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ORIGINAL RESEARCH article

Front. Phys., 21 February 2023

Sec. Statistical and Computational Physics

Volume 11 - 2023 | https://doi.org/10.3389/fphy.2023.1130765

New concepts on level graphs of vague graphs with application in medicine

Xiaolong ShiXiaolong Shi1Wubian Jiang
Wubian Jiang2*Aysha KhanAysha Khan3Maryam AkhoundiMaryam Akhoundi4
  • 1Institute of Computing Science and Technology, Guangzhou University, Guangzhou, China
  • 2Renmin Hospital of Wuhan University Outpatient Management Service Department, Wuhan, China
  • 3Department of Mathematics, Prince Sattam Bin Abdulaziz University, Al-Kharj, Saudi Arabia
  • 4Clinical Research Development Unit of Rouhani Hospital, Babol University of Medical Sciences, Babol, Iran

Vague graphs (VGs), belonging to the fuzzy graph (FG) family, have good capabilities when facing problems that cannot be expressed by FGs. When an element membership is unclear, neutrality is a good option that can be well-supported by a VG. The previous definitions limitations in FG have led us to offer new definitions in VGs. Therefore, this study introduces the notion of vague edge graph (VEG) ˆζ=(V,N), in which V is a crisp vertex set and N is a vague relation (VR) on M, presenting some of its properties. Using λ-level graphs (LGs) and (λ, δ)-LGs, we characterize VG ζ = (M, N), where M is a vague set (VS) on V and N is a VR on V. Medical diagnosis is one of the most sensitive and important issues in the medical sciences. If it is not done properly, the patient will suffer irreparable damage. Therefore, an application of VG in the diagnosis of the disease is expressed.

1 Introduction

After the introduction of fuzzy sets (FSs) [1], the fuzzy set theory is included as a large research field. Since then, the theory of FSs has become a vigorous area of research in different disciplines, including life sciences, management science, statistic, graph theory, and automata theory. Graphs from ancient times to the present day have played a very important role in various fields, including computer science and social networks, so that, with the help of the nodes and edges of a graph, the relationships between objects and elements in a social group can be easily introduced.

A fuzzy graph (FG) is one of the most widely used topics in fuzzy theory, which has been studied by many researchers. One of the advantages of FG is its flexibility in reducing time and costs in economic issues, which has been welcomed by all managers of institutions and companies. Gau and Buehrer [2] organized the FS theory by presenting the VS notion by changing the value of an element in a set with a subinterval of [0,1]. A VS is more initiative and helpful due to the existence of false membership degrees. Kauffman [3] introduced FGs using Zadeh’s fuzzy relation (FR) [4, 5]. However, Rosenfeld [6] presented another detailed definition, such as paths, cycles, and connectedness. Mordeson and Chang-Shyh [7] defined operations on FGs. References [8, 9] introduced certain types of product bipolar FGs and some operations and densities of m-polar FGs. Das et al. [10] presented generalized neutrosophic competition graphs. Bhattacharya [11] identified some remarks on FGs. Mordeson and Nair [12] studied several concepts of FGs. Mahapatra [13] introduced radio FGs and frequency assignment in radio stations. References [14–16] investigated new definitions of vague graphs, and references [17–20] defined several concepts on VGs and neutrosophic competition graphs. Shoaib et al. [21] studied complex Pythagorean FGs.

VG is a type of FG. VGs have a variety of applications in other sciences, including biology, psychology, management, and medicine. They are used to find the most effective person in an organization or institution. Likewise, a VG can focus on determining the uncertainty combined with the inconsistent and indeterminate information of any real-world problem in which FGs may not lead to adequate results. The nodes in this graph represent the individuals, and the edges show the extent of the relationship between employees. Furthermore, VGs play a very important role in the field of medical sciences and are used to diagnose diseases and reduce the costs of hospitals and medical clinics using the concept of domination and covering. Ramakrishna [22] recommended the VG notion and evaluated some of its features. Borzooei and Rashmanlou [23, 24] introduced new concepts in VGs. Sunitha and Vijayakumar [25] presented a complement of FGs. Kosari et al. and Kou et al. [26, 27] studied new results in VG structures. References [28–30] defined dominating and equitable dominating sets in VGs. Shi and Kosari [31] investigated the global dominating set in product-VGs. Shao et al. [32] introduced a bondage set and bondage number in intuitionistic FG. VG is used to illustrate real-world phenomena using vague models in a variety of fields, including technology, social networking, and biological networks. Therefore, in this study, we presented the notion of VEG and introduced some of its properties. Likewise, we characterized VG ζ = (M, N), where M is a VS and N is a VR. Some operations, including CP, LP, SP, and cross-product on VGs, have been defined. Finally, an application of VG in medical diagnosis has been given.

2 Preliminaries

In this section, we introduce some basic concepts of VGs.

A graph is an ordered pair ζ* = (V, E), where V is the set of nodes of ζ* and E∈̃V2 is the set of edges of ζ*. Two nodes p and q in a graph G* are said to be neighbors in G*, if {p, q} is in an edge of G*.

Definition 2.1. A fuzzy graph (FG) is a pair ς = (τ, ν) with a set X [12]; then τ is a fuzzy set (FS) in X, and ν is a fuzzy relation (FR) in X × X, so that

γ(pq)≤min{τ(p),τ(q)},

for all pq ∈ X × X.

Definition 2.2. A VS is a pair (tM, fM) on set X [2], where tM and fM are real-valued functions, which can be presented on V → [0, 1] so that tM(p) + fM(p) ≤ 1 and ∀p ∈ X.

Definition 2.3. A VG is defined as a pair ζ = (M, N) [22],where M = (tM, fM) is a VS on V and N = (tN, fN) is a VS on E ⊆ V × V so that for each pq ∈ E, tN (pq) ≤ tM(p) ∧ tM(q) and fN (pq) ≥ fN(p) ∨ fN(q).

Definition 2.4. A VEG on a non-empty set V is an ordered pair of the form ˆζ=(V,N), where V is a crisp vertex set (CVS) and N is a VR on V so that tN (pq) ≤ min{tM(p), tM(q)}, fN (pq) ≥ max{fM(p), fM(q)}, and 0 ≤ tN (pq) + fN (pq) ≤ 1, for all pq ∈ E.

We consider VEGs with CVS, that is, VGs ˆζ=(V,N), that is, tM(p) = 1, fM(p) = 0, ∀p ∈ V, and edges with true membership and false membership degrees in [0,1].

Example 2.5. Consider a simple graph (SG) ζ* = (V, E) [24]so that V = {p, q, s} and E = {pq, qs, ps}. Let N be a VR on V described by N = {(pq, 0.4, 0.2), (qs, 0.5, 0.2), (ps, 0.3, 0.2)}. Clearly, ˆζ=(V,N) is a VEG with CVS and VS of edges (see Figure 1).

