# A Novel Description on Vague Graph with Application in Transportation Systems

Authors: Zheng Kou; Saeed Kosari; Maryam Akhoundi
Journal: Journal of Mathematics
 (2021)

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## Abstract

Fuzzy graph (FG) models embrace the ubiquity of existing in natural and man-made structures, specifically dynamic processes in physical, biological, and social systems. It is exceedingly difficult for an expert to model those problems based on a FG because of the inconsistent and indeterminate information inherent in real-life problems being often uncertain. Vague graph (VG) can deal with the uncertainty associated with the inconsistent and determinate information of any real-world problem, where FGs many fail to reveal satisfactory results. Regularity definitions have been of high significance in the network heterogeneity study, which have implications in networks found across biology, ecology, and economy; so, adjacency sequence (AS) and fundamental sequences (FS) of regular vague graphs (RVGs) are defined with examples. One essential and adequate prerequisite has been ascribed to a VG with maximum four vertices is that it should be regular based on the adjacency sequences concept. Likewise, it is described that ifζ and its principal crisp graph (CG) are regular, then all the nodes do not have to have the similar AS. In the following, we obtain a characterization of vague detour (VD) g-eccentric node, and the concepts of vague detour g-boundary nodes and vague detour g-interior nodes in a VG are examined. Finally, an application of vague detour g-distance in transportation systems is given.


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## Body

## 1. Introduction

The graph concept stands as one of the most dominant and widely employed tools for the multiple real-world problem representation, modeling, and analyses. To represent the objects and the relations between them, the graph vertices and edges are applied, respectively. FG-models are beneficial mathematical tools for addressing the combinatorial problems in various fields involving research, optimization, algebra, computing, environmental science, and topology. Thanks to the natural existence of vagueness and ambiguity, fuzzy graphical models are strikingly better than graphical models. Rosenfeld [1] proposed the idea of FG in 1975. Akin to the set theory, the historical past of the FG theory is the fuzzy set theory developed by Zadeh [2] in 1965. The notion of vague set theory, the generalization of Zadeh’s fuzzy set theory, was introduced by Gau and Buehrer [3] in 1993. The VSs describe more possibilities than fuzzy sets. VS is more effective for the existence of the false membership degree. Kauffman [4] represented FGs based on Zadeh’s fuzzy relation [5, 6]. Akram et al. [7, 8] described several concepts and results of FGs. Samanta et al. [9, 10] represented fuzzy competition graphs and some remarks on bipolar fuzzy graphs. Gani et al. [11, 12] investigated on irregular and regular fuzzy graphs. Sunitha et al. [13, 14] studied complement of a fuzzy graph. VG notion was introduced by Ramakrishna in [15]. Borzooei et al. [16, 17] investigated new concepts of VGs. Gary Chartrand [18] discussed the concepts of detour center of a graph. The notion of detour number, detour set, detour nodes, and detour basis in a graph were established by Gary Chatrand, G. L. Johns, and P. Zhang [19]. Interior nodes and boundary nodes are discussed by G. Chatrand, D. Erwin, G. L. Johns, and P. Zhang [20]. Fuzzy detour g-distance was given by J. P. Linda and M. S. Sunitha [21]. Ghorai et al. [22, 23] defined detour g-interior nodes in bipolar fuzzy graphs and characterization of regular bipolar fuzzy graphs. The idea of strong arcs in FG was given by Bhutani and Rosenfeld [24], and types of arc in FG were given by Mathew and Sunitha [25]. The notion of bridge, trees, cycles, and end nodes were described by Rosenfeld [1]. Rosenfeld and Bhutani [24] represented the notion of g-distance in FG. Also, the notions of g-boundary node, g-interior node, and g-eccentric node were defined by Linda and Sunitha [26].A VG 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. Also, 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.The concepts of regularity play an important part in both graph theory and application in the vague environment. The highly regular graphs characterization has also been applied to the question of heterogeneity, yet all of these fail to shed enough light on real-world situations; hence, in this paper, AS and FS of RVGs are defined with examples. One essential and adequate prerequisite has been ascribed to a VG with maximum four vertices is that it should be regular based on the AS concept. Detection of nodes on the network boundary is necessary for correct operation in many wireless applications. Also, nodes close to network boundary are often assumed to provide the best candidate for beacon nodes in virtual co-ordinate construction. Focusing on these applications, a research is carried out for boundary nodes and interior ones in vague graph. A characterization of vague detour g-eccentric node and the concepts of vague detour g-boundary nodes and vague detour g-interior nodes in a VG are examined. Finally, an application of vague detour g-distance in transportation systems is given. Recently, some research has been conducted by the authors in the continuation of previous works related to VGs, bipolar fuzzy graphs, and intuitionistic fuzzy graphs which are mentioned in [27–38]. Shoaib et al. [39] introduced new concepts of pythagorean fuzzy graphs.


