IFG χ=N,M is an IF tree if χ has an IF spanning subgraph H=N,M′, which is a tree, where for all arcs vi,vj not in H, TMvi,vj<CONNTHvi,vj,FMvi,vj>CONNFHvi,vj.Example 2.
Here, we have takenTNu,FNu=1,0 for all u∈N. As CONNTHv2,v3=0.7 and CONNFHv2,v3=0.3, we see that TMv2,v3=0.6<CONNTHv2,v3 and FMv2,v3=0.4>CONNFHv2,v3. Thus, χ is an IF tree, and H is spanning tree of χ. The graph is shown in Figure 2.Figure 2
An IF treeχ and its spanning tree H.Definition 13.
(see [49]). The CI of an IFG, χ=N,M, is defined by CIχ=ℐχ+ℱℐχ, where ℐχ and ℱℐχ are T-connectivity and F-connectivity indices of χ, respectively, given as follows:(2)ℐχχ=∑v,v′∈VχTNvTNv′CONNTχv,v,ℱℐχ=∑v,v′∈VχFNvFNv′CONNFχv,v′.Theorem 1.
(see [49]). Let χ=N,M be a complete IFG, and χ=N,M with N∗=w1,w2,w3,…,wn such that r1≤r2≤…≤rn and s1≥s2≥s3≥…≥sn, where ri=TNwi and si=FNwi. Then,(3)CIχ=∑i=1n−1ri2∑j=i+1nrj+∑i=1n−1si2∑j=i+1nsj.
FGs contain less information than IFGs. There are certain situations where we need information from both sides. IFGs are more useful in such type of situations. So, to fulfill these requirements, we aim to establish the concept of WI for IFGs settings. We have made the extension in the results for WI of FGs. One thing is interesting to see that we have two WIs, one for truth membership values and other for falsity membership values, which produce a comparison between them. To get single value, we take the sum of both the values of these indices.


