Abstract
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) , 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) [], 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 [] 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 [] introduced FGs using Zadeh’s fuzzy relation (FR) [, ]. However, Rosenfeld [] presented another detailed definition, such as paths, cycles, and connectedness. Mordeson and Chang-Shyh [] defined operations on FGs. References [, ] introduced certain types of product bipolar FGs and some operations and densities of m-polar FGs. Das et al. [] presented generalized neutrosophic competition graphs. Bhattacharya [] identified some remarks on FGs. Mordeson and Nair [] studied several concepts of FGs. Mahapatra [] introduced radio FGs and frequency assignment in radio stations. References [–] investigated new definitions of vague graphs, and references [–] defined several concepts on VGs and neutrosophic competition graphs. Shoaib et al. [] 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 [] recommended the VG notion and evaluated some of its features. Borzooei and Rashmanlou [, ] introduced new concepts in VGs. Sunitha and Vijayakumar [] presented a complement of FGs. Kosari et al. and Kou et al. [, ] studied new results in VG structures. References [–] defined dominating and equitable dominating sets in VGs. Shi and Kosari [] investigated the global dominating set in product-VGs. Shao et al. [] 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 is the set of edges of ζ*. Two nodes p and q in a graph are said to be neighbors in , if {p, q} is in an edge of .
Definition 2.1.A fuzzy graph (FG) is a pairς = (τ, ν) with a setX; thenτis a fuzzy set (FS) inX,andνis a fuzzy relation (FR) inX × X, so thatfor allpq ∈ X × X.
Definition 2.2.A VS is a pair (tM, fM) on setX, wheretMandfMare real-valued functions, which can be presented onV → [0, 1] so thattM(p) + fM(p) ≤ 1 and∀p ∈ X.
Definition 2.3.A VG is defined as a pairζ = (M, N) ,whereM = (tM, fM) is a VS onVandN = (tN, fN) is a VS onE ⊆ V × Vso that for eachpq ∈ E,tN (pq) ≤ tM(p) ∧ tM(q) andfN (pq) ≥ fN(p) ∨ fN(q).
Definition 2.4.A VEG on a non-empty setVis an ordered pair of the form, whereVis a crisp vertex set (CVS) andNis a VR onVso thattN (pq) ≤ min{tM(p), tM(q)},fN (pq) ≥ max{fM(p), fM(q)}, and 0 ≤ tN (pq) + fN (pq) ≤ 1, for allpq ∈ E.
We consider VEGs with CVS, that is, VGs , that is, tM(p) = 1, fM(p) = 0, ∀p ∈ V, and edges with true membership and false membership degrees in [0,1].
Example 2.5Consider a simple graph (SG)ζ* = (V, E) so thatV = {p, q, s} andE = {pq, qs, ps}. LetNbe a VR onVdescribed byN = {(pq, 0.4, 0.2), (qs, 0.5, 0.2), (ps, 0.3, 0.2)}. Clearly,is a VEG with CVS and VS of edges (see Figure 1).
FIGURE 1
3 Vague graphs by level graphs
Definition 3.1.Suppose thatM = (tm, fM) is a VS onV. Then, the setM(λ,δ) = {p ∈ V|tM(p) ≥ λ, fM(p) ≤ δ}, where (λ, δ) ∈ [0, 1] × [0, 1] andλ + δ ≤ 1 is named the (λ, δ)-level set ofM. LetN = (tN, fN) be a VR onV. Then, the setN(λ,δ) = {pq ∈ V × V|tN (pq) ≥ λ, fM(pq) ≤ δ}, where (λ, δ) ∈ [0, 1] × [0, 1] andλ + δ ≤ 1 is called (λ, δ)-LG. In the case ofλ = δ, whereλ ≤ 1, we write LG byζαinstead ofζ(λ,δ). Note that
Remark 3.2.The level graphζ(λ,δ) = (M(λ,δ), N(λ,δ)) is a subgraph ofζ* = (V, E).
Example 3.3Consider an SGζ* = (V, E) so thatV = {p, q, r, s} andE = {pq, qr, rs, ps, pr, qs}. From Figure 2, we get thatζ = (M, N) is a VG.Takeλ = 0.5. We haveM0.5 = {s, r} andN0.5 = {rs}. Obviously, the 0.5-LGζ0.5is a subgraph ofζ*.Now, we takeλ = 0.2 andδ = 0.3. By Definition 3.1, we haveM(0.2,0.3) = {p, r, s} andN(0.2,0.3) = {ps}. Clearly, (0.2,0.3)-LGζ(0.2,0.3)is a subgraph ofζ*.