## 2. Preliminaries

A graph denotes a pairG∗=V,E satisfying E⊆V×V. The elements of V and E are the nodes and edges of the graph G∗, correspondingly. An FG has the form of G=λ,ν, where λ:V⟶0,1 and ν:V×V⟶0,1 as is defined by νab≤λa∧λb and ∀a,b∈V and ν is a symmetric fuzzy relation on λ and ∧ denotes the minimum.The FS of an FGG=λ,ν ([40]) is described as follows:(1)fSG=λa>0,a∈V∪νa,b>0,a,b∈V,in which the elements are taken from the interval 0,1.Definition 1 (see [3]).
A VSM is a pair tM,fM on set V that tM and fM are real-valued functions which can be defined on V⟶0,1, so that tMa+fMa≤1, ∀a∈V.Definition 2 (see [15]).
A pairζ=M,N is called to be a VG on a CG G∗, where M=tM,fM is a VS on V and N=tN,fN is a VS on E⊆V×V so that tMab≤mintMa,tM<b and fNab≥maxfMa,fMb, for every edge ab∈E.Definition 3 (see [16]).
The complement of a VGζ is a VG ζ¯=M¯,N¯, where M¯=M and N¯=tN¯,fN¯ are defined by the following:(2)tN¯ab=mintMa,tMb−tNab,fN¯ab=fNab−maxfMa,fMb,for allab∈E.Definition 4 (see [17]).
Letζ=M,N be a VG.(i)
Then, OND ofa∈V is defined by dega=degta,degfa, that degta=∑a≠bab∈EtNab and degfa=∑a≠bab∈EfNab. If all the nodes of ζ have the same OND m1,m2, then, ζ is called to be m1,m2-regular.(ii)
The CND ofa∈V in ζ is denoted as dega=degta,degfa, where degta=degta+tMa and degfa=degfa+fMa. If every node of ζ has equal closed neighborhood degree n1,n2, then ζ is called to be n1,n2-totally regular.Definition 5 (see [17]).
Letζ be a VG. If tNab=mintMa,tMb and fNab=maxfMa,fMb, ∀a,b∈V, then ζ is called a complete-VG and if tNab=mintMa,tMb and fNab=maxfMa,fMb,∀ab∈E, then, ζ is called a strong-VG of the CG G∗=V,E.Definition 6 (see [7]).
LetLG∗=Y,Z be a line graph of a CG G∗. Suppose ζ=M,N be a VG of G∗. Then, a vague line graph Lζ=M1,N1 of ζ is described as follows:iM1 and N1 are vague subsets of Y and Z, respectivelyiitM1Rm=tNm=tNumvmiiifM1Rm=fNm=fNumvmivtN1RmRn=mintNm,tNn=mintNumvm,tNunvnvfN1RmRn=maxfNm,fNn=maxfNumvm,fNunvn, ∀Rm,Rn∈Y and RmRn∈ZDefinition 7 (see [16]).
Letζ be a VG and a,b∈V.(i)
A pathP:a=a0,a1,…,ak=b in ζ is a sequence of distinct nodes so that tNai−1ai>0, fNai−1ai>0, i=1,2,…,k and the length of the path is k.(ii)
IfP:a=a0,a1,…,ak=b be a path of length k between a and b, then, tNabk and fNabk are defined as tNabk=suptNaa1∧tNa1a2∧⋯∧tNak−1b and fNabk=inffNaa1∨fNa1a2∨⋯∨fNak−1b. tN∞ab,fN∞ab is said to be the strength of connectedness between two nodes a and b in ζ, where tN∞ab=suptNkab and fN∞ab=inffNkab. If tNab≥tN∞ab and fNab≤fN∞ab, then, the arc ab is said to be a strong arc. A path a−b is strong path if all arcs on the path are strong.Definition 8 (see [16]).
A connected-VGζ is said to be a vague tree if ζ has a vague spanning subgraph S=A,E which is also a vague tree and for all arcs a,b not in S, tNab<tN∞ab and fNab>fN∞ab. The vague spanning subgraph S of ζ is a maximum spanning subgraph of ζ if ζ has no vague spanning subgraph different from S contains S.Notations are shown in Table1.Table 1
Some basic notations.
NotationMeaningFGFuzzy graphVSVague setVGVague graphASAdjacency sequenceFSFundamental sequenceRVGRegular vague graphCGCrisp graphONDOpen neighborhood degreeCNDClosed neighborhood degreeTMVTrue membership valueFMVFalse membership valueFFSFirst fundamental sequenceSFSSecond fundamental sequenceVLGVague line graphCVGConnected vague graphRGRegular graphMVMembership valueVENVague end nodeVDVague detour