## 3. Wiener Index for Intuitionistic Fuzzy Graphs

The concepts of geodesic path andWI for IFGs are presented in this section. Some relevant illustrations are given for these concepts, and some bounds of WI are investigated.Definition 14.
A pathP between two vertices is known as geodesic if:(1)
P is strong(2)
There does not exist any strong pathQ shorter than P
The T weight and F weight are the sum of truth and falsity membership values assigned to all the arcs in the geodesic.Example 3.
In Figure1, all arcs except v3,v4 are strong. The strongest path between v3 and v4 is P:v3v2v1v4, which is mentioned already in Example 1, and therefore, CONNTχV3,V4=0.2 and CONNFχV3,V4=0.7. Moreover, the strong paths between v3 and v4 are v3v2v1v4,v3v1v4 and v3v2v4. The latter two of them are shortest, and hence, both are geodesic.Definition 15.
TheWI of an IFG, χ=N,M is defined as follows:(4)WIχ=∑ζ,ζ′∈N∗TNζ,FNζTNζ′,FNζ′dsζ,ζ′=∑ζ,ζ′∈N∗TNζ,FNζTNζ′,FNwTdsζ,ζ′,Fdsζ,ζ′=∑ζ,ζ′∈N∗TNζTNζ′Tdsζ,ζ′+∑ζ,ζ′∈N∗FNζFNζ′Fdsζ,ζ′=FWIχ+FWIχ,where Tdsζ,ζ′ is the minimum T weight sum and Fdsζ,ζ′ is the maximum F weight sum of the geodesics from ζ to w and both lie on the same geodesic.Remark 1.
In 3.3, if bothTdsv,v′ and Fdsv,v′ do not lie on the same geodesic, we left it for the user to choose the geodesic according to his will. In that case, one of the two geodesics must be compromised.Example 4.
From Figure1, we have the following:(5)ℐχ=0.50.80.5+0.50.60.5+0.50.20.2+0.80.60.6+0.80.20.2+0.60.20.7=0.774,FWIχ=0.40.20.4+0.40.30.3+0.40.70.7+0.20.30.4+0.20.70.7+0.30.71=0.596,Thus,WIχ=ℐχ+FWIχ=0.774+0.596=1.37.Note 1.
It may be noted thatFWIχ<ℐχ, which shows that this problem has more truth level.Theorem 2.
Letχ=N,M be an IFG with N∗=n and TNw,FNw=λ,μ for all w∈Vχ with 0≤λ+μ≤1. Let p=∧TMw,w′|w,w′∈M∗andw,w′is not aδ−arc and q=∧FMw,w′|w,w′∈M∗andw,w′is not aδ−arc. Then,(6)WIχ≥pλ2+qμ2nn−1−M∗.Proof.
Letχ=N,M be an IFG. Suppose that p=∧TMw,w′|w,w′∈M∗andw,w′is not aδ−arc. Let w,w′∈N∗. If w,w′∈M∗, then Tdsw,w′≥p. The number of unordered pairs v,w satisfying Tdsw,w′≥p is M∗, and the number of unordered vertex pairs v,v′ satisfying Tdsv,v′≥2p is n2−M∗. Then,(7)ℐχ=∑c,d∈N∗TNcTNdTdsc,d=λ2∑c,d∈N∗Tdsc,d=λ2∑w,w′∈N∗Tdsw,w′+∑v,v′∈N∗Tdsv,v′≥λ2M∗p+n2−M∗2p=pλ2M∗+2nn−12−2M∗=pλ2nn−1−M∗.
Now, letq=∧FMw,w′|w,w′∈M∗andw,w′is not aδ−arc. Then, as argued before, we have the following:(8)FWIχ≥qμ2nn−1−M∗.
Thus, we have the following:(9)WIχ=ℐχ+FWIχ≥pλ2+qμ2nn−1−M∗.Theorem 3.
Letχ1=N1,M1 and χ2=N2,M2 be two IFGs such that χ1 is isomorphic to χ2. Then, WIχ1=WIχ2.Proof.
Suppose thatχ=N1,M1 and χ2=N2,M2 are two isomorphic IFGs. Then, there exists a bijective mapping f:N1∗⟶N2∗ such that TN1w=TN2fw and FN1w=FN2fw for each w∈N1∗ and TM1w,w′=TM2fw,fw′,FM1w,w′=FM2fw,fw′ for every edge w,w′∈M1∗. Suppose Pw,w′ be the path for each w,w′∈N1∗ serving as Tdsw,w′ and Fdsw,w′. Then, there will be the corresponding path Pfw,fw′′ in χ2 serving as Tdsfw,fw′ and Fdsfw,fw′ such that TM1v,v′=TM2fv,fv′,FM1v,v′=FM2fv,fv′ for each edge v,v′∈Pw,w′. Since all properties are preserved between both the graphs, Tdsw,w′=Tdsfw,fw′ and Fdsw,w′=Fdsfw,fw′. Then,(10)ℐχ1=∑w,w′∈N1∗TN1wTN1w′Tdsw,w′=∑fw,fw′∈N2∗TN2fwTN2fw′Tdsfw,fw′=ℐχ2,FWIχ1=∑w,w′∈N1∗FN1wFN1w′Fdsw,w′=∑fw,fw′∈N2∗FN2fwFN2fw′Fdsfw,fw′=FWIχ2.
Hence,ℐχ1+FWIχ1=ℐχ2+FWIχ2, which implies that WIχ1=WIχ2.