FIGURE 1
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FIGURE 1. Vague edge graph ˆζ=(V,N).

3 Vague graphs by level graphs

Definition 3.1. Suppose that M = (tm, fM) is a VS on V. Then, the set M(λ,δ) = {p ∈ V|tM(p) ≥ λ, fM(p) ≤ δ}, where (λ, δ) ∈ [0, 1] × [0, 1] and λ + δ ≤ 1 is named the (λ, δ)-level set of M. Let N = (tN, fN) be a VR on V. Then, the set N(λ,δ) = {pq ∈ V × V|tN (pq) ≥ λ, fM(pq) ≤ δ}, where (λ, δ) ∈ [0, 1] × [0, 1] and λ + δ ≤ 1 is called (λ, δ)-LG. In the case of λ = δ, where λ ≤ 1, we write LG by ζα instead of ζ(λ,δ). Note that

M(λ,δ)={p∈V|tM(p)≥λ}∩{p∈V|fM(p)≤δ}=U(t;λ)∩L(f;δ),N(λ,δ)={pq∈V×V|tN(pq)≥λ}∩{pq∈V×V|fN(pq)≤δ}=U(t;λ)∩L(f;δ).

Remark 3.2. The level graph ζ(λ,δ) = (M(λ,δ), N(λ,δ)) is a subgraph of ζ* = (V, E).

Example 3.3. Consider an SG ζ* = (V, E) so that V = {p, q, r, s} and E = {pq, qr, rs, ps, pr, qs}. From Figure 2, we get that ζ = (M, N) is a VG.Take λ = 0.5. We have M0.5 = {s, r} and N0.5 = {rs}. Obviously, the 0.5-LG ζ0.5 is a subgraph of ζ*.Now, we take λ = 0.2 and δ = 0.3. By Definition 3.1, we have M(0.2,0.3) = {p, r, s} and N(0.2,0.3) = {ps}. Clearly, (0.2,0.3)-LG ζ(0.2,0.3) is a subgraph of ζ*.

FIGURE 2
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FIGURE 2. Vague graph ζ.

Theorem 3.4. ζ = (M, N) is a VG if ζ(λ,δ) is a crisp graph for each pair (λ, δ) ∈ [0, 1] × [0, 1] and λ + δ ≤ 1.Proof. Suppose ζ is a VG. For each (λ, δ) ∈ [0, 1] × [0, 1], we take pq ∈ N(λ,δ). Then, tN (pq) ≥ λ and fN (pq) ≤ δ. Since ζ is a VG, it follows that

λ≤tN(pq)≤min(tM(p),tM(q)),δ≥fN(pq)≥max(fM(p),fM(q)).

It shows that λ ≤ tM(p), λ ≤ tM(q), δ ≥ fM(p), and δ ≥ fM(q); that is, p, q ∈ M(λ,δ). Hence, ζ(λ,δ) is a graph for each (λ, δ) ∈ [0, 1] × [0, 1]. Conversely, suppose ζ(λ,δ) is a graph for all (λ, δ) ∈ [0, 1] × [0, 1]. For each pq∈̃V2, let fN (pq) = δ and tN (pq) = λ. Then, pq ∈ N(λ,δ). Since ζ(λ,δ) is a graph, we have p, q ∈ M(λ,δ), so tM(p) ≥ λ, tM(q) ≥ λ, fM(p) ≤ δ, and fM(q) ≤ δ. Therefore,

tN(pq)=λ≤min(tM(p),tM(q)),fN(pq)=δ≥max(fM(p),fM(q)),

that is, ζ = (M, N) is a VG. Definition 3.5. Suppose ζ1 = (M1, N1) and ζ2 = (M2, N2) are two VGs of ζ*1=(V1,E1) and ζ*2=(V2,E2), respectively. The Cartesian product (CP) ζ1 × ζ2 is the pair (M, N) of VSs defined on the CP ζ*1×ζ*2 so that

(i){tM(p1,p2)=min(tM1(p1),tM2(p2))fM(p1,p2)=max(fM1(p1),fM2(p2)),∀(p1,p2)∈V1×V2,(ii){tN((p,p2)(p,q2))=min(tM1(p),tN2(p2q2))fM((p,p2)(p,q2))=max(fM1(p),fN2(p2q2)),∀p∈V1andp2q2∈E2,(iii){tN((p1,r)(q1,r))=min(tN1(p1q1),tM2(r))fM((p1,r)(q1,r))=max(fN1(p1q1),fM2(r)),∀r∈V2andp1q1∈E1.

Theorem 3.6. ζ = (M, N) is the CP of ζ1 and ζ2 if and only if each pair (λ, δ) ∈ [0, 1] × [0, 1] and λ + δ ≤ 1, (λ, δ)-LG ζ(λ,δ) is the CP of (ζ1)(λ,δ) and (ζ2)(λ,δ).Proof. Assume ζ = (M, N) is the CP of ζ1 and ζ2. For each (λ, δ) ∈ [0, 1] × [0, 1], if (p, q) ∈ M(λ,δ), then

min(tM1(p),tM2(q))=tM(p,q)≥δ

and

max(fM1(p),fM2(q))=fM(p,q)≤λ.

Hence, p∈(M1)(λ,δ) and q∈(M2)(λ,δ); that is (p,q)∈(M1)(λ,δ)×(M2)(λ,δ). Therefore, M(λ,δ)⊆(M1)(λ,δ)×(M2)(λ,δ).Now if (p,q)∈(M1)(λ,δ)×(M2)(λ,δ), then p∈(M1)(λ,δ) and q∈(M2)(λ,δ). It follows min(tM1(p),tM2(q))≥δ and max(fM1(p),fM2(q))≤λ. Since (M, N) is the CP of ζ1 and ζ2, tM(p, q) ≥ δ and fM(p, q) ≤ λ; that is, (p, q) ∈ M(λ,δ). So, (M1)(λ,δ)×(M2)(λ,δ)⊆M(λ,δ). Thus, (M1)(λ,δ)×(M2)(λ,δ)=M(λ,δ). Now, we prove N(λ,δ) = E, where E is the edge set of the CP of (ζ1)(λ,δ)×(ζ2)(λ,δ) and ∀(λ, δ) ∈ [0, 1] × [0, 1]. Suppose (p1, p2) (q1, q2) ∈ N(λ,δ). Then, tN((p1,p2)(q1,q2))≥δ and tN((p1,p2)(q1,q2))≤λ. Since (M, N) is the CP of ζ1 and ζ2, one of the following cases holds:(i) p1 = q1 and p2q2 ∈ E2.(ii) p2 = q2 and p1q1 ∈ E1.For case (i), we have

tN((p1,p2)(q1,q2))=min(tM1(p1),tM2(p2q2))≥δ,fN((p1,p2)(q1,q2))=max(fM1(p1),fN2(p2q2))≤λ.

