# Connectivity Index in Vague Graphs with Application in Construction

Authors: Huiqin Jiang; Yongsheng Rao
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

The vague graph (VG), which has recently gained a place in the family of fuzzy graph (FG), has shown good capabilities in the face of problems that cannot be expressed by fuzzy graphs and interval-valued fuzzy graphs. Connectivity index (CI) in graphs is a fundamental issue in fuzzy graph theory that has wide applications in the real world. The previous definitions’ limitations in the connectivity of fuzzy graphs directed us to offer new classifications in vague graph. Hence, in this paper, we investigate connectivity index, average connectivity index, and Randic index in vague graphs with several examples. Also, one of the motives of this research is to introduce some special types of vertices such as vague connectivity enhancing vertex, vague connectivity reducing vertex, and vague connectivity neutral vertex with their properties. Finally, an application of connectivity index in the selected town for building hospital is presented.


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## Body

## 1. Introduction

The FG concept serves as one of the most dominant and extensively employed tools for multiple real-world problem representations, modeling, and analysis. To specify the objects and the relations between them, the graph vertices or nodes and edges or arcs are applied, respectively. Graphs have long been used to describe objects and the relationship between them. Many of the issues and phenomena around us are associated with complexities and ambiguities that make it difficult to express certainty. These difficulties were alleviated by the introduction of fuzzy sets by Zadeh [1]. This concept established well-grounded allocation membership degree to elements of a set. Actually, fuzzy set theory is one of the best and most powerful tools for modeling problems in examining the relationship between uncertainties in the real world. Rosenfeld [2] proposed the idea of FG in 1975. Kauffman [3] represented FGs based on Zadeh’s fuzzy relation [4, 5]. Bhutani and Rosenfeld [6] introduced the concept of strong edges. Bhattacharya [7] presented some observations on FGs, and some operations on FGs were described by Mordeson and Peng [8]. The existence of a single degree for a true membership could not resolve the ambiguity on uncertain issues, so the need for a degree of membership was felt. Afterward, to overcome the existing ambiguities, Gau and Buehrer [9] gave false membership degrees and defined a vague set as the sum of degrees not greater than 1. Ramakrishna in [10] proposed the vague graph concept. New VG concept was introduced and analyzed by Borzooei et al. [11–13]. The vague graph, which has recently gained a place in the family of FG, has shown good capabilities in the face of problems that cannot be expressed by FGs. A vague graph is referred to as a generalized structure of an FG that conveys more exactness, adaptability, and compatibility to a system when coordinated with systems running on FGs. Furthermore, a VG is able to concentrate on determining the uncertainly coupled with the inconsistent and indeterminate information of any real-world problem, where FGs may not lead to adequate results. Connectivity index is one of the most important topics that has many applications in dynamic detection of competition condition in parallel programming, finding phylogenetic trees based on protein domain information, and social networks. The connectivity index problem can also be used to model many real-world situations in the fields of circuit design, telecommunications, network flow, and so on. Binu et al. [14] introduced connectivity index of a fuzzy graph. Mathew and Sunitha [15] defined several types of arc in fuzzy graph. Naeem et al. [16] investigated connectivity indices of intuitionistic fuzzy graphs (IFGs). Poulik and Ghorai [17] studied indices of graphs under bipolar fuzzy environment. Sebastian et al. [18] presented connectivity parameters in generalized fuzzy graphs. Several concepts and results in VGs were proposed and investigated by Akram et al. [19, 20]. Ghorai et al. [21] studied regular product vague graphs and product vague line graphs. References [22, 23] introduced competition-FGs and some remarks on bipolar-FGs. References [24–29] investigated several concepts on FGs and VGs. Kou et al. [30] studied g-eccentric node and vague detour g-boundary nodes in VGs. Rashmanlou et al. [31–33] described categorical properties and paired domination in IFGs. Kosari et al. [34] introduced the restrained K-rainbow reinforcement number of graphs.In this paper, we investigated connectivity index, average connectivity index, and Randic index in VGs with several examples. Likewise, we introduced some special types of nodes such as vague connectivity enhancing node, vague connectivity reducing node, and vague connectivity neutral node with their properties. Finally, an application of connectivity index in construction is presented.