## 4. Relationship between the Connectivity Index and Weiner Index

In the forthcoming example, it can be noted thatCIχ=WIχ, but this is not always the case. Examples 5 and 6 are given to show this matter.Example 5.
Since every arc in Figure3 is strong, therefore CONNTχv,w=Tdsv,w and CONNFχv,w=Fdsv,w. We see that for all v,w∈Vχ,CONNTχv,w=TMv,w and CONNFχv,w=FMv,w. Thus, Tdsv,w=TMv,w and Fdsv,w=FMv,w. Then,(11)ℐχ=0.50.80.5+0.50.60.5+0.50.20.2+0.80.60.6+0.80.20.2+0.60.20.2=0.714,FWIχ=0.40.20.5+0.40.30.5+0.40.70.7+0.20.30.4+0.20.70.7+0.30.70.7=0.565.
Thus,WIχ=ℐχ+FWIχ=0.714+0.565=1.279. Obviously, CIχ=1.279, and thus, WIχ=CIχ.Figure 3
IFG withWIχ=1.279.Note 2.
In last example, we can observe that(1)
Every arc is strong(2)
Every pair of vertices is connected by an edge(3)
There is noδ-arcExample 6.
Considerχ is a graph depicted in Figure 4. After calculations, we get ℐχ=0.714,ℱℐχ=0.565, and CIχ=0.714+0.565=1.279. Since each arc is strong,(12)CONNχv1,v2=dsv1,v2=0.5,0.5,CONNχv1,v4=dsv1,v4=0.2,0.7,CONNχv2,v3=dsv2,v3=0.6,0.4,CONNχv3,v4=dsv3,v4=0.2,0.7.
The pathsv1v2v3 and v1v4v3 are strong between v1 and v3 and of equal length. The sum of membership values of v1v2v3 and v1v4v3 is 1.1,0.9 and 0.4,1.4. Thus dsv1,v3=0.4,1.4. This implies that Tdsv1,v3=0.4 and Fdsv1,v3=1.4. Similarly, we have Tdsv2,v4=0.7 and Fdsv2,v4=1.2.(13)ℐχ=0.50.80.5+0.50.60.4+0.50.20.2+0.80.60.6+0.80.20.7+0.60.20.2=0.764,FWIχ=0.40.20.5+0.40.31.4+0.40.70.7+0.20.30.4+0.20.71.2+0.30.70.7=0.743.
Hence,WIχ=0.764+0.743=1.507.Figure 4
IFG withℐχ=1.279 and ℐχ=1.507.Note 3.
Note that in Example6, CIχ<WIχ although every arc is strong. This inequality does not happen always.Theorem 4.
Letχ=N,M be an IFG with the following two conditions.(1)
χ has no δ-arcs(2)
For every pairζ,w∈N∗,ζ,w∈M∗
Then,(14)CIχ=WIχ.Proof.
Considerχ=N,M is an IFG satisfying (i) and (ii). Since χ has no δ-arcs, all arcs will be either α- or β-strong. In each of the case, we have for all ζ,ζ′∈N∗:(15)TMζ,w=CONNTχζ,ζ′=Tdsζ,ζ′,FMζ,ζ′=CONNFχζ,ζ′=Fdsζ,ζ′.
So, we have the following:(16)ℐχ=∑ζ,ζ′∈N∗TNζTNζ′CONNTχζ,ζ′=∑ζ,ζ′∈N∗TNζTNζ′TMζ,ζ′=∑ζ,ζ′∈N∗TNζTNζ′Tdsζ,ζ′=ℐχ,ℱℐχ=∑ζ,ζ′∈N∗FNζFNζ′CONNFχζ,ζ′=∑ζ,ζ′∈N∗FNζFNζ′FMζ,ζ′=∑ζ,ζ′∈N∗FNζFNζ′Fdsζ,ζ′=FWIχ.Hence,CIχ=ℐχ+ℱℐχ=ℐχ+FWIχ=WIχ.Theorem 5.
Letχ=N,M be a complete IFG with N∗=n. Also, r1≤r2≤r3≤⋯≤rn such that TNwi=ti for i=1,2,3,…,n and s1≥s2≥s3≥⋯≥sn be such that FNwi=si for i=1,2,3,…,n. Then,(17)WIχ=∑i=1n−1ri2∑j=i+1nrj+∑i=1n−1si2∑j=i+1nsj.Proof.