## 3. New Concepts in Regular Vague Graphs

Definition 9.
The AS of a nodea in a VG ζ is described by ASa=ASat,ASaf, where ASat=t1,t2,…,tn and ASaf=f1,f2,…,fn signify the sequence of TMVs and FMVs of the edges neighbor to a arranged in ascending order, respectively.Example 1.
Consider the VGζ as Figure 1. The AS of the nodes are(3)ASa=ASat,ASaf=0.1,0.2,0.5,0.7,ASb=ASbt,ASbf=0.1,0.2,0.2,0.5,0.6,0.6,ASc=ASct,AScf=0.1,0.1,0.6,0.7,ASd=ASdt,ASdf=0.1,0.2,0.2,0.6,0.7,0.8,ASe=ASet,ASef=0.1,0.2,0.7,0.8.Figure 1
Vague graphζ and its AS.Remark 1.
(i)
The elements number inASat or ASaf is the degree of a in G∗.(ii)
The sum of all elements inASat and the sum of all elements in ASaf is dega in VG ζ, i.e.,(4)dega=∑ti∈ASatti,∑ti∈ASaffi.Remark 2.
Ifζ is a RVG, then G∗ does not have to be regular and all the nodes do not need to have the same AS.Now, we present an example that proves the correctness of the above remark.Example 2.
Consider the VGζ in Figure 2.
In this example,ζ is a 0.4,0.8-RVG, but G∗ is not regular and the nodes in ζ do not share the same AS.(5)ASb=0.1,0.1,0.2,0.2,0.2,0.4≠ASa=0.2,0.2,0.4,0.4.
Hence, in a RVG, all the nodes do not need to have the similar AS.Figure 2
Rvgζ.Remark 3.
Ifζ and ζ∗ are RGs, then all the nodes do not need to have the similar AS.Example 3.
Consider the VGζ as Figure 3.
Here,dega=degb=degc=degd=dege=0.4,1.5. So, ζ is a RVG. Correspondingly, we observe that G∗ is regular. But adsa=0.1,0.1,0.2,0.4,0.5,0.6≠adsf=0.1,0.1,0.2,0.4,0.4,0.7.Figure 3
Regular graphsζ and G∗ with different adjacency sequence.Definition 10.
The FS of a VGζ is defined as fsζ=Fstζ,fsfζ, where(6)fstζ=tMa>0;a∈V∪tNab>0:a,b∈V,fsfζ=fMa>0;a∈V∪fNab>0:a,b∈V.
The elements offstζ and fsfζ are ordered in descending and ascending order, respectively. The node part of the FS is characterized as the FFS and the edge part of the FS is labeled as the SFS and described by ffsζ and sfsζ, respectively.Theorem 1.
Letζ be a VG. If n≤4, then ζ is a RVG iff G∗, i.e., RG and all the nodes have the similar AS, n being the number of nodes in ζ.Proof.
SupposeG∗ is a RG and all the nodes have the similar AS. Then, ζ is a RVG.
On the contrary, letζ be a d=d1,d2-RVG. If there is no edge among the nodes or if the number of nodes is one, then there is nothing to prove. Thus, we suppose that E≠∅. Then, p>1 and d≠0. Consider the three cases for n=2,3,4.Case 1.