So, tM1(p1)≥δ, fM1(p1)≤λ, tN2(p2q2)≥δ, and fN2(p2q2)≤λ. It follows that p1=q1∈(M1)(λ,δ) and p2q2∈(N2)(λ,δ); that is, (p1, p2) (q1, q2) ∈ E.Similarly, for case (ii), we get (p1, p2) (q1, q2) ∈ E. Thus, N(λ,δ) ⊆ E. For each (p, p2) (p, q2) ∈ E, tM1(p)≥δ, fM1(p)≤λ, tN2(p2q2)≥δ, and fN2(p2q2)≤λ. Since (M, N) is the CP of ζ1 and ζ2, we have

tN((p,p2)(p,q2))=min(tM1(p),tN2(p2q2))≥δ,fM((p,p2)(p,q2))=max(fM1(p),fN2(p2q2))≤λ.

Therefore, (p, p2) (p, q2) ∈ N(λ,δ).In the same way, for each (p1, r) (q1, r) ∈ E, we get (p1, r) (q1, r) ∈ N(λ,δ). So, E ⊆ N(λ,δ) and N(λ,δ) = E.The converse part is obvious.Definition 3.7. Let ζ1 and ζ2 be two VGs of ζ*1=(V1,E1) and ζ*2=(V2,E2), respectively. The composition (Co) ζ1 [ζ2] is the pair (M, N) of VSs defined on the Co ζ*1[ζ*2] so that

(i){tM(p1,p2)=min(tM1(p1),tM2(p2)),fM(p1,p2)=max(fM1(p1),fM2(p2)),∀(p1,p2)∈V1×V2.(ii){tN((p,p2)(p,q2))=min(tM1(p),tN2(p2q2)),fN((p,p2)(p,q2))=max(fM1(p),fN2(p2q2)),∀p∈V1and∀p2q2∈E2.(iii){tN((p1,r)(q1,r))=min(tN1(p1q1),tM2(r)),fN((p1,r)(q1,r))=max(fN1(p1q1),fM2(r)),∀r∈V2and∀p1q1∈E1.(iv){tN((p1,p2)(q1,q2))=min(tM2(p2),tM2(q2),tN1(p1q1)),fN((p1,p2)(q1,q2))=max(fM2(p2),fM2(q2),fN1(p1q1)),∀p2,q2∈V2and∀p1q1∈E1thatp2≠q2.

Theorem 3.8. ζ = (M, N) is the Co of VGs ζ1 and ζ2 if, for every (λ, δ) ∈ [0, 1] × [0, 1] and λ + δ ≤ 1, (λ, δ)-LG ζ(λ,δ) is the Co of (ζ1)(λ,δ) and (ζ2)(λ,δ).Proof. Let ζ = (M, N) be the Co of ζ1 and ζ2. By the definition of ζ1 [ζ2] and the same argument as in the proof of Theorem 3.6, we have M(λ,δ)=(M1)(λ,δ)×(M2)(λ,δ). Now, we prove N(λ,δ) = E, where E is the edge set of the co (ζ1)(λ,δ)[(ζ2)(λ,δ)], for all (λ, δ) ∈ [0, 1] × [0, 1]. Assume (p1, p2) (q1, q2) ∈ N(λ,δ). Then, tN ((p1, p2) (q1, q2)) ≥ δ and fN ((p1, p2) (q1, q2)) ≤ λ. Since ζ = (M, N) is the Co ζ[ζ2], one of the following conditions hold:(i) p1 = q1 and p2q2 ∈ E2.(ii) p2 = q2 and p1q1 ∈ E1.(iii) p2 ≠ q2 and p1q1 ∈ E1.For cases (i) and (ii), the same as cases (i) and (ii) in the proof of Theorem 3.6, we get (p1, p2) (q1, q2) ∈ E. For case (iii), we have

tN((p1,p2)(q1,q2))=min(tM2(p2),tM2(q2),tN1(p1q1))≥δ,fN((p1,p2)(q1,q2))=max(fM2(p2),fM2(q2),fN1(p1q1))≤λ.

So, tM2(p2)≥δ, tM2(q2)≥δ, tN1(p1q1)≥δ, fM2(p2)≤λ, fM2(q2)≤λ, and fN1(p1q1)≤λ. It follows that p2q2∈(M2)(λ,δ) and p1q1∈(N1)(λ,δ); that is, (p1, p2) (q1, q2) ∈ E. Thus, N(λ,δ) ⊆ E. For each (p1, p2) (q1, q2) ∈ E, tM1(p)≥δ, fM1(p)≤λ, tN2(p2q2)≥δ, and fN2(p2q2)≤λ. Since ζ = (M, N) is the Co of ζ1 [ζ2], we get

tN((p,p2)(p,q2))=min(tM1(p),tN2(p2q2))≥δ,fM((p,p2)(p,q2))=max(fM1(p),fN2(p2q2))≤λ.

So, (p, p2) (p, q2) ∈ N(λ,δ). Similarly, for each (p1, r) (q1, r) ∈ E, we get (p, p2) (p, q2) ∈ N(λ,δ). For each (p1, p2) (q1, q2) ∈ E, where p2 ≠ q2 and p1 ≠ q1, tN1(p1q1)≥δ, fN1(p1q1)≤λ, tM2(q2)≥δ, fM2(q2)≤λ, tM2(p2)≥δ, and fM2(p2)≥λ. Since ζ = (M, N) is the Co of G1[G2], we have

tN((p1,p2)(q1,q2))=min(tM2(p2),tM2(q2),tN1(p1q1))≥δ,fN((p1,p2)(q1,q2))=max(fM2(p2),fM2(q2),fN1(p1q1))≤λ,

and then (p1, p2) (q1, q2) ∈ N(λ,δ). Hence, E ⊆ N(λ,δ). Thus, E = N(λ,δ).Conversely, suppose (M(λ,δ), N(λ,δ)), where (λ, δ) ∈ [0, 1] × [0, 1], is the Co of (ζ1)(λ,δ) and ((M2)(λ,δ),(N2)(λ,δ)). In the same way, by the same arguments as in the proof of Theorem 3.6, we get

tN((p1,p2)(q1,q2))=min(tM2(p2),tM2(q2),tN1(p1q1)),fN((p1,p2)(q1,q2))=max(fM2(p2),fM2(q2),fN1(p1q1)),

∀p2, q2 ∈ V2 (p2 ≠ q2) and ∀ p1q1 ∈ E1.This completes the proof.Definition 3.9. Let ζ1 = (M1, N1) and ζ2 = (M2, N2) be two VGs. The union ζ1 ∪ ζ2 is defined as the pair (M, N) of VSs described on the union of graphs ζ*1 and ζ*2 so that

(i)tM(p)={tM1(p)ifp∈V1,p∉V2,tM2(p)ifp∈V2,p∉V1,max(tM1(p),tM2(p))ifp∈V1∩V2,(ii)fM(p)={fM1(p)ifp∈V1,p∉V2,fM2(p)ifp∈V2,p∉V1,min(fM1(p),fM2(p))ifp∈V1∩V2,(iii)tN(pq)={tN1(pq)ifpq∈E1,pq∉E2,tN2(pq)ifpq∈E2,pq∉E1,max(tN1(pq),tN2(pq))ifpq∈E1∩E2,(iv)fN(pq)={fN1(pq)ifpq∈E1,pq∉E2,fN2(pq)ifpq∈E2,pq∉E1,min(fN1(pq),fN2(pq))ifpq∈E1∩E2.