FIGURE 2
ζ = (M, N) is a VG ifζ(λ,δ)is a crisp graph for each pair (λ, δ) ∈ [0, 1] × [0, 1] andλ + 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
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 , 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,
that is, ζ = (M, N) is a VG.
Definition 3.5.Supposeζ1 = (M1, N1) andζ2 = (M2, N2) are two VGs ofand, respectively. The Cartesian product (CP)ζ1 × ζ2is the pair (M, N) of VSs defined on the CPso that
ζ = (M, N) is the CP ofζ1andζ2if and only if each pair (λ, δ) ∈ [0, 1] × [0, 1] andλ + δ ≤ 1, (λ, δ)-LGζ(λ,δ)is the CP ofand.
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), thenandHence, and ; that is . Therefore, .
Now if , then and . It follows and . 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, . Thus, . Now, we prove N(λ,δ) = E, where E is the edge set of the CP of and ∀(λ, δ) ∈ [0, 1] × [0, 1]. Suppose (p1, p2) (q1, q2) ∈ N(λ,δ). Then, and . 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
So, , , , and . It follows that and ; 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, , , , and . Since (M, N) is the CP of ζ1 and ζ2, we haveTherefore, (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.Letζ1andζ2be two VGs ofand, respectively. The composition (Co)ζ1 [ζ2] is the pair (M, N) of VSs defined on the Coso that
ζ = (M, N) is the Co of VGsζ1andζ2if, for every (λ, δ) ∈ [0, 1] × [0, 1] andλ + δ ≤ 1, (λ, δ)-LGζ(λ,δ)is the Co ofand.
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 . Now, we prove N(λ,δ) = E, where E is the edge set of the co , 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
So, , , , , , and . It follows that and ; that is, (p1, p2) (q1, q2) ∈ E. Thus, N(λ,δ) ⊆ E. For each (p1, p2) (q1, q2) ∈ E, , , , and . Since ζ = (M, N) is the Co of ζ1 [ζ2], we get
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, , , , , , and . Since ζ = (M, N) is the Co of , we have
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 and . In the same way, by the same arguments as in the proof of Theorem 3.6, we get∀p2, q2 ∈ V2 (p2 ≠ q2) and ∀ p1q1 ∈ E1.
This completes the proof.
Definition 3.9.Letζ1 = (M1, N1) andζ2 = (M2, N2) be two VGs. The unionζ1 ∪ ζ2is defined as the pair (M, N) of VSs described on the union of graphsandso that
Letζ1 = (M1, N1) andζ2 = (M2, N2) be two VGs andV1 ∩ V2 =∅. Then,ζ = (M, N) is the union ofζ1andζ2if each (λ, δ)-LGζ(λ,δ)is the union ofand.
Proof. Let ζ = (M, N) be the union of VGs ζ1 and ζ2. We show that , 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 and , which shows . Similarly, p ∈ V2 − V1 shows . Hence, . Therefore, .
Now, let . Then, , , or , . For the first case, we get and , which shows p ∈ M(λ,δ). For the second case, we get and . Hence, p ∈ M(λ,δ). Thus, .
To prove , 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 and . Hence, . Similarly, pq ∈ E2 − E1 gives . So, . If , then or . For the first case, and . Therefore, pq ∈ N(λ,δ). In the second case, we get pq ∈ N(λ,δ). Thus, . The converse part is clear.
Definition 3.11.Letζ1 = (M1, N1) andζ2 = (M2, N2) be two VGs. The joinζ1 + ζ2is the pair (A, B) of VSs defined onso that
Supposeζ1 = (M1, N1) andζ2 = (M2, N2) are two VGs andV1 ∩ V2 =∅. Then,ζ = (M, N) is the join ofζ1andζ2if each (λ, δ)-LGζ(λ,δ)is the ofand.
Proof. Let ζ = (M, N) be the join of VGs ζ1 and ζ2. Then, by the definition and the proof of Theorem 3.10, , for all (λ, δ) ∈ [0, 1] × [0, 1]. We prove that , for all (λ, δ) ∈ [0, 1] × [0, 1], where is the set of all edges joining the nodes of and .
From the proof of Theorem 3.10, it follows that . If , then , , , and . So,andIt follows that pq ∈ N(λ,δ). Thus, . For each pq ∈ N(λ,δ), if pq ∈ E1 ∪ E2, then , by the proof of Theorem 3.10. If p ∈ V1 and q ∈ V2, thenMoreover,So, and . Thus, . Hence, . Conversely, suppose every LG ζ(λ,δ) is the join of and . From the proof of Theorem 3.10, we have
Assume p ∈ V1, q ∈ V2, , , tN (pq) = t, and fN (pq) = w. Then, , , and pq ∈ N(w,t). It shows pq ∈ N(λ,δ), , and . Hence, tN (pq) ≥ r, fN (pq) ≤ λ, , , , and . Thus,
So, , and, as described.