Supposeχ=N,M is a complete IFG. Then, χ does not have any δ-arc. By the previous theorem, CIχ=WIχ. By Theorem 1, we obtain the required result.Corollary 1.
Letχ=N,M be a complete IFG. Then, WIχ=CIχ.Theorem 6.
Ifχ=N,M is an IF tree and N∗≥3, then WIχ=CIχ.Proof.
Assume thatχ=N,M is an IF tree with N∗≥3. Then, a unique strong path exits between every two vertices. Indeed, it is also unique and strongest. For all v,w∈N∗,TdSv,w=∑c,d∈PTMc,d and FdSv,w=∑c,d∈PFMc,d, while CONNTχv,w=minc,d∈PTMc,d and CONNFχv,w=maxc,d∈PFMc,d. So, we have the following:(18)CONNTχv,w≤TdSv,wandCONNFχv,w≤FdSv,w.
Ifv,w is a strong arc, then CONNTχv,w=Tdsv,w and CONNFχv,w=FdSv,w, but if TMv,w=0 or v,w∈M∗ is not strong, then CONNTχv,w<TdSv,w and CONNFχv,w<FdSv,w. Thus, ∑CONNTχv,w<∑Tdsv,w, which implies that ℐχ<ℐχ. Similarly, we have ∑CONNFχv,w<∑Fdsv,w, which gives ℱℐχ<FWIχ. Hence:(19)ℐχ+ℱℐχ<ℐχ+FWIχCIχ<WIχ.Corollary 2.
Letχ=N,M be an IF tree. Then, CIχ≤WIχ.Proof.
Supposeχ=N,M is an IF tree and N∗=2. Then, χ has only one arc joining its two vertices. Obviously, this arc is strong, and therefore, CIχ<WIχ. By Theorem 6 with N∗≥3, we have CIχ<WIχ. By combining both CIχ<WIχ and CIχ<WIχ, we get CIχ<WIχ as desired.Example 7.
From Figure2, we calculate connectivity index as follows:(20)ℐχ=∑u,v∈VχTNuTNvCONNTχu,v=0.920.5+0.5+0.5+0.5+0.7+0.8+0.9+0.7+0.7+0.8=5.346,ℱℐχ=∑u,v∈VχFNuFNvCONNFχu,v=0.120.5+0.5+0.5+0.5+0.3+0.2+0.1+0.3+0.3+0.2=0.034.
Thus,CIχ=ℐχ+ℱℐχ=5.346+0.034=5.38. All the arcs except v2,v3 are strong. Now, we calculate WI as follows:(21)χ=∑u,v∈VχTNuTNvTdsu,v=0.920.5+2.9+2.2+1.4+2.4+1.7+0.9+0.7+1.5+0.8=12.15,ℱχ=∑u,v∈VχFNuFNvFdsu,v=0.120.5+1.1+0.8+0.6+0.6+0.3+0.1+0.3+0.5+0.2=0.05.
So,WIχ=ℐχ+FWIχ=12.15+0.05=12.2. Clearly, we can observe that WIχ>CIχ.Theorem 7.
IfH=N′,M′ is an partial IF subgraph of an IFGχ=N,M, then CONNTHv,w≤CONNTχv,w and CONNFHv,w≥CONNFχv,w.Proof.
(i) By the definition of partial IF subgraph, we haveTM′vi,vj≤TMvi,vj for each i,j. If ST′ denotes the T strength of a path P′ in H and ST denotes the T strength of a path P in χ, then minTM′vi,vj≤minTMvi,vj implies ST′≤ST. So, we have maxST′≤maxST for all possible paths P′ in H and P in χ between V and W. Thus, we obtain CONNTHv,w≤CONNTχv,w.
(ii) Again by the definition of partial IF subgraph, we haveFM′vi,vj≥FMvi,vj, which implies CONNFHv,w≥CONNFχv,w.Example 8.