Theorem 3.10. Let ζ1 = (M1, N1) and ζ2 = (M2, N2) be two VGs and V1 ∩ V2 =∅. Then, ζ = (M, N) is the union of ζ1 and ζ2 if each (λ, δ)-LG ζ(λ,δ) is the union of (ζ1)(λ,δ) and (ζ2)(λ,δ).Proof. Let ζ = (M, N) be the union of VGs ζ1 and ζ2. We show that M(λ,δ)=(M1)(λ,δ)∪(M2)(λ,δ), for each (λ, δ) ∈ [0, 1] × [0, 1]. Suppose p ∈ M(λ,δ). Then, p ∈ V1 − V2 or p ∈ V2 − V1. If p ∈ V1 − V2, then tM1(p)=tM(p)≥δ and fM1(p)=fM(p)≤λ, which shows p∈(M1)(λ,δ). Similarly, p ∈ V2 − V1 shows p∈(M2)(λ,δ). Hence, p∈(M1)(λ,δ)∪(M2)(λ,δ). Therefore, M(λ,δ)⊆(M1)(λ,δ)∪(M2)(λ,δ).Now, let p∈(M1)(λ,δ)∪(M2)(λ,δ). Then, p∈(M1)(λ,δ), p∉(M2)(λ,δ), or p∈(M2)(λ,δ), p∉(M1)(λ,δ). For the first case, we get tM1(p)=tM(p)≥δ and fM1(p)=fM(p)≤λ, which shows p ∈ M(λ,δ). For the second case, we get tM2(p)=tM(p)≥δ and fM2(p)=fM(p)≤λ. Hence, p ∈ M(λ,δ). Thus, (M1)(λ,δ)∪(M2)(λ,δ)⊆M(λ,δ).To prove N(λ,δ)=(N1)(λ,δ)∪(N2)(λ,δ), for all (λ, δ) ∈ [0, 1] × [0, 1], suppose pq ∈ N(λ,δ). Then, pq ∈ E1 − E2 or pq ∈ E2 − E1. For pq ∈ E1 − E2, we get tN1(pq)=tN(pq)≥δ and fN1(pq)=fN(pq)≤λ. Hence, pq∈(N1)(λ,δ). Similarly, pq ∈ E2 − E1 gives pq∈(N2)(λ,δ). So, N(λ,δ)⊆(N1)(λ,δ)∪(N2)(λ,δ). If pq∈(N1)(λ,δ)∪(N2)(λ,δ), then pq∈(N1)(λ,δ)−(N2)(λ,δ) or pq∈(N2)(λ,δ)−(N1)(λ,δ). For the first case, tN1(pq)=tN(pq)≥δ and fN1(pq)=fN(pq)≤λ. Therefore, pq ∈ N(λ,δ). In the second case, we get pq ∈ N(λ,δ). Thus, (N1)(λ,δ)∪(N2)(λ,δ)⊆N(λ,δ). The converse part is clear.Definition 3.11. Let ζ1 = (M1, N1) and ζ2 = (M2, N2) be two VGs. The join ζ1 + ζ2 is the pair (A, B) of VSs defined on ζ*1+ζ*2 so that

(i)tM(p)={tM1(p)ifp∈V1andp∉V2,tM2(p)ifp∈V2andp∉V1,max(tM1(p),tM2(p))ifp∈V1∩V2,(ii)fM(p)={fM1(p)ifp∈V1andp∉V2,fM2(p)ifp∈V2andp∉V1,min(fM1(p),fM2(p))ifp∈V1∩V2,(iii)tN(pq)={tN1(pq)ifpq∈E1andpq∉E2,tN2(pq)ifpq∈E2andpq∉E1,max(tN1(pq),tN2(pq))ifpq∈E1∩E2,min(tM1(p),tM2(q))ifpq∈E′,(iv)fN(pq)={fN1(pq)ifpq∈E1andpq∉E2,fN2(pq)ifpq∈E2andpq∉E1,min(fN1(pq),fN2(pq))ifpq∈E1∩E2,max(fN1(p),fN2(q))ifpq∈E′.

Theorem 3.12. Suppose ζ1 = (M1, N1) and ζ2 = (M2, N2) are two VGs and V1 ∩ V2 =∅. Then, ζ = (M, N) is the join of ζ1 and ζ2 if each (λ, δ)-LG ζ(λ,δ) is the of (ζ1)(λ,δ) and (ζ2)(λ,δ).Proof. Let ζ = (M, N) be the join of VGs ζ1 and ζ2. Then, by the definition and the proof of Theorem 3.10, M(λ,δ)=(M1)(λ,δ)∪(M2)(λ,δ), for all (λ, δ) ∈ [0, 1] × [0, 1]. We prove that N(λ,δ)=(N1)(λ,δ)∪(N2)(λ,δ)∪E′(λ,δ), for all (λ, δ) ∈ [0, 1] × [0, 1], where E′(λ,δ) is the set of all edges joining the nodes of (M1)(λ,δ) and (M2)(λ,δ).From the proof of Theorem 3.10, it follows that (N1)(λ,δ)∪(N2)(λ,δ)⊆N(λ,δ). If pq∈E′(λ,δ), then tM1(p)≥δ, fM1(p)≤λ, tM2(q)≥δ, and fM2(q)≤λ. So,

tN(pq)=min(tM1(p),tM2(q))≥δ

and

fN(pq)=max(fM1(p),fM2(q))≤λ.

It follows that pq ∈ N(λ,δ). Thus, (N1)(λ,δ)∪(N2)(λ,δ)∪E′(λ,δ)⊆N(λ,δ). For each pq ∈ N(λ,δ), if pq ∈ E1 ∪ E2, then pq∈(N1)(λ,δ)∪(N2)(λ,δ), by the proof of Theorem 3.10. If p ∈ V1 and q ∈ V2, then

min(tM1(p),tM2(q))=tN(pq)≥δ.