## 2. Preliminaries

A FGG=V,v,η is a nonempty set V together with a pair of functions v:V⟶0,1 and η:V×V⟶0,1 such that ηab≤minva,vb, for all a,b∈V. Here, η is a symmetric fuzzy relation on V × V.Definition 1 (see [9]).
A VSM is a pair tM,fM on set V where tM and fM are taken as real valued functions which can be defined on V⟶0,1, so that tMa+fMa≤1, for all a∈V.Definition 2 (see [10]).
G=M,N is called a VG on a crisp graph G∗, in which M=tM,fM is a VS on V and N=tN,fN is a VS on E⊆V×V so that tNab≤mintMa,tMb and fNab≥maxfMa,fMb, ∀ab∈E.Definition 3 (see [11]).
LetG=M,N be a VG.(i)
A VGG′=M′,N′ is said to be a PVSG of G if tM′a≤tMa, fM′a≥fMa, ∀a∈V and tN′ab≤tNab, fN′ab≥fNab, for each edge ab∈G.(ii)
A VGG′=M′,N′ is said to be a VSG of G if tMa=tM′a, fMa=fM′a, ∀a∈V′ and tN′ab=tNab, fN′ab=fNab, for each edge ab∈G′.Definition 4 (see [11]).
LetG=M,N be a VG.(i)
A pathp:a=a0,a1,…,ak−1,ak=b in G is a sequence of distinct nodes where tNai−1ai>0, fNai−1ai>0, i=1,2,…,k. The length of P is k.(ii)
Ifp:a=a0,a1,…,ak−1,ak=b is a path between a and b of length k, then tNabk=suptNaa1∧tNa1a2∧…∧tNak−1b and fNabk=inffNaa1∨fNa1a2∨…∨fNak−1b.
CONNGa,b=CONNGta,b,CONNGfa,b=tNab∞,fNab∞ is called the strength of connectedness between any two nodes a and b in G where CONNGta,b=supk∈NtNabk and CONNGfab=infk∈NfNabk. If tNab>0 and fNab>0, ∀a,b in G, then the VG G is called a CVG. If tNab=mintMa,tMb and fNab=maxfMa,fMb, ∀a,b∈V, then G is named a complete VG.Definition 5 (see [11]).
For a VGG, if tNab≥tNab∞ and fNab≤fNab∞, then the arc ab is named a strong arc of G.Definition 6 (see [11]).
LetG=M,N be a CVG.(i)
G is called a VT if G has a VSSG G′=A′,B′ which is also a tree and for all arcs ab not in G, tN′ab∞<tNab∞ and fN′ab∞>fNab∞.(ii)
An edgeab is said to be a VB if tNab∞>tN′ab∞ and fNab∞<fN′ab∞.(iii)
G is called a vague cycle if G∗ is a cycle and there does not exist unique edge ab in G for which tNab=mintNa1b1∣a1b1∈E and fNab=maxfNa1b1∣a1b1∈E.
All the basic notations are shown in Table1.Table 1
Some basic notations.
NotationMeaningFGFuzzy graphVGVague graphCVGConnected vague graphVSVague setMVMembership valuePVSGPartial vague subgraphVSGVague subgraphVTVague treeACIAverage connectivity indexVCRNVague connectivity reducing nodeVCENVague connectivity enhancing nodeVCNNVague connectivity neutral nodeCEVGConnectivity enhancing vague graphCRVGConnectivity reducing vague graphNVGNeutral vague graphONDOpen neighborhood degreeRIRandic indexCIConnectivity index


## 3. Connectivity Index of a Vague Graph

Definition 7.
The CI of a VGG denoted by CIVGG is defined as(1)CIVGG=CIVGtG,CIVGfG=∑a,b∈VtMatMb⋅CONNGta,b,∑a,b∈VfMafMb⋅CONNGfa,b,where CIVGtG and CIVGfG, respectively, denote the true-CI and false-CI of G.Example 1.
Consider the CVGG of the graph G∗ shown in Figure 1, where V=a,b,c,d,e and E=a,b,b,c,c,d,d,e,a,e,b,d. Here, we suppose that the true-MV and false-MV of every node are 0.2 and 0.5, respectively.