Using Figure5, we have prepared the following table after routine computations. From Table 1, we see that WI of χ has been increased and decreased by deleting α-strong edges. We also have made the observation that WI of χ is increased by removing β-strong edges, but by deleting the δ-arc v1,v2, the strength of connectedness between every pair of vertices will remain the same. So, that is the reason we have WIχ=WIχ−v1,v2.Figure 5
AnIFG with WIχ=2.798.Table 1
Values of wiener index forχ and χ−vi,vj.
ArcsTypes of arcsGraphsℐℱℐℐ——χ0.7642.0342.798v1,v2δ-weakχ−v1,v20.7642.0342.798v2,v3β-strongχ−v2,v30.8122.2023.014v2,v4β-strongχ−v2,v40.9212.4963.416v3,v4α-strongχ−v3,v40.7642.3863.15v4,v5α-strongχ−v4,v50.1920.5480.74v1,v5α-strongχ−v1,v50.3320.7841.116Theorem 8.
Letχ=N,M be an IFG. For c,d∈N∗, let Pc,d be the path with minimum sum of truth values and maximum total of falsity values between the entire shortest strong paths among c&d. Let v,w∈M∗.(1)
Ifv,w is an α-strong or β-strong arc and v,w is not a part of any Pc,d for c,d∈N∗ with c,d≠v,w, then WIχ−v,w≠WIχ(2)
Ifv,w is a δ-edge, then WIχ−u,v=WIχProof.
Supposeχ=N,M is an IFG and v,w∈M∗. (i) Let v,w be an α-strong or β-strong arc. Then, v,w arc itself is the shortest strong path between v and w. Therefore, Tdsv,w=TMv,w and Fdsv,w=FMv,w.Case 1.
Ifv,w is a β-strong arc, then one or more strong paths connecting v and w having length of 2 or more will exist. Suppose that v,w is not a part of any Pc,d for c,d∈N∗ with c,d≠v,w. Then, one of the following will hold according to the path Pc,d in χ−v,w:(1)
Tdsv,w|χ−v,w>Tdsv,w|χ=TMv,w and Fdsv,w|χ−v,w>Fdsv,w|χ=FMv,w(2)
Tdsv,w|χ−v,w<Tdsv,w|χ=TMv,w and Fdsv,w|χ−v,w<Fdsv,w|χ=FMv,w(3)
Tdsv,w|χ−v,w>Tdsv,w|χ=TMv,w and Fdsv,w|χ−v,w<Fdsv,w|χ=FMv,w(4)
Tdsv,w|χ−v,w<Tdsv,w|χ=TMv,w and Fdsv,w|χ−v,w>Fdsv,w|χ=FMv,w
Lets,t≠v,w. By supposition, v,w is not a part of any Ps,t. So, we have Tdsv,w|χ−v,w=Tdsv,w|χ and Fdss,t|χ−v,w=Fdss,t|χ. So, either WIχ−v,w<WIχ or WIχ−v,w>WIχ implies that WIχ−v,w≠WIχ.Case 2.
Letv,w be an α-strong arc. If v and w are not connected by any path on χ−v,w, then Tdsv,w|χ−v,w=0=Fdsv,w|χ−v,w. Therefore, Tdsv,w|χ−v,w<Tdsv,w|χ and Fdsv,w|χ−v,w<Fdsv,w|χ. If v and w are connected in χ−v,w, then there exits one or more strong paths connecting v and w of length of 2 or more. So, using the case (1), we get WIχ−v,w≠WIχ.
(ii) Ifv,w∈M∗ is a δ-arc, then no geodesic from c to d for any c,d∈N∗ will contain the arc v,w. Therefore, the deletion of v,w from χ has no effect on Tdsc,d and Fdsc,d for any c,d∈N∗. So, we must have WIχ−v,w=WIχ.Theorem 9.