Moreover,

max(fM1(p),fM2(q))=fN(pq)≤λ.

So, p∈(M1)(λ,δ) and q∈(M2)(λ,δ). Thus, pq∈E′(λ,δ). Hence, N(λ,δ)⊆(N1)(λ,δ)∪(N2)(λ,δ)∪E′(λ,δ). Conversely, suppose every LG ζ(λ,δ) is the join of (ζ1)(λ,δ) and ((M2)(λ,δ),(N2)(λ,δ)). From the proof of Theorem 3.10, we have

(i){tM(p)=tM1(p)ifp∈V1tM(p)=tM2(p)ifp∈V2(ii){fM(p)=fM1(p)ifp∈V1fM(p)=fM2(p)ifp∈V2(iii){tN(pq)=tN1(pq)ifpq∈E1tN(pq)=tN2(pq)ifpq∈E2(iv){fN(pq)=fN1(pq)ifpq∈E1fN(pq)=fN2(pq)ifpq∈E2.

Assume p ∈ V1, q ∈ V2, min(tM1(p),tM2(q))=r, max(fM1(p),fM2(q))=s, tN (pq) = t, and fN (pq) = w. Then, p∈(M1)(λ,δ), q∈(M2)(λ,δ), and pq ∈ N(w,t). It shows pq ∈ N(λ,δ), p∈(M1)(w,t), and q∈(M2)(w,t). Hence, tN (pq) ≥ r, fN (pq) ≤ λ, tM1(p)≥t, fM1(p)≤w, tM2(q)≥t, and fM2(q)≥w. Thus,

tN(pq)≥δ=min(tM1(p),tM2(q))≥t=tN(pq),fN(pq)≥λ=max(fM1(p),fM2(q))≤w=fN(pq).

So, tN(pq)=min(tM1(p),tM2(q)), andfw(pq)=max(fM1(p),fM2(q)), as described.Definition 3.13. Let ζ1 = (M1, N1) and ζ2 = (M2, N2) be two VGs. The cross product ζ1∗ζ2 is the pair (M, N) of VSs defined on the cross product ζ∗1∗ζ∗2 so that

(i){tA(p1,p2)=min(tA1(p1),tA2(p2)),fA(p1,p2)=max(fA1(p1),fA2(p2)),∀(p1,p2)∈V1×V2,(ii){tN((p1,p2)(q1,q2))=min(tN1(p1q1),tN2(p2q2)),fN((p1,p2)(q1,q2))=max(fN1(p1q1),fN2(p2q2)),∀p1q1∈E1,and∀p2q2∈E2.

Theorem 3.14. Suppose ζ1 = (M1, N1) and ζ2 = (M2, N2) are two VGs. Then, ζ = (M, N) is the cross product of ζ1 and ζ2 if each LG ζ(λ,δ) is the cross product of (ζ1)(λ,δ) and (ζ2)(λ,δ).Proof. Let ζ = (M, N) be the cross product of ζ1 and ζ2. Then, by the definition of the CP and the proof of Theorem 3.6, we have M(λ,δ)=(M1)(λ,δ)×(M2)(λ,δ) and ∀(λ, δ) ∈ [0, 1] × [0, 1]. We prove that

N(λ,δ)={(p1,p2)(q1,q2)|p1q1∈(N1)(λ,δ),p2q2∈(N2)(λ,δ)},

∀(λ, δ) ∈ [0, 1] × [0, 1]. If (p1, p2) (q1, q2) ∈ N(λ,δ), then

tN((p1,p2)(q1,q2))=min(tN1(p1q1),tN2(p2q2))≥δ,fN((p1,p2)(q1,q2))=max(fN1(p1q1),fN2(p2q2))≤λ.

Hence, tN1(p1q1)≥δ, tN2(p2q2)≥δ, fN1(p1q1)≤λ, and fN2(p2q2)≤λ. Thus, p1q1∈(N1)(λ,δ) and p2q2∈(N2)(λ,δ). Now, if p1q1∈(N1)(λ,δ) and p2q2∈(N2)(λ,δ), then tN1(p1q1)≥δ, fN1(p1q1)≤λ, tN2(p2q2)≥δ, and fN2(p2q2)≤λ. So, we have

tN((p1,p2)(q1,q2))=min(tN1(p1q1),tN2(p2q2))≥δ,fN((p1,p2)(q1,q2))=max(fN1(p1q1),fN2(p2q2))≤λ

because ζ = (M, N) is the cross product of ζ1∗ζ2. Therefore, (p1, p2) (q1, q2) ∈ N(λ,δ). The converse part is clear. Definition 3.15. Let ζ1 = (M1, N1) and ζ2 = (M2, N2) be two VGs. The lexicographic product (LP) ζ1•ζ2 is the pair (M, N) of VSs defined on the LP G*1•G*2 so that

(i){tM(p1,p2)=min(tM1(p1),tM2(p2)),fM(p1,p2)=max(fM1(p1),fM2(p2)),∀(p1,p2)∈V1×V2,(ii){tN((p,p2)(p,q2))=min(tM1(p),tN2(p2q2)),fN((p,p2)(p,q2))=max(fM1(p),fN2(p2q2)),∀p∈V1,and∀p2q2∈E2,(iii){tN((p1,p2)(q1,q2))=min(tN1(p1q1),tN2(p2q2)),fN((p1,p2)(q1,q2))=max(fN1(p1q1),fN2(p2q2)),∀p1q1∈E1,and∀p2q2∈E2.