Ifn=2, then, ζ have two nodes a1 and a2. So, G∗ is 1-regular, and a1 and a2 have the same AS ASa1=tNa1a2,fNa1a2.Case 2.
Ifn=3, then there are three nodes a1,a2,a3 with edges a1a2,a2a3,a1a3 in ζ (see Figure 4). Let the edge-MV a1a2, a2a3, and a1a3 be tNa1a2,fNa1a2, tNa2a3,fNa2a3, and tNa1a3,fNa1a3, respectively.
Sinceζ is d=d1,d2-RVG, dai=d1,d2, for i=1,2,3. Hence, we have(7)tNa1a2+tNa2a3=d1,tNa2a3+tNa1a3=d1,tNa1a2+tNa1a3=d1,fNa1a2+fNa2a3=d2,fNa2a3+fNa1a3=d2,fNa1a2+fNa1a3=d2.
Simplifying the above equations, we have(8)tNa1a2=tNa2a3=tNa1a3=d12,fNa1a2=fNa2a3=fNa1a3=d22.
So, each node has the similar AS, i.e.,ASai=d1/2,d1/2,d2/2,d2/2, for i=1,2,3 and G∗ is 2-RG. Here, G∗ is a cycle and each edge has the MV d1/2,d2/2.Figure 4
ζ is a d1,d2-regular vague graph with G∗ cyclic and 2-RG.Case 3.
Ifn=4, then, there are four nodes a1,a2,a3,a4 of ζ with edges a1a2,a1a4,a1a3,a2a3,a2a4,a3a4 (see Figure 5). Suppose the MV of the edges is tNa1a2,fNa1a2, tNa1a4,fNa1a4, tNa1a3,fNa1a3, tNa2a3,fBa2a3, tNa2a4,fNa2a4, and tNa3a4,fNa3a4, respectively. Since ζ is d1,d2-RVG, dai=d1,d2, for i=1,2,3,4. Hence, we have(9)tNa1a2+tNa1a3+tNa1a4=d1,tNa2a1+tNa2a3+tNa2a4=d1,tNa3a1+tNa3a2+tNa3a4=d1,tNa4a1+tNa4a2+tNa4a3=d1,fNa1a2+fNa1a3+fNa1a4=d2,fNa2a1+fNa2a3+fNa2a4=d2,fNa3a1+fNa3a2+fNa3a4=d2,fNa4a1+fNa4a2+fNa4a3=d2.
On simplification, we obtain(10)tNa1a3=tNa2a4,tNa1a2=tNa3a4,tNa1a4=tBa2a3,fNa1a3=fNa2a4,fNa1a2=fNa3a4,fNa1a4=fNa2a3.
LettNa1a3=tNa2a4=a, tNa1a2=tNa3a4=b, tNa1a4=tNa2a3=c, and fNa1a3=fNa2a4=d, fNa1a2=fNa3a4=e, and fNa1a4=fNa2a3=f.
Since thatE≠∅, at least one of the edges-MV should be nonzero. If all the edges-MVs are non-zero, then G∗ is 3-RG and every node of AS has the elements a,b,c,d,e,f in ascending order.
If any two edges-MVs are nonzero, thenG∗ is 2-RG and each node AS has those two nonzero MVs in the ascending order. If one edge-MVs is nonzero, then G∗ is 1-RG and each node AS has that nonzero MV.Figure 5
ζ is a d1,d2-regular vague graph with G∗ 3-RG or 2-RG or 1-RG.Theorem 2.
Letζ=M,N be a d1,d2-RVG of the q-regular CG G∗ and every node has the same AS r1,r2,…,rq,s1,s2,…,sq, ri∈0,1 and si∈0,1, i=1,2,…,q. Then, the VLG Lζ of ζ is regular iff either N is constant or N has exactly three values so that r1=r2=⋯=rq−1, ri<rq≤1, for i=1,2,…,q−1 and s2=s3=⋯=sq−1=sq, 0≤s1<si, for i=2,…,q−1.Proof.