Letχ=N,M be an IFG with the given condition. If u,v∈M∗ is β-strong, then a unique strong cycle of strength TMu,v,FMu,v exits through u,v. Then, WIχ−u,v>WIχ.Proof.
Supposeχ=N,M is an IFG with the given condition. Further, suppose u,v∈M∗ is β-strong arc. Then, by the given condition, there exits only one strong cycle C through u,v having strength TMu,v,FMu,v, which implies that TMc,d≥TMu,v and FMc,d≥FMu,v for all arcs c,d in C.
Any strong path fromu to v in χ−u,v will have two or more arcs with CONNTχc,d≥TMu,v and CONNFχc,d≥FMu,v for each arc c,d in the path. Thus,(22)Tdsu,v|χ−uv>Tdsu,v|χ&Fdsu,v|χ−uv>Fdsu,v|χ.
Supposeu,u0,u1,…,uk,v is the strong path from u to v having strength TMu,v,FMu,v. Let a,b∈N∗ such that u,v is an arc of the shortest strong path P1:a,a0,a1,…,ap,u,v,bq,…,b0,b from a to b with the following:(23)∑x,y∈P1TMx,y,∑x,y∈P1FMx,y=Tdsa,b,Fdsa,b.
Also,P1 and P2:a,a0,a1,…,ap,u,u0,…,uk,v,b are only two strong paths exist from a⟶b. Now, we prove the assertion. If possible, there will be a strong path Q between a⟶b other than P1 and P2. Let the strong cycle be C:u,u0,u1,…,uk,v,u. Now, consider three cases.Case 3.
Figure6 shows that P1 and P2 are internally disjoint with Q. Then, Q together with P1 makes a strong cycle through u,v. This contradicts the assumption that C:u,u0,u1,…,uk,v,u is the only strong cycle through u,v.Figure 6
Sketch of Case I.Case 4.
Figure7 shows that Q is not internally disjoint with either P1 or P2. Suppose Q and P1 are not internally disjoint. Let Q:a,a0,…,a1,c0,c1,…,cm,bd,…,b0,b. Then, a1,…,ap,u,v,bq,bq−1,…,bd,cm,…,c1,c0,a1 is a strong cycle through u,v different from C, which is again contradiction to the assumption that C is the only strong cycle through u,v. The case when Q have common vertices with P2 also provides a contradiction.Figure 7
Sketch of Case II.Case 5.
P1 and P2 are not internally disjoint with Q (see Figure 8).
Suppose that bothP1 and P2 are not internally disjoint with Q. Let Q:a,a0,…,ax,d0,d1,…,dr,uy,uy−1,…,uk,v.
Then, the pathu,ap,ap−1,…,ax,dr,…,d1,d0,uy,…,uk,v,u acts as a strong cycle through u,v other than C, which is impossible. Thus, there are only two strong paths P1 and P2 between a and b. Thus,(24)Tdsa,b|χ−uv>Tdsa,b|χ&Fdsa,b|χ−uv>Fdsa,b|χ.
Ifp,q∈N∗ such that u,v is not an arc of any strong path from p to q, then Tdsp,q|χ−uv=Tdsp,q|χ and Fdsp,q|χ−uv=Fdsp,q|χ. Hence, WIχ−uv>WIχ.Figure 8
Sketch of Case III.