Here,(2)CONNGta,b=0.1,CONNGta,c=0.1,CONNGta,d=0.1,CONNGta,e=0.2,CONNGtb,c=0.2,CONNGtb,d=0.2,CONNGtb,e=0.1,CONNGtc,d=0.2,CONNGtc,e=0.1,CONNGtd,e=0.1.CONNGfa,b=0.6,CONNGfa,c=0.6,CONNGfa,d=0.6,CONNGfa,e=0.6,CONNGfb,c=0.5,CONNGfb,d=0.5,CONNGfb,e=0.6,CONNGfc,d=0.5,CONNGfc,e=0.6,CONNGfd,e=0.6.CIVGtG=0.004+0.004+0.004+0.008+0.008+0.008+0.004+0.008+0.004+0.004=0.056,CIVGfG=0.15+0.15+0.15+0.15+0.125+0.125+0.15+0.125+0.15+0.15=1.425.
Hence,CIVGG=CIVGtG,CIVGfG=0.056,1.425Figure 1
Connected VGG.Theorem 1.
For a PVSGG′=M′,N′of a VGG=M,N,CIVGtG′≤CIVGtGandCIVGfG′≥CIVGfG.Proof.
LetG′=M′,N′ be a PVSG of G=M,N. Hence, tM′a≤tMa, fM′a≥fMa, ∀a∈G′ and tN′ab≤tNab, fN′ab≥fNab, ∀ab∈G′. Since CONNG′ta,b and CONNG′fa,b lie on one or many edges of G′ and CONNGta,b and CONNGfa,b lie on one or many edges of G and tN′ab≤tNab, fN′ab≥fNab, ∀ab∈G′, and hence we have(3)CONNG′ta,b≤CONNGta,b,CONNG′fa,b≥CONNGfa,b,∀a,b∈G′.
Since(4)fM′afM′b≥0,fMafMb≥0.
So,(5)tM′atM′bCONNG′ta,b≤tMatMbCONNGta,b,fM′afM′bCONNG′fa,b≥fMafMbCONNGfa,b.
Therefore,(6)∑a,b∈V′tM′atM′bCONNG′ta,b≤∑a,b∈VtMatMbCONNGta,b(7)∑a,b∈V′fM′afM′bCONNG′fa,b≥∑a,b∈VfMafMbCONNGfa,b.
Therefore,CIVGtG′≤CIVGtG and CIVGfG′≤CIVGfG.Remark 1.
For a VSGG′=M′,N′ of a VG G, CIVGtG′≤CIVGtG and CIVGfG′≤CIVGfG.Example 2.
LetG′ be a CVSG of the VG G as in Example 2 (see Figure 2). Here,(8)CIVGtG′=0.004+0.004+0.008+0.008+0.004+0.004=0.032,CIVGfG′=0.15+0.15+0.15+0.125+0.15+0.15=0.875.
So,CIVGG′=0.032,0.875. Therefore, Example 2 shows that CIVGtG′≤CIVGtG and CIVGfG′≤CIVGfG.Figure 2
Connected VSGG′.Theorem 2.
IfG′=M′,N′ is a VSG of a CVG G=M,N, where V′=V−s, s∈V, then CIVGtG′<CIVGtG and CIVGfG′<CIVGfG.Proof.
LetG have n nodes, i.e., V=n and s∈V. Then, V′=V−s i.e., V′=n−1. Suppose V=a1,a2,…,an=s. Hence, V′=a1,a2,…,an−1. Since G′ is VSG of G, G′ must be a PVSG of G. So,(9)CIVGtG=CIVGtG′+∑i=1n−1tMantMaiCONNGtan,ai,CIVGfG=CIVGfG′+∑i=1n−1fManfMaiCONNGfan,ai.
Thus,CIVGtG′<CIVGtG and CIVGfG′<CIVGfG.Theorem 3.
IfG is a complete VG and V=a1,a2,…,an so that tMai=pn and fMai=qi, i=1,2,…,n, where p1≤p2≤…≤pn and q1≥q2≥…≥qn, then CIVGtG=∑i=1n−1pi2∑j=i+1npj and CIVGfG=∑i=1n−1qi2∑j=i+1nqj.Proof.