Definition 3.13.Letζ1 = (M1, N1) andζ2 = (M2, N2) be two VGs. The cross productζ1∗ζ2is the pair (M, N) of VSs defined on the cross productso that
Supposeζ1 = (M1, N1) andζ2 = (M2, N2) are two VGs. Then,ζ = (M, N) is the cross product ofζ1andζ2if each LGζ(λ,δ)is the cross product ofand.
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 and ∀(λ, δ) ∈ [0, 1] × [0, 1]. We prove that∀(λ, δ) ∈ [0, 1] × [0, 1]. If (p1, p2) (q1, q2) ∈ N(λ,δ), then
Hence, , , , and . Thus, and . Now, if and , then , , , and . So, we have
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.Letζ1 = (M1, N1) andζ2 = (M2, N2) be two VGs. The lexicographic product (LP)ζ1•ζ2is the pair (M, N) of VSs defined on the LPso that
Supposeζ1 = (M1, N1) andζ2 = (M2, N2) are two VGs. Then,ζ = (M, N) is LP ofζ1andζ2if,∀(λ, δ) ∈ [0, 1] × [0, 1] andλ + 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 and ∀(λ, δ) ∈ [0, 1] × [0, 1]. We prove that , ∀(λ, δ) ∈ [0, 1] × [0, 1], where is the subset of the edge set of the direct product (DP) , and is the edge set of the cross product . 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 , by the definition of cross product and the proof of Theorem 3.14. Hence, . 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 . Thus, .
Conversely, assume and ∀(λ, δ) ∈ [0, 1] × [0, 1]. It is clear that has the same vertex set as the CP . Now, by the proof of Theorem 3.6, we get
∀(p1, p2) ∈ V1 × V2. For p ∈ V1 and p2q2 ∈ E2, let , , tN ((p, p2) (p, q2)) = δ1, and fN ((p, p2) (p, q2)) = λ1. Then, according to the definitions of CP and LP, we haveBy the same reasoning as proof of Theorem 3.6, we get
Now, assume that tN ((p1, p2) (q1, q2)) = δ1, fN ((p1, p2) (q1, q2)) = λ1, , and , for p1q1 ∈ E1 and p2q2 ∈ E2. Then, according to the definitions of the cross product and LP, we deriveSimilar to the proof of Theorem 3.14, we have
which completes the proof.
Lemma 3.17Letζ1 = (M1, N1) andζ2 = (M2, N2) be two VGs so thatV1 = V2,M1 = M2, andE1 ∩ E2 =∅. Then,ζ = (M, N) is the union ofζ1andζ2ifζ(λ,δ)is the union ofand,∀(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, . Now, we prove that , for all (λ, δ) ∈ [0, 1] × [0, 1]. For each , we get and . So, pq ∈ N(λ,δ). Thus, . In the same way, we get . Then, . For each pq ∈ N(λ,δ), pq ∈ E1, or pq ∈ E2. If pq ∈ E1, then . Thus, . If pq ∈ E2, then we get . Therefore, .Conversely, assume (λ, δ)-LG ζ(λ,δ) is the union of and . Let tM(p) = δ, fM(p) = λ, , and , for some p ∈ V1 = V2. Then, p ∈ M(λ,δ) and . So, and p ∈ M(λ,δ) because and . Thus, , , tM(p) ≥ t, and fM(p) ≤ w. Hence, , , , and . Therefore, and because M1 = M2, V1 = V2, and M = M1 = M1 ∪ M2. In the same way, we deriveDefinition 3.18.Assumeζ1 = (M1, N1) andζ2 = (M2, N2) are two vague pair of graphsand, respectively. The strong product (SP)ζ1 ⊠ ζ2is the pair (M, N) of VSs defined on the SPso that
Letζ1 = (M1, N1) andζ2 = (M2, N2) be two VGs. Then,ζ = (M, N) is the SP ofζ1andζ2ifζ(λ,δ), where (λ, δ) ∈ [0, 1] × [0, 1] andλ + δ ≤ 1 is the SP ofand.
Proof. By definitions of SP, cross product, and CP, we get ζ1 ⊠ ζ2 = (ζ1 × ζ2) ∪ (ζ1∗ζ2) and , and ∀(λ, δ) ∈ [0, 1] × [0, 1]. By Theorems 