Let the VLGLζ of ζ be t1,t2-regular. Let Ne=ri,si for some i=1,2,…,q, where e=ab be any edge of ζ. Then, every a and b is neighbor with q−1 edges with MVs r1,s1,r2,s2,…,ri−1,si−1,ri+1,si+1,…,rq,sq. So, the node e in Lζ is neighbor with 2q−1 edges with MVs r1∧ri,s1∨si,r2∧ri,s2∨si,…,ri−1∧ri,si−1∨si,ri+1∧ri,si+1∨si,…,rq∧ri,sq∨si, each appearing twice. So, dLζe=dLζte,dLζfe, where(11)dLζte=2∑j≠irj∧ri=2∑j=1i−1rj∧ri+2∑j=i+1qrj∧ri=2∑j=1i−1rj+2∑j=i+1qri,sincer1≤r2≤⋯≤rq=2∑j=1i−1rj+2q−iri,
and(12)dLζfe=2∑j≠isj∨si=2∑j=1i−1sj∨si+2∑j=i+1qsj∨si=2∑j=1i−1si+2∑j=i+1qsj,sinces1≤s2≤⋯≤sq=2i−1si+2∑j=i+1qsj.
This holds for every edgee of ζ. Since Lζ is t1,t2-regular, dLζe=t1,t2 for each node set e in Lζ. Thus, 2∑j=1i−1rj+2q−iri=t1 and 2i−1si+2∑j=i+1qsj=t2, i.e.,(13)∑j=1i−1rj+q−iri=t12i−1si+∑j=i+1qsj=t22,for alli=1,2,…,q.
By equation (13), we have(14)q−1r1=t12∑j=2qsj=t22,(15)r1+q−2r2=t12s2+∑j=3qsj=t22,(16)r1+r2+q−3r3=t122s3+∑j=4qsj=t22,(17)r1+r2+⋯+rq−2+rq−1=t12q−2sq−1+sq=t22,(18)r1+r2+⋯+rq−1=t12q−1sq=t22.
Simplifying equations (13)–(18), we have the following:(19)r1=r2=⋯=rqs2=s3=sq−1=sq.
Ifr1=rq and s1=sq, then N is constant; otherwise, N has three values so that r1=r2=⋯=rq−1 and s2=s3=⋯=sq−1=sq.
On the contrary, supposeN is constant or N have just three values so that r1=r2=⋯=rq−1, ri<rq≤1, for i=1,2,…,q−1, and s2=s3=⋯=sq−1=sq, 0≤s1<si, for i=2,…,q−1. If N is constant, let Nab=k1,k2, ∀ab∈E. Then, k1=ri and k2=si, ∀i=1,2,…,q. So, rj∧ri=k1 and sj∨si=k2, ∀j≠i. Otherwise, suppose rj=k1, ∀j=1,2,…,q−1 and sj=k2, ∀j=2,…,q. Next, rj=k1<rq and s1<sj=k2.
So,rj∧ri=k1 and sj∨si=k2, ∀j≠i. Therefore, in all the cases, rj∧ri=k1 and sj∨si=k2, ∀j≠i. Thus, for any node e=ab∈Lζ, we have(20)dLζe=dLζte,dLζfe=2∑j≠irj∧ri,2∑j≠isj∨si=2∑j≠ik1,2∑j≠ik2=2q−1k1,2q−1k2.
Hence,Lζ is a 2q−1k1,2q−1k2-regular.Remark 4.
The RVG complement does not have to be regular. This can be observed in the following example.Example 4.
Consider the VGζ as Figure 6.
Here,dega=degb=degc=0.2,1.2. So, ζ is 0.2,1.2-regular. But in its complement ζ¯, dega=0.6,0.4, degb=0.4,0.5, degc=0.6,0.5.
So,ζ¯ is not regular.Figure 6
Vague graphζ and its complement ζ¯.