Leta1∈V, tMa1=p1, and fMa1=q1. Since G is complete, CONNGtai,aj=tNai,aj and CONNGfai,aj=fNai,aj. Then, CONNGta1,aj=tMa1=p1, CONNGfa1,aj=fMa1=q1, j>1. Since p1≤p2≤…≤pn and q1≥q2≥…≥qn, then tMa1≤tMa2≤…≤tMan and fMa1≥fMa2≥…≥fMan. Therefore,(10)tMa1tMajCONNGta1aj=p1pjp1=p12pj,fMa1fMajCONNGfa1aj=q1qjq1=q12qj.
It shows that(11)∑j=2ntMa1tMajCONNGta1,aj=∑j=2np12pj,∑j=2nfMa1fMajCONNGfa1,aj=∑j=2nq12qj.
Therefore,(12)∑i=1n−1∑j=2ntMaitMajCONNGtai,aj=∑i=1n−1∑j=2npi2pj,∑i=1n−1∑j=2nfMaifMajCONNGfai,aj=∑i=1n−1∑j=2nqi2qj.
Thus,CIVGtG=∑i=1n−1∑j=i+1npj and CIVGfG=∑i=1n−1∑j=i+1nqj.Example 3.
Consider the complete VGG shown in Figure 3 where V=a,b,c. Here, tMvi=pi and fMvi=qi, for i=1,2,3.
tMa=p1=0.1, fMa=q1=0.4, tMb=p2=0.2, fMb=q2=0.3, tMc=p3=0.3, fMc=q3=0.2.
Hence,p1≤p2≤p3 and q1≥q2≥q3.(13)CIVGtG=∑i=12pi2∑j=i+13pj=p12p2+p12p3+p22p3=0.120.2+0.120.3+0.220.3=0.017,(14)CIVGfG=∑i=12qi2∑j=i+13qj=q12q2+q12q3+q22q3=0.048+0.032+0.018=0.098.
Thus,CIVGG=CIVGtG,CIVGfG=0.017,0.098.
Now we define average connectivity index (ACI) of a VG.Figure 3
A complete VGG.Definition 8.
The ACI of a VGG=M,N denoted by ACIVGG is defined as(15)ACIVGG=ACIVGtG,ACIVGfG=1n2∑a,b∈VtMatMbCONNGta,b,∑a,b∈VfMafMbCONNGfa,b.Example 4.
From Example2, for VG G, the ACI of G is(16)ACIVGG=ACIVGtG,ACIVGfG=0.05652,1.42552=0.05610,1.42510=0.0056,0.1425.Definition 9.
LetG=M,N be a CVG. A node a is called a(i)
VCRN ifACVGtG−a<ACVGtG and ACIVGfG−a>ACIVGfG.(ii)
VCEN ifACIVGtG−a>ACIVGtG and ACIVGfG−a<ACIVGfG.(iii)
VCNN ifACIVGtG−a=ACIVGtG and ACIVGfG−a=ACIVGfG.Example 5.
Consider the VGG shown in Figure 4. We have(17)CONNGta,b=0.1,CONNGta,c=0.1,CONNGta,d=0.1,CONNGtb,c=0.2,CONNGtb,d=0.2,CONNGtc,d=0.2,CONNGfa,b=0.4,CONNGfa,c=0.5,CONNGfa,d=0.4,CONNGfb,c=0.5,CONNGfb,d=0.4,CONNGfc,d=0.5.
Hence,(18)CIVGtG=0.004+0.004+0.004+0.008+0.008+0.008=0.036,CIVGfG=0.036+0.045+0.045+0.036+0.045+0.036+0.045=0.243.,ACIVGtG=0.0366=0.006,ACIVGfG=0.040,ACIVGtG−a=0.008,ACIVGfG−a=0.039,ACIVGtG−b=0.005,ACIVGfG−b=0.042,ACIVGtG−c=0.005,ACIVGfG−c=0.036,ACIVGtG−d=0.005,ACIVGfG−d=0.042.
Hence,a is a VCEN of G and b and d are VCRN of G.Figure 4
A connected VGG.Theorem 4.
LetG=M,N be a complete VG with V=n≥3, and p1=CIVGtG/CIVGfG−a, p2=CIVGfG/CIVGfG−a, a∈V. The node a is a(i)
VCEN if and only ifp1≤n/n−2 and p2>n/n−2.(ii)
VCRN if and only ifp1>n/n−2 and p2<n/n−2.(iii)
VCNN if and only ifp1=n/n−2 and p2=n/n−2.Proof.