Theorem 3.16. Suppose ζ1 = (M1, N1) and ζ2 = (M2, N2) are two VGs. Then, ζ = (M, N) is LP of ζ1 and ζ2 if ζ(λ,δ)=(ζ1)(λ,δ)•(ζ2)(λ,δ), ∀(λ, δ) ∈ [0, 1] × [0, 1] and λ + δ ≤ 1.Proof. Let ζ = (M, N) = G1•G2. According to the definition of CP ζ1 × ζ2 and the proof of Theorem 3.6, we get M(λ,δ)=(M1)(λ,δ)×(M2)(λ,δ) and ∀(λ, δ) ∈ [0, 1] × [0, 1]. We prove that N(λ,δ)=E(λ,δ)∪E′(λ,δ), ∀(λ, δ) ∈ [0, 1] × [0, 1], where E(λ,δ)={(p,p2)(p,q2)|p∈V1,p2q2∈(N2)(λ,δ)} is the subset of the edge set of the direct product (DP) ζ(λ,δ)=(ζ1)(λ,δ)×(ζ2)(λ,δ), and E′(λ,δ)={(p1,p2)(q1,q2)|p1q1∈(N1)(λ,δ),p2q2∈(N2)(λ,δ)} is the edge set of the cross product (ζ1)(λ,δ)∗(ζ2)(λ,δ). For each (p1, p2) (q1, q2) ∈ N(λ,δ), p1 = q1, p2q2 ∈ E2, or p1q1 ∈ E1, p2q2 ∈ E2. If p1 = q1 and p2q2 ∈ E2, then (p1, p2) (q1, q2) ∈ E(λ,δ), by the definition of the CP and the proof of Theorem 3.6. If p1q1 ∈ E1 and p2q2 ∈ E2, then (p1,p2)(q1,q2)∈E′(λ,δ), by the definition of cross product and the proof of Theorem 3.14. Hence, N(λ,δ)⊆E(λ,δ)∪E′(λ,δ). From the definition of CP and the proof of Theorem 3.6, we get E(λ,δ) ⊆ N(λ,δ). In addition, from definition of cross product and proof of Theorem 3.14, we get E′(λ,δ)⊆N(λ,δ). Thus, E(λ,δ)∪E′(λ,δ)⊆N(λ,δ).Conversely, assume ζ(λ,δ)=(M(λ,δ),N(λ,δ))=(ζ1)(λ,δ)•(ζ2)(λ,δ)) and ∀(λ, δ) ∈ [0, 1] × [0, 1]. It is clear that (ζ1)(λ,δ)•(ζ2)(λ,δ)) has the same vertex set as the CP (ζ1)(λ,δ)×(ζ2)(λ,δ)). Now, by the proof of Theorem 3.6, we get

tM((p1,p2))=min(tM1(p1),tM2(p2)),fM((p1,p2))=max(fM1(p1),fM2(p2)),

∀(p1, p2) ∈ V1 × V2. For p ∈ V1 and p2q2 ∈ E2, let min(tM1(p),tN2(p2q2))=δ, max(fM1(p),fN2(p2q2))=λ, tN ((p, p2) (p, q2)) = δ1, and fN ((p, p2) (p, q2)) = λ1. Then, according to the definitions of CP and LP, we have

(p,p2)(p,q2)∈(N1)(λ,δ)•(N2)(λ,δ)⇔(p,p2)(p,q2)∈(N1)(λ,δ)×(N2)(λ,δ).

By the same reasoning as proof of Theorem 3.6, we get

tN((p,p2)(p,q2))=min(tM(p),tN2(p2q2)),fN((p,p2)(p,q2))=max(fM(p),fN2(p2q2)).

Now, assume that tN ((p1, p2) (q1, q2)) = δ1, fN ((p1, p2) (q1, q2)) = λ1, min(tN1(p1q1),tN2(p2q2))=δ, and max(fN1(p1q1),fN2(p2q2))=λ, for p1q1 ∈ E1 and p2q2 ∈ E2. Then, according to the definitions of the cross product and LP, we derive

(p1,p2)(q1,q2)∈(N1)(λ,δ)•(N2)(λ,δ)⇔(p1,p2)(q1,q2)∈(N1)(λ,δ)∗(N2)(λ,δ).

Similar to the proof of Theorem 3.14, we have

tN((p1,p2)(q1,q2))=min(tN1(p1q1),tN2(p2q2)),fN((p1,p2)(q1,q2))=max(fN1(p1q1),fN2(p2q2)),

which completes the proof.

Lemma 3.17. Let ζ1 = (M1, N1) and ζ2 = (M2, N2) be two VGs so that V1 = V2, M1 = M2, and E1 ∩ E2 =∅. Then, ζ = (M, N) is the union of ζ1 and ζ2 if ζ(λ,δ) is the union of (ζ1)(λ,δ) and (ζ2)(λ,δ), ∀(λ, δ) ∈ [0, 1] × [0, 1].Proof. Assume ζ = (M, N) is the union of VGs ζ1 and ζ2. Then, according to the definition of union and as V1 = V2 and M1 = M2, we get M = M1 = M2. Then, M(λ,δ)=(M1)(λ,δ)∪(M2)(λ,δ). Now, we prove that N(λ,δ)=(N1)(λ,δ)∪(N2)(λ,δ), for all (λ, δ) ∈ [0, 1] × [0, 1]. For each pq∈(N1)(λ,δ), we get tN(pq)=tN1(pq)≥δ and fN(pq)=fN1(pq)≤λ. So, pq ∈ N(λ,δ). Thus, (N1)(λ,δ)⊆N(λ,δ). In the same way, we get (N2)(λ,δ)⊆N(λ,δ). Then, ((N1)(λ,δ)∪(N2)(λ,δ))⊆N(λ,δ). For each pq ∈ N(λ,δ), pq ∈ E1, or pq ∈ E2. If pq ∈ E1, then fN1(pq)=fN(pq)≤λ. Thus, pq∈(N1)(λ,δ). If pq ∈ E2, then we get pq∈(N2)(λ,δ). Therefore, N(λ,δ)⊆(N1)(λ,δ)∪(N2)(λ,δ).Conversely, assume (λ, δ)-LG ζ(λ,δ) is the union of (ζ1)(λ,δ) and ((M2)(λ,δ),(N2)(λ,δ)). Let tM(p) = δ, fM(p) = λ, tM1(p)=δ1, and fM1(p)=λ1, for some p ∈ V1 = V2. Then, p ∈ M(λ,δ) and p∈(M1)(λ,δ). So, p∈(M1)(λ,δ) and p ∈ M(λ,δ) because M(λ,δ)=(M1)(λ,δ) and (M1)(λ,δ)=M(λ,δ). Thus, tM1(p)≥r, fM1(p)≤α, tM(p) ≥ t, and fM(p) ≤ w. Hence, tM1(p)≥tM(p), fM1(p)≤fM(p), tM(p)≥tM1(p), and fM(p)≤fM1(p). Therefore, tM(p)=tM1(p) and fM(p)=fM1(p) because M1 = M2, V1 = V2, and M = M1 = M1 ∪ M2. In the same way, we derive

(i){tN(pq)=tN1(pq)ifpq∈E1tN(pq)=tN2(pq)ifpq∈E2(ii){fN(pq)=fN1(pq)ifpq∈E1fN(pq)=fN2(pq)ifpq∈E2.