## 4. Vague Detourg-Distance and Vague Detour g-Periphery

Definition 11.
The length of ax−y strong path p between x and y in a CVG ζ=M,N is said to be a vague detour (VD) g-distance if there is no other strong path longer than p between x and y, and we show this by VDgx,y. Any x−y strong path whose length is VDgx,y is named a x−y vague g-detour.Example 5.
Letζ=M,N be a CVG of the graph G∗, where V=x,y,z,w,k and E=x,y,z,w,y,z,x,w,x,z,x,k,k,w. For the VG ζ in Figure 7, it is seen that all arcs except k,w and y,w are strong arcs and the VD g-distance of two nodes are given as follows:(21)VDgx,y=1,VDgy,k=1,VDgw,k=4,VDgz,w=2,VDgx,z=2,VDgx,w=2,VDgy,w=3,VDgk,z=4,VDgy,z=3,VDgk,x=2.Figure 7
Connected vague graphζ.Definition 12.
The length of any smallest strong path fromx to y is named the vague geodesic distance, described by Vdgx,y.The VDg-eccentricity eVDga for a node a is maxV⋅Dga,x, ∀x∈ζ. The set of all VD g-eccentricity nodes of a, described by aV⋅Dg∗. The VD g-radius of ζ, denoted by radV⋅Dgζ, is defined as min eV⋅Dgx, ∀x∈ζ. If eV⋅Dga=radV⋅Dgζ, then the node a is the VD g-central node of ζ. The VD g-diameter of ζ, denoted by diamV⋅Dgζ, is defined as maxeV⋅Dgx, ∀x∈ζ. If eV⋅Dga=diamV⋅Dgζ, then the node a is named the VD g-peripheral node of ζ.Example 6.
For the connected-VGζ in Figure 7, eV⋅Dgx=2, eV⋅Dgy=3, eV⋅Dgk=4, eV⋅Dgw=4, eV⋅Dgz=4, and radV⋅Dgζ=1, diamV⋅Dgζ=4.Definition 13.
The vague subgraph of the VGζ=M,N, whose nodes are only the VD g-peripheral nodes is named a VD g-periphery of ζ, and it is denoted by PerV⋅Dgζ.Definition 14.
If every node of a CVGζ=M,N is a VD g-eccentric node, then ζ is said to be a VD g-eccentric vague graph. The vague subgraph of ζ formed by the set of all vague g-eccentric nodes of ζ is named a VD g-eccentric vague subgraph of ζ, it is described by ECCV⋅DgζExample 7.
For the VG of Figure8, all nodes x,y,z, and w are VD g-periphery nodes since eV⋅Dgx=eV⋅Dgy=eV⋅Dgz=eV⋅Dgw=2, diamV⋅Dgζ=2. Also, we have(22)xV⋅Dg∗=y,yV⋅Dg∗=x,zV⋅Dg∗=w,wV⋅Dg∗=z.Figure 8
Connected vague graphζ.Theorem 3.
A VGζ is a VD g-self-centered if and only if each node of ζ is a VD g-eccentric.Proof.
Assumeζ is a VD g-self centered graph and let y be a node in ζ. Let x∈yV⋅Dg∗. So, eV⋅Dgy=V⋅Dgx,y. Since ζ is a VD g-self centered vague graph, eV⋅Dgx=eV⋅Dgy=V⋅Dgx,y, and this implies that y∈xV⋅Dg∗. So, y is a VD g-eccentric vertex of ζ. Conversely, let each vertex of ζ is a VD g-eccentric node. If possible, suppose ζ be not VD g-self-centered vague graph. Then, radV⋅Dgζ≠diamV⋅Dgζ, and there exists a node s∈ζ so that eV⋅Dgs=diamV⋅Dgζ. Also, let p∈sV⋅Dg∗ Let u be a s−p VD in ζ. So, there must have a node q on u for which the node q is not a VD g-eccentric node of u. Also, q cannot be a VD g-eccentric node of each other node. Again, if q be a VD g-eccentric node of a node x (say), it means q∈xV⋅Dg∗. Then, there exists an extension of a x−q vague g-detour up to s or up to p. But this contradicts the facts that q∈xV⋅Dg∗. Therefore, radV⋅Dgζ=diamV⋅Dgζ and ζ is a VD g-self centered vague graph.Theorem 4.
Ifζ is a VD g-self-centered graph, then radV⋅Dgζ=diamV⋅Dgζ=n−1, where n is the number of nodes of ζ.Proof.
Letζ be a VD g-self-centered graph. If possible, let diamV⋅Dgζ=l<n−1. Let u1 and u2 be two distinct VD g-peripheral path. Let r∈u1, s∈u2. So, there exists a strong path between r and s because of connectedness of ζ. Then, there exists nodes on u1 and u2, whose eccentricity>l, but this is impossible because diamV⋅Dgζ=l. Hence, u1 and u2 are not distinct. Since u1 and u2 are arbitrary, so there exists a node t in ζ such that t is a common in all VD peripheral paths. So, eV⋅Dgt<l, which is impossible, because ζ is a VD g-self-centered. Therefore, diamV⋅Dgζ=n−1=radV⋅Dgζ.Corollary 1.