(i) Assume thata is a VCEN. Then, Definition 9 shows that(19)ACIVGtG<ACIVGtG−a,ACIVGfG>ACIVGfG−a.
So,CIVGtG/n/2<CIVGtG−a/n−1/2 and CIVGfG/n/2<CIVGfG−a/n−1/2. Hence, CIVGtG/CIVGtG−a<n/2/n−1/2 and CIVGfG/CIVGfG−a>n/2/n−1/2. Thus, p1<n/n−2 and p2>n/n−2.
Conversely, letp1<n/n−2 and p2>n/n−2. So, CIVGtG/CIVGtG−a<n/n−2 and CIVGfG/CIVGfG−a>n/n−2. Hence, CIVGtG/n/2<CIVGtG−a/n−1/2 and CIVGfG/n/2>CIVGfG−a/n−1/2. Thus, ACIVGtG<ACIVGtG−a and ACIVGfG>ACIVGfG−a. Therefore, a is a VCEN of G. In the same way, (ii) and (iii) can be proved.Theorem 5.
LetG=M,N be a complete VG with V=n≥3. Suppose Lt=∑a∈V−btMatMbCONNGta,b, Lf=∑a∈V−bfMafMbCONNGfa,b, and b is an end node of G. Then,(i)
Lt<2/n−2 and Lf>2/2−n iff b is a VCEN.(ii)
Lt>2/n−2 and Lf<2/2−n iff b is a VCRN.(iii)
Lt=2/n−2 and Lf=2/2−n iff b is a VCNN.Proof.
(i) LetLt<2/2−n and Lf>2/2−n. Now,(20)CIVGtG=CIVGtG−b+∑a∈V−btMatMbCONNGta,b,CIVGfG=CIVGfG−b+∑a∈V−bfMafMbCONNGfa,b.
So,CIVGtG=CIVGtG−b+Lt and CIVGfG=CIVGfG−b+Lf. Hence, CIVGtG/n/2=CIVGtG−a/n/2+Lt/n/2 and CIVGfG/n/2=CIVGfG−a/n/2+Lf/n/2. Thus, ACIVGtG<CIVGtG−b/n−1/2⋅n−2/n+2/n−2/n/2 and ACIVGfG>CIVGfG−b/n−1/2⋅n−2/n+2/2−n/n/2. Therefore,(21)ACIVGtG<ACIVGtG−b−2nACIVGtG−b−2n−1n−2,ACIVGfG>ACIVGfG−b−2nACIVGfG−b+2n−1n−2.
So,ACIVGtG<ACIVGtG−b and ACIVGfG>ACIVGfG−b.
In the same way, (ii) and (iii) can be proved.Definition 10.
LetG=M,N be a CVG.(i)
G is named CEVG, if it has at least one VCEN.(ii)
G is named CRVG, if it has no VCRN.(iii)
G is named NVG, if all nodes of G are VCNN.Example 6.
For the VGG in Example 5, the node a is a VCEN of G, so G is a CEVG, but G is neither a CRVG nor a NVG.Theorem 6.
For each pair of numbersL1,L2, where L1 is a positive true real number and L2 is a positive false real number, there always exist VG G=M,N so that CIVGtG=L1 and CIVGfG=L2, V=n.Proof.
SupposetMa=1, ∀a∈V, and fMa=0, ∀a∈V. Now, we consider a path in G so that tNab=L1/n/2 and fNab=L2/n/2 for all edges a,b∈G. Hence, we get(22)CIVGtG=∑a,b∈VtMatMbCONNGta,b=n2L1n2=L1,CIVGfG=∑a,b∈VfMafMbCONNGfa,b=n2L2n2=L2.Definition 11.
The OND or degree of a nodec in a VG G is described as dega=degtc,degfc, where degtc=∑c≠dcd∈EtNcd and degfc=∑c≠dcd∈EfNcd. If degc=t1,t2, ∀c∈V, then G is named t1,t2-regular.
Now, we introduce the RI of a VG with examples.Definition 12.
The RI of a VGG=M,N is shown by RIVGG and described as(23)RIVGG=RIVGtG,RIVGfG=∑i≠jaiaj∈EtMaitMajdegtaidegtaj−1/2,∑i≠jaiaj∈EfMaifMajdegfaidegfaj−1/2.Example 7.