## 5. Application: Transport Network Flow

An application of Wiener index is proposed in this section. The Wiener index is applied to a transport network flow to know different situations created by the removal or closing of certain roads.Consider a transport network flow as shown in the Figure9. In it, vertices and edges represent cities and roads. The values given to the vertices and edges are incoming and outgoing flows of traffic. The maximum values for incoming and outgoing flows are 25 and 20, and vertices are fuzzified by dividing their values by 45 (sum of 25 and 20). The edges are also fuzzified by the same technique. The fuzzified network is shown in Figure 10.Figure 9
A transport network.Figure 10
Intuitionistic fuzzified transport network flow.In this network, each arc is strong. By simple calculations, we have prepared the following table.From Table2, we see that cities v1,v4,v5,v8 become isolated after the removal of arcs or roads v1,v2,v3,v4,v5,v6, and v7,v8. The people of these cities will be disconnected from the cities v2,v3,v6 and v7. The deletion of these edges reduces the overall Wiener index of the network, but still path number for incoming flow is greater than the path number for outgoing flow in other places. From all of these observations, we have arrived at the result that a given network has more incoming flow.Table 2
ℐχ,ℐχ−vi,vj, and ℐ.
ArcsTypes of arcsGraphsℐℱℐℐℐ——χ0.980.8111.7910.78v1,v2Strong arcχ−v1,v20.4520.6731.1250.586v2,v3α-strongχ−v2,v31.0480.9431.9910.806v3,v4Strong arcχ−v3,v40.6640.6211.2850.584v3,v6β-strongχ−v3,v61.270.7672.0370.78v5,v6Strong arcχ−v5,v60.7280.5621.290.569v6,v7β-strongχ−v6,v71.330.7872.1170.78v7,v8Strong arcχ−v7,v80.7490.3631.1120.458v2,v7α-strongχ−v2,v71.3080.9912.2990.792Table2 shows that WIχ−v2,v3>WIχ. This is why that elimination of v2,v3 makes some routes longer than before. For example, the geodesic route from v1 to v3 was v1v2v3, which is of length 2, but now it has become v1v2v7v6v3 of 4 in length. Besides its length, its sum of weights has increased also. People who used to travel from city v1 to v3 will now have to spend more money and time. Similarly, the routes from v1 to v4 and from v2 to v3 are also affected by this elimination. All other routes are not affected by this removal, for example, the route from v2 to v5. There are two geodesics between v2 and v5 of the same length and the same sum of incoming and outgoing flows. So, alternative route with the same geodetic distances is available after the removal of v2,v3 and the people who used to travel between these cities will not be affected much. The removal of v2,v3 is shown in the Figure 11. Similar information can be obtained for other three removals v3,v6,v6,v7, and v2,v7. The purpose of this application is to analyze the results produced by the closing or removal of certain routes connecting different cities.Figure 11
χ−v2,v3.By comparison between the Wiener index and the connectivity index, we can observe from Table2 that the values of both the indices do not agree with each other for χ and χ−vi,vj, although each road is strong. The reason is that there is no road between every two cities; for example, the cities v2 and v6 are not connected directly by a road. Another thing to note is that the values for both the indices are lower than before by deleting strong roads; that is, the behavior of both the indices is same, and the values become higher than before in case of deleting α-strong roads, but the values of the Wiener index rise than before while the values of the connectivity index remain the same by deleting β-strong roads.


## 6. Conclusion

Mainly, our goal for the setting of this study was to introduce the concept ofWI for IFGs, which deals with two types of degrees named as truth membership degree and falsity membership degree.(i)
We have purposed the definition ofWI for IFGs structure with numeral examples.(ii)
Some bounds ofWI are investigated for IFGs settings.(iii)
We have discussed the relationship betweenCI and WI with examples. Also, some related results are provided.(iv)
An application in transport network flow is given.In the future, we will do this work for T-spherical fuzzy graphs and picture fuzzy graphs. We will also extend this study for interval-valuedIFGs and explore its applications.


### 6.1. Advantages

Our study has the following advantages:(i)
The main advantage of our work lies in the fact that we have developedWI for IFGs settings, which are defined by two membership degrees.(ii)
FGs are defined by only one degree that is membership degree and therefore failed to give more information in some situations. Instead of FGs, IFGs give two types of information that is membership and nonmembership and hence are more suitable in various situations.(iii)
Our study and results are in fact of the generalization of the corresponding results ofFGs. Our results will be shifted into the corresponding results of FGS by ignoring the second membership grades. In this way, the corresponding results of FGs will be taken as a special case of the results for IFGs developed by us.(iv)
Our work shows thatIFGs have more information than FGs.


## 6.1. Advantages

Our study has the following advantages:(i)
The main advantage of our work lies in the fact that we have developedWI for IFGs settings, which are defined by two membership degrees.(ii)
FGs are defined by only one degree that is membership degree and therefore failed to give more information in some situations. Instead of FGs, IFGs give two types of information that is membership and nonmembership and hence are more suitable in various situations.(iii)
Our study and results are in fact of the generalization of the corresponding results ofFGs. Our results will be shifted into the corresponding results of FGS by ignoring the second membership grades. In this way, the corresponding results of FGs will be taken as a special case of the results for IFGs developed by us.(iv)
Our work shows thatIFGs have more information than FGs.



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