For a CVGζ, perV⋅Dgζ=ζ if and only if the VD g-eccentricity of every node of ζ is n−1, n=numberofnodesinζ.Proof.
LetPerV⋅Dgζ=ζ. Then, eV⋅Dgp=diamV⋅Dgζ, ∀p∈ζ. So, each node of ζ is a VD g-periphery node of ζ. So, ζ is a self-centered vague graph and diamV⋅Dgζ=radV⋅Dgζ=n−1. Therefore, the VD g-eccentricity of every node of ζ is n−1.
Conversely, let the VDg-eccentricity of every node of ζ is n−1. So, diamV⋅Dgζ=radV⋅Dgζ=n−1. All nodes of ζ are VD g-peripheral nodes, and hence PerV⋅Dgζ=ζ.Corollary 2.
For a CVGζ, ECCV⋅Dgζ=ζ if and only if the VD g-eccentricity of every node of ζ is n−1, n=numberofnodesinζ.Proof.
LetECCV⋅Dgζ=ζ. So, all nodes of ζ are VD g-eccentric node. Hence, ζ is self-centered vague graph and diamV⋅Dgζ=n−1. So, the VD g-eccentricity of every node of ζ is n−1.
Conversely, let the VDg-eccentricity of every node of ζ is n−1. So, radV⋅Dgζ=diamV⋅Dgζ=n−1. Therefore, all nodes of ζ are VD g-peripheral nodes as well as VD g-eccentric node. So, ECCV⋅Dgζ=ζ.Theorem 5.
In a CVGζ, a node x is a VD g-eccentric node if and only if x is a VD g-peripheral node.Proof.
Letx be a VD g-eccentric node of ζ and let x∈yV⋅Dg∗. Let z and w be two VD g-peripheral nodes, then V⋅Dgz,w=diamV⋅Dgζ=ksay. Let p1 and p2 be any z−w and y−x vague g-detour in ζ, respectively. There arise two cases:Case 1: if x is not internal node in ζ, i.e., there is only one node, say s which is adjacent to x. So, s∈P2. Since ζ is connected, s is connected to a node of p1, say p′. Therefore, either s′∈p2 or s′∈p1∩p2. Thus, in any case, the path from y to z or y to w through s and s′ is longer than p2. But it is impossible, since x is a VD g-eccentric node of y. So, eV⋅Dgy=diamV⋅Dgζ, i.e., b is a VD g-peripheral node of ζ.Case 2: if x is an internal node in ζ, then ∃ a connection between x to z and x to w because of connectedness of ζ. Then, y−x vague g-detour can be extended to z or w. This is impossible because x is a VD g-eccentric node of y. Hence, eV⋅Dgy=diamV⋅Dgζ, i.e., x is a VD g-peripheral node of ζ. Conversely, we suppose that x be a VD g-peripheral node of ζ. So, ∃ a VD g-peripheral node, say y (distinct from x). Therefore, x is a VD g-eccentric node of y.Definition 15.
In a CVGζ, a node y is said to be a VD g-boundary node of a node x, if V⋅Dgx,y≥V⋅Dgx,z, for each z in ζ, where z is a neighbor of y.
The set of all VDg-boundary nodes of x described by xV⋅Dg′.Example 8.
Consider the CVGζ in Figure 9.
In this example, we have(23)xV⋅Dg′=t,s,yV⋅Dg′=x,t,s,zV⋅Dg′=x,t,s,wV⋅Dg′=x,t,s,tV⋅Dg′=x,s,sV⋅Dg′=x,t.
Here,x, t, and s are the VD g-boundary nodes of ζ.Figure 9
Connected vague graphζ.Theorem 6.
A CVGζ is a vague tree if and only if ζ is a vague g-detour graph.Proof.
Letζ be a vague tree. Then, between any two nodes in ζ, there is exactly one vague strong path. So, V⋅Dgx,y=V⋅dgx,y, for any two nodes x,y in ζ. Hence, ζ is a vague g-detour graph.
On the contrary, letζ be a vague g-detour graph, which has n nodes. Then, V⋅Dgx,y=V⋅dgx,y, for any two nodes x,y in ζ. If n=z, then ζ is a vague tree. Let n≥3. If possible, let ζ be not a vague tree. So, there exists two nodes s,t in ζ for which there is at least two strong paths between s and t. Let Q1 and Q2 be two s−t vague strong paths. So, Q1∪Q2 has a cycle Csay in ζ. If nodes x and y are adjacent nodes in ζ, then we have V⋅dgx,y=1 and V⋅Dgx,y>1. This contradicts the fact that V⋅Dgx,y=V⋅dgx,y. Hence, ζ is a vague tree.Definition 16.
A nodex in a VG ζ is called a VEN of ζ if y is the only strong neighbor of x, where y∈ζ.Example 9.
For the VGζ in Figure 9, the nodes x, t, and s are VENs of ζ.Theorem 7.
A nodey in a vague tree ζ is a VD g-boundary node if and only if y is a VEN.Proof.
Let a nodey be a VD g-boundary node for a node x in a vague tree ζ. Let E be a maximum vague spanning tree in ζ, which is unique in ζ.