Consider the VGG shown in Figure 5. Here, dega=0.3,1.3, degb=0.6,1.8, degc=0.4,1.5, degd=0.4,1.7, and dege=0.3,1.1.(24)∑i≠jvivj∈EtMvitMvjdegtvidegtvj−1/2=16.667+20.449+28.901+10.214+17.699+14.450+9.624+16.667=134.671,(25)∑i≠jvivj∈EfMvifMvjdegfvidegfvj−1/2=2.669+2.532+2.746+1.756+1.807+1.905+1.835+1.889=17.139.
Therefore,RIVGG=RIVGtG,RIVGfG=134.671,17.139.Figure 5
VGG.Theorem 7.
LetG=M,N be a connected VG and G′=M′,N′ so that V′=V−cn, cn∈V with V=n. Then, RIVGtG≥RIVGtG′ and RIVGfG≥RIVGfG′.Proof.
G has n nodes, i.e., V=c1,c2,…,cn. Then, V′=c1,c2,…,cn−1. So, V′ should be a subset of V, so G′ is a VSG of G. Hence, tMci=tM′ci, fMci=fM′ci, tNcicj=tN′cicj, and fNcicj=fN′cicj, ∀ci∈V′ and ∀cicj∈E′.
Now,degtci=∑cicj∈EcjtNcicj and degfci=∑cicj∈EcjfNcicj, i.e., degtci and degfci are sum of the true-MVs and false-MVs of the edges incident in ci in G, respectively. Then, degfcidegfcj is s positive real number. So, deg′tci=∑cicj∈E′cjtNcicj and deg′fci=∑cicj∈E′cjfNcicj, i.e., deg′tci and deg′fci are sum of the T-MVs and false-MVs of the edges incident in ci in G′, respectively. Then, deg′fcideg′fcj is a positive real number. Thus,(26)tMcitMcjdegtcidegtcj−1/2≥0,tM′citM′cjdeg′tcideg′tcj−1/2≥0,fMcifMcjdegfcidegfcj−1/2≥0,fM′cifM′cjdeg′fcideg′fcj−1/2≥0.
So,(27)∑1≤i≠j≤ncicj∈EtMcitMcjdegMtcidegMtcj−1/2≥∑1≤i≠j≤n−1cicj∈E′tM′citM′cjdeg′tcideg′tcj−1/2∑1≤i≠j≤ncicj∈EfMcifMcjdegMfcidegMfcj−1/2≥∑1≤i≠j≤n−1cicj∈E′fM′cifM′cjdeg′fcideg′fcj−1/2.
Therefore,RIVGtG≥RIVGtG′ and RIVGfG≥RIVGfG′.Example 8.
Consider VGG′=M′,N′ of Figure 6 and VG G of Figure 5. Clearly, tMvi≥tM′vi, fMvi≤fM′vi, ∀vi∈V′, and tNvivj≥tN′vivj, fNvivj≤fN′vivj, ∀vivj∈E′. Hence, G′ is a VSG of the VG G.(28)RIVGtG′=20.449+20.449+33.333+12.5+20.449+20.449=127.629,RIVGfG′=3.140+2.532+3.414+2.067+2.247+2.788=16.188.Therefore, RIVGtG≥RIVGtG′ and RIVGfG≥RIVGfG′.Figure 6
Vague subgraphG′ of the vague graph G of Figure 5.Remark 2.
LetG=M,N be a connected VG and G′=M′,N′ so that E′=E−cicj, cicj∈E. Then, RIVGtG≥RIVGtG′ and RIVGfG≥RIVGfG′.Proof.
AssumeG has n nodes, i.e., V=n and ci∈V. Let V=c1,c2,…,cn. So, V′=c1,c2,…ci−1,ci+1,…,cn. Then, V′ must be a subset of V, and so G′ is a VGS of G. So, RIVGtG≥RIVGtG′ and RIVGfG≥RIVGfG′.Theorem 8.
LetG be a complete VG so that M is a constant function. Then, RIVGG=n/2ν12,n/2ν22, where n=V and ν1,ν2=tMci,fMci, ci∈V.Proof.