Definition 3.18. Assume ζ1 = (M1, N1) and ζ2 = (M2, N2) are two vague pair of graphs ζ*1 and ζ*2, respectively. The strong product (SP) ζ1 ⊠ ζ2 is the pair (M, N) of VSs defined on the SP ζ*1⊠ζ*2 so that

(i){tM(p1,p2)=min(tM1(p1),tM2(p2)),fM(p1,p2)=max(fM1(p1),fM2(p2)),∀(p1,p2)∈V1×V2,(ii){tN((p,p2)(p,q2))=min(tM1(p),tN2(p2q2)),fN((p,p2)(p,q2))=max(fM1(p),fN2(p2q2)),∀p∈V1and∀p2q2∈E2,(iii){tN((p1,r)(q1,r))=min(tN1(p1q1),tM2(r)),fN((p1,r)(q1,r))=max(fN1(p1q1),fM2(r)),∀r∈V2and∀p1q1∈E1,(iv){tN((p1,p2)(q1,q2))=min(tN1(p1q1),tN2(p2q2)),fN((p1,p2)(q1,q2))=max(fN1(p1q1),fN2(p2q2)),∀p1q1∈E1and∀p2q2∈E2.

Theorem 3.19. Let ζ1 = (M1, N1) and ζ2 = (M2, N2) be two VGs. Then, ζ = (M, N) is the SP of ζ1 and ζ2 if ζ(λ,δ), where (λ, δ) ∈ [0, 1] × [0, 1] and λ + δ ≤ 1 is the SP of (ζ1)(λ,δ) and (ζ2)(λ,δ).Proof. By definitions of SP, cross product, and CP, we get ζ1 ⊠ ζ2 = (ζ1 × ζ2) ∪ (ζ1∗ζ2) and (ζ1)(λ,δ)⊠(ζ2)(λ,δ)=((ζ1)(λ,δ)×(ζ2)(λ,δ))∪((ζ1)(λ,δ)∗(ζ2)(λ,δ)), and ∀(λ, δ) ∈ [0, 1] × [0, 1]. By Theorems 3.14 and 3.6, and Lemma 3.17, we have

ζ=ζ1⊠ζ2⇔ζ=(ζ1×ζ2)∪(ζ1∗ζ2) ⇔ζ(λ,δ)=(ζ1×ζ2)(λ,δ)∪(ζ1∗ζ2)(λ,δ) ⇔ζ(λ,δ)=((ζ1)(λ,δ)×(ζ2)(λ,δ))∪((ζ1)(λ,δ)∗(ζ2)(λ,δ)) ⇔ζ(λ,δ)=(ζ1)(λ,δ)⊠(ζ2)(λ,δ),∀(λ,δ)∈[0,1]×[0,1].

4 Application of vague graph in medical sciences

In this section, we introduce a distance measure on a VS and use it to diagnose a disease for a group of people who suffer from certain symptoms.

Definition 4.1. Suppose that Z = {q1, q2, … , qn} is the universe of discourse. Let M = {(qi, tM(qi), fM(qi): qi ∈ Z} and N = {(qi, tN (qi), fN (qi): qi ∈ Z} be two VSs. The new distance measure is defined as

D(M,N)=2n∑ni=1sin{π6|tM(qi)−tN(qi)|}+sin{π6|fM(qi)−fN(qi)|}1+sin{π6|tM(qi)−tN(qi)|}+sin{π6|fM(qi)−fN(qi)|}.

Clearly, D (M, N) has all four conditions of a distance measure.

Assume {E1, E2, … , En} is a set of diseases and {T1, T2, … , Tn} is a set of n number of patients. Suppose that {R1(tEi1,fEj1),R2(tEi2,fEj2),…,Rl(tEil,fEjl)} is the symptoms of the diseases Ei, and {R1(tTj1,fTj1),R2(tTj2,fTj2),…,Rl(tTjl,fTjl)} is the symptoms of patient Tj given in VSs. So, we have

d(Ei,Tj)=2l∑lh=1sin{π6|tEih−tTjh}+sin{π6|fEih−fTjh}1+sin{π6|tEih−tTjh}+sin{π6|fEih−fTjh},

where i = 1, 2, … , m and j = 1, 2, … , n. The distance between each pair of diseases and patients can be expressed as the following matrix:

T1T2⋯TnE1E2⋮Em(d(E1,T1)d(E1,T2)⋯d(E1,Tn)d(E2,T1)d(E2,T2)⋯d(E2,Tn)⋮d(Em,T1)d(Em,T2)⋯d(Em,Tn))

Note that if the distance between the two VSs is less, their similarity will be greater. This is true for a patient and the type of illness they have.

Consider a set of symptoms R, a set of diagnoses E, and a set of patients T. Assume that T = {Safari, Najafi, Ahmadi, Rahmani}, R = {Jaundice, Nausea, Heart Burn, Constipation, Chronic Diarrhea}, and E = {Cholecystitis, Migraine, Dyspepsia, Diverticulitis, Inflammatory bowel disease}. We intend to make the right diagnosis for each patient. Tables 1 and 2 show the relation between symptoms and diseases, as well as patients and symptoms, respectively.

TABLE 1
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TABLE 1. Symptoms–diseases VR.

TABLE 2
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TABLE 2. Patient–symptoms VR.

Now, we show the patients and symptoms as VSs as follows:

CH={⟨JA,(0.7,0.2)⟩,⟨NA,(0.1,0.4)⟩,⟨HB,(0.2,0.3)⟩,⟨CO,(0.6,0.3)⟩,⟨CD,(0.2,0.3)⟩}MI={⟨JA,(0.2,0.2)⟩,⟨NA,(0.7,0.3)⟩,⟨HB,(0.3,0.4)⟩,⟨CO,(0.2,0.4)⟩,⟨CD,(0.3,0.5)⟩}DY={⟨JA,(0.2,0.5)⟩,⟨NA,(0.2,0.4)⟩,⟨HB,(0.7,0.1)⟩,⟨CO,(0.3,0.4)⟩,⟨CD,(0.2,0.6)⟩}DI={⟨JA,(0.6,0.2)⟩,⟨NA,(0.3,0.5)⟩,⟨HB,(0.3,0.5)⟩,⟨CO,(0.7,0.2)⟩,⟨CD,(0.4,0.5)⟩}IBD={⟨JA,(0.3,0.5)⟩,⟨NA,(0.3,0.2)⟩,⟨HB,(0.5,0.4)⟩,⟨CO,(0.2,0.6)⟩,⟨CD,(0.7,0.2)⟩}.
Safari={〈JA,(0.3,0.6)〉,〈NA,(0.7,0.2)〉,〈HB,(0.4,0.5)〉,〈CO,(0.3,0.2)〉,〈CD,(0.2,0.4)〉}Najafi={〈JA,(0.3,0.4)〉,〈NA,(0.2,0.5)〉,〈HB,(0.4,0.4)〉,〈CO,(0.3,0.5)〉,〈CD,(0.7,0.1)〉}Ahmadi={〈JA,(0.8,0.1)〉,〈NA,(0.4,0.3)〉,〈HB,(0.5,0.2)〉,〈CO,(0.6,0.3)〉,〈CD,(0.3,0.4)〉}Rahmani={〈JA,(0.2,0.3)〉,〈NA,(0.3,0.5)〉,〈HB,(0.8,0.2)〉,〈CO,(0.3,0.4)〉,〈CD,(0.3,0.5)〉}.