## 5. Application of Vague Detourg-Distance in the Transportation System

Today, the issue of transportation plays a very important role in human life. If this is done faster and easier, then it can affect the quality of life and human health. Unfortunately, in the past, due to the lack of sufficient vehicles and bad road conditions, many problems were created for human beings, one of the most important of which is the lack of timely transfer of patients to private hospitals and clinics for treatment. Therefore, in this paper, we intend to express the importance and application of vague detourg-distance in the transportation of a patient to the most appropriate hospital in the shortest possible time. For this purpose, we consider four hospitals in Iran (Babol City) named Yahya Nejad (B), Beheshti (C), Mehregan (D), and Rouhani (E), which are shown in the graph with the symbols of B, C, D, and E. Suppose that a patient lives in location A and must be transported by ambulance to one of these four hospitals in the shortest possible time. In this vague graph, the nodes represent the hospitals and the edges also shows the amount of traffic generated at a certain hour of the day. The weight of the nodes and edges is shown in Tables 2 and 3. The location of hospitals is shown in Figure 10.Table 2
Weight of nodes.
ABCDEtA0.20.20.30.50.4fA0.20.30.50.20.3Table 3
Weight of edges.
ζA−Yahya NejadA−BeheshtiA−MehregantB,fB0.2,0.50.1,0.80.2,0.4ζA−RouhaniBeheshti−MehreganMehregan−RouhanitB,fB0.2,0.40.3,0.70.3,0.7ζYahya Nejad−BeheshtitB,fB0.2,0.7Figure 10
Location of hospitals in Babol City.The nodeD0.5,0.2 shows that this hospital has 50% of the medical facilities and equipment needed to treat a patient, but does not have 20% of the necessary tools. The edge AD indicates that 40% of this route has traffic and congestion caused by vehicles, but 20% of it is free of cars and vehicles. The vague detour g-distance for Figure 11 is as follows:(24)VDgA,B=4,VDgA,C=3,VDgA,D=3,VDgA,E=4,VDgB,C=4,VDgB,D=3,VDgB,E=3,VDgC,D=4,VDgC,E=3,VDgD,E=4.Figure 11
Vague graphζ.It is clear thatVDgA,D has the lowest value, so we conclude that it can be the best choice because, firstly, Mehregan hospital has the most medical equipment and facilities compared to other hospitals, and secondly, A-Mehregan route has the most empty space for patient transfer to medical centers by ambulance. Therefore, governments should provide conditions for patients to be transported to hospitals without stress due to the congestion of roads and intercity traffic, so that they can be treated as soon as possible.


## 6. Conclusion

A VG is an indiscriminately comprehensive structure of an FG that offers higher precision, adaptability, and compatibility to a system when coordinated with systems running on FGs. VGs are so useful tool to examine many issues such as networking, social systems, geometry, biology, clustering, medical science, and traffic plan. Eccentric nodes andg-boundary nodes are a practical interest in several areas. In wireless networking, boundary nodes are used to find efficient routes within ad hoc mobile networks. They have also been used in document summarization and in designing secure systems for electrical grids. Hence, in this paper, the adjacency sequence of RVGs is defined with examples. Also, a characterization of vague detour g-eccentric node and the concepts of vague detour g-boundary nodes, vague detour g-interior nodes in a VG are examined. In our future work, we will introduce vague incidence graphs and study the concepts of perfect dominating set, regular perfect dominating set, and independent perfect dominating set on vague incidence graph. Likewise, we will try to define the average connectivity index, parameter and the new concepts of vague connectivity-enhancing node, vague detour g-interior node, and vague detour g-boundary node of a vague incidence tree using maximum vague spanning tree.



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*Source: 4800499-2021-12-02.xml*