SinceM is constant and tMci=ν1, fMci=ν2, ci∈V, then tMci=ν1, fMci=ν2, ∀ci∈V. Since G is a complete VG, tNcicj=mintMci,tMcj=ν1 and fNcicj=maxfMci,fMcj=ν2, ∀cicj∈E. Again, G is complete and V=n, and thus there are nn−1/2 nodes and nn−1/2 edges and each node in G is neighbor to n−1 nodes. Therefore,(29)degtci=∑cjcicj∈EtNcicj=ν1ν1…ν1=n−1ν1,degfci=∑cjcicj∈EfNcicj=ν2ν2…ν2=n−1ν2.
Hence,(30)RIVGtG=∑i≠jcicj∈EtMcitMcjdegtcidegtcj−1/2=nn−12ν1ν1n−1ν1n−1ν1−1/2=n2ν12,RIVGfG=∑i≠jcicj∈EfMcifMcjdegfcidegfcj−1/2=nn−12ν2ν2n−1ν2n−1ν2−1/2=n2ν22.
Therefore,RIVGG=RIVGtG,RIVGfG=n/2ν12,n/2ν22.Example 9.
LetG be a VG as shown in Figure 7. Here, V=a,b,c,d,e, n=V=5, and tMvi=0.3=ν1, fMvi=0.5=ν2, 1≤i≤5. Hence, M is a constant function.(31)E=ab,ac,ad,ae,bc,bd,be,cd,ce,de,and tNvivj=0.3, fNvivj=0.5, for all edges vivj∈E. Now, tNvivj=0.3=mintMvi,tMvj and fNvivj=0.5=maxfMvi,fMvj. So, G is a complete VG. Using Theorem 8,(32)RIVGtG=n2ν12=520.32=27.77,RIVGfG=n2ν22=520.52=10.Figure 7
A VGG.


## 4. Connectivity Index in the Selected Town for Building Hospital

Today, the treatment and medical services issue sits among the most important issues for every country. Governments are always trying to provide the best possible services and medical facilities for patients. In the past, unfortunately, as a result of the lack of medical services and the lack of hospitals and clinics, patients were not transferred to these medical centers for the required time and so they lost their lives. This crisis was because of the governments’ constraints for revenue and financial problems. So, governments decided to build hospitals in the cities so that patients could seek treatment without stress as soon as possible. But determining the right place to build a hospital is very important because firstly, it must be built in a place where there is no traffic and crowds nearby, and secondly, patients living in neighboring cities can easily access it. Therefore, in this paper, by using the concept of connectivity index, we try to determine the most appropriate places to establish a hospital. For this purpose, we consider six cities in China (Guangdong state), namely, Shaoguan, Qingyuan, Heyuan, Zhaoqing, Foshan, and Guangzhou, and the attributing symbols for each city are given in the graph asA, B, C, D, E, and F, respectively. Location of cities is shown in Figure 8. Also, distance between cities is shown in Table 2.Figure 8
Location of cities.Table 2
Distance between cities.
CitiesShaoguan-FoshanFoshan-ZhaoqingDistance202 km69 kmCitiesShaoguan-ZhaoqingShaoguan-QingyuanDistance225 km134 kmCitiesQingyuan-HeyuanHeyuan-ZhaoqingDistance168 km293 kmCitiesZhaoqing-GuangzhouDistance81 kmIn this vague graph, the nodes representing cities and edges also represent the quality of roads as well as the amount of traffic of cars during most hours of the day. Weight of nodes and edges is shown in Tables3 and 4.Table 3
Weight of nodes inG.
GABCDEFtM,fM0.5,0.30.6,0.20.5,0.40.3,0.50.6,0.30.7,0.2Table 4
Weight of edges inG.