Here, we calculate the vague distance between the disease and the patients based on their symptoms.

d(CH,Safari)=25{sinπ6|0.7−0.3|+sinπ6|0.2−0.6|1+sinπ6|0.7−0.3|+sinπ6|0.2−0.6| +sinπ6|0.1−0.7|+sinπ6|0.4−0.2|1+sinπ6|0.1−0.7|+sinπ6|0.4−0.2| +sinπ6|0.2−0.4|+sinπ6|0.3−0.5|1+sinπ6|0.2−0.4|+sinπ6|0.3−0.5| +sinπ6|0.6−0.3|+sinπ6|0.3−0.2|1+sinπ6|0.6−0.3|+sinπ6|0.3−0.2| +sinπ6|0.2−0.2|+sinπ6|0.3−0.4|1+sinπ6|0.2−0.2|+sinπ6|0.3−0.4|} =25{0.2875+0.2857+0.1666+0.1666+0.0476}=0.3808.
d(CH,Najafi)=25{sinπ6|0.7−0.3|+sinπ6|0.2−0.4|1+sinπ6|0.7−0.3|+sinπ6|0.2−0.4| +sinπ6|0.1−0.2|+sinπ6|0.4−0.5|1+sinπ6|0.1−0.2|+sinπ6|0.4−0.5| +sinπ6|0.2−0.4|+sinπ6|0.3−0.4|1+sinπ6|0.2−0.4|+sinπ6|0.3−0.4| +sinπ6|0.6−0.3|+sinπ6|0.3−0.5|1+sinπ6|0.6−0.3|+sinπ6|0.3−0.5| +sinπ6|0.2−0.7|+sinπ6|0.3−0.1|1+sinπ6|0.2−0.7|+sinπ6|0.3−0.1|} =25{0.2307+0.0909+0.1304+0.2+0.2592}=0.3644.
d(CH,Ahmadi)=25{sinπ6|0.7−0.8|+sinπ6|0.2−0.1|1+sinπ6|0.7−0.8|+sinπ6|0.2−0.1| +sinπ6|0.1−0.4|+sinπ6|0.4−0.3|1+sinπ6|0.1−0.4|+sinπ6|0.4−0.3| +sinπ6|0.2−0.5|+sinπ6|0.3−0.2|1+sinπ6|0.2−0.5|+sinπ6|0.3−0.2| +sinπ6|0.6−0.6|+sinπ6|0.3−0.3|1+sinπ6|0.6−0.6|+sinπ6|0.3−0.3| +sinπ6|0.2−0.3|+sinπ6|0.3−0.4|1+sinπ6|0.2−0.3|+sinπ6|0.3−0.4|} =25{0.0909+0.1666+0.1666+0.0909}=0.206.
d(CH,Rahmani)=25{sinπ6|0.7−0.2|+sinπ6|0.2−0.3|1+sinπ6|0.7−0.2|+sinπ6|0.2−0.3| +sinπ6|0.1−0.3|+sinπ6|0.4−0.5|1+sinπ6|0.1−0.3|+sinπ6|0.4−0.5| +sinπ6|0.2−0.8|+sinπ6|0.3−0.2|1+sinπ6|0.2−0.8|+sinπ6|0.3−0.2| +sinπ6|0.6−0.3|+sinπ6|0.3−0.4|1+sinπ6|0.6−0.3|+sinπ6|0.3−0.4| +sinπ6|0.2−0.3|+sinπ6|0.3−0.5|1+sinπ6|0.2−0.3|+sinπ6|0.3−0.5|} =25{0.2307+0.1304+0.2592+0.1666+0.1304}=0.3669.

In the same way, we have

d(MI,Safari)=0.2239,d(MI,Najafi)=0.3255d(MI,Ahmadi)=0.3215,d(MI,Rahmani)=0.2340,d(DY,Safari)=0.3164,d(DY,Najafi)=0.300,d(DY,Ahmadi)=0.3564,d(DY,Rahmani)=0.1454,d(DI,Safari)=0.3452,d(DI,Najafi)=0.3427,d(DI,Ahmadi)=0.2570,d(DI,Rahmani)=0.3056,d(IBD,Safari)=0.3057,d(IBD,Najafi)=0.1601,d(IBD,Ahmadi)=0.3928,d(IBD,Rahmani)=0.3401.

The distance matrix for the aforementioned values is as follows:

SafariNajafiAhmadiRahmaniCholecystitisMigraineDyspepsiaDiverticulitisInflammatorybowldisease(0.38080.36440.2060.36690.22390.32550.32150.32400.31640.3000.35640.14540.34520.34270.25700.30560.30570.16010.39280.3401)

As the distance between the patient and the mentioned diseases decreases, the probability of the patient suffering from that disease increases, so we conclude that Safari suffers from migraine, Najafi suffers from inflammatory bowel disease, Ahmadi suffers from cholecystitis, and Rahmani suffers from dyspepsia.

5 Conclusion

VGs are important in other sciences, including psychology, life sciences, medicine, and social studies, and can help researchers with optimization and save time and money. Likewise, VGs, belonging to the FG family, have good abilities because they face problems that cannot be explained by FGs. Hence, in this study, we introduced the notion of VEG and presented some of its properties. Moreover, we characterized VG ζ = (M, N), where M is a VS and N is a VR. Some operations have been defined, such as CP, cross product, LP, and SP on VGs. Finally, an application of VG in medical sciences has been presented. In our future work, we will introduce some connectivity indices, such as the Wiener index, harmonic index, Zagreb index, and Randic index in VGs, and investigate some of their properties.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material. Further inquiries can be directed to the corresponding author.

Author contributions

All authors have made a substantial, direct, and intellectual contribution to the work and approved it for submission.

Funding

This work was supported by the National Key R and D Program of China (Grant 2019YFA0706 338402) and the National Natural Science Foundation of China under grants 62172302, 62072129, and 61876047.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: vague set, vague edge graph, (λ, δ)-level graph, lexicographic product, cross product, strong product

Citation: Shi X, Jiang W, Khan A and Akhoundi M (2023) New concepts on level graphs of vague graphs with application in medicine. Front. Phys. 11:1130765. doi: 10.3389/fphy.2023.1130765

Received: 23 December 2022; Accepted: 18 January 2023;
Published: 21 February 2023.

Edited by:

Yilun Shang, Northumbria University, United Kingdom

Reviewed by:

Sovan Samanta, Tamralipta Mahavidyalaya, India
Madhumangal Pal, Vidyasagar University, India

Copyright © 2023 Shi, Jiang, Khan and Akhoundi. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Wubian Jiang, jwb65659@163.com

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.