GAEADABtN,fN0.4,0.50.2,0.60.5,0.5GEDDFBCtN,fN0.2,0.60.3,0.60.5,0.4GCDtN,fN0.2,0.5The vertexB0.6,0.3 shows that the city of Qingyuan has 60% of the necessary facilities and equipment to build a hospital, of which the necessary manpower is also part of this equipment. Clearly, it does not have 30% of the equipments for construction. The AB edge shows that the Shaoguan-Qingyuan route has 50% quality and the necessary road and transportation standards (road quality and traffic signs), and the amount of car traffic on this route is equal to 50%. For the vague graph G shown in Figure 9, we have(33)CONNGtA,E=0.4,CONNGtA,D=0.2,CONNGtA,F=0.2,CONNGtA,B=0.5,CONNGtA,C=0.5,CONNGtB,E=0.4,CONNGtB,D=0.2,CONNGtB,C=0.5,CONNGtB,F=0.2,CONNGtC,D=0.2,CONNGtC,E=0.4,CONNGtC,F=0.2,CONNGtD,E=0.2,CONNGtD,F=0.3,CONNGtE,F=0.2,CONNGfA,E=0.5,CONNGfA,D=0.5,CONNGfA,F=0.6,CONNGfA,B=0.5,CONNGfA,C=0.5,CONNGfB,E=0.5,CONNGfB,D=0.5,CONNGfB,C=0.4,CONNGfB,F=0.6,CONNGfC,D=0.5,CONNGfC,E=0.5,CONNGfC,F=0.6,CONNGfD,E=0.5,CONNGfD,F=0.6,CONNGfE,F=0.6,CIVGtG=0.12+0.03+0.07+0.15+0.125+0.144+0.036+0.15+0.084+0.03+0.12+0.07+0.036+0.063+0.084=1.312,CIVGfG=0.045+0.075+0.036+0.03+0.06+0.03+0.05+0.032+0.024+0.1+0.06+0.048+0.075+0.06+0.036=0.761.ACIVGtG=1.31215=0.0874,ACIVGfG=0.76115=0.0507.Figure 9
VGG.In the same way, we have(34)CIVGtG−A=1.817,CIVGfG−A=0.515,CIVGtG−B=0.748,CIVGfG−B=0.595,CIVGtG−C=0.817,CIVGfG−C=0.461,CIVGtG−D=1.117,CIVGfG−D=0.401,CIVGtG−E=0.808,CIVGfG−E=0.515,CIVGtG−F=0.941,CIVGfG−F=0.557.ACIVGtG−A=0.1817,ACIVGtG−B=0.0748,ACIVGtG−C=0.0817,ACIVGtG−D=0.1117,ACIVGtG−E=0.0808,ACIVGtG−F=0.0941,ACIVGfG−A=0.0515,ACIVGfG−B=0.0595,ACIVGfG−C=0.0461,ACIVGfG−D=0.0401,ACIVGfG−E=0.0515,ACIVGfG−F=0.0557.Hence,(35)ACIVGtG−B<ACIVGtGACIVGfG−B>ACIVGfG,ACIVGtG−D>ACIVGtGACIVGfG−D<ACIVGfG,ACIVGtG−E<ACIVGtGACIVGfG−E>ACIVGfG.Clearly, the verticesB and E are VCRN and the vertex D is VCEN. If we remove the node D, then the true-CI is strictly increased and false-CI is strictly decreased. Also, by deleting vertex D, vertex F is automatically deleted (because it is completely disconnected).Therefore, if the government wants to build a hospital in one of these six cities for the treatment of patients, it should avoid the cities of Zhaoqing and Guangzhou. Also, the cities of Qingyuan and Foshan are the best choices for building a hospital because firstly, they have the most facilities and human resources for construction, and secondly, the communication roads between other cities with these two cities are of better quality and have the least traffic and congestion compared to other routes.


## 5. Conclusion

The vague graphs can amplify flexibility and precision to model complex real-time problems better than a fuzzy graph. They have several applications in many decision-making processes among solution choice, weather forecasting, prognosis risks in business, and so on, and one of the most important features of VGs that has many applications in real problems is the concept of connectivity index. Connectivity index has many applications in psychology, medical sciences, social groups, and computer networks. Therefore, in this paper, we examined connectivity index, average connectivity index, and Randic index in VGs with several examples. Likewise, some special types of nodes such as vague connectivity enhancing vertex, vague connectivity reducing vertex, and vague connectivity neutral vertex are presented. Finally, an application of connectivity index in construction has been introduced. In our future work, we will define the domination of the VGs in terms of strong edges and examine their properties. Also, we will study domination in terms of independent sets, and since many of the phenomena surrounding us are hybrid, we also discuss the domination concept on its fuzzy operations.



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*Source: 9082693-2022-02-15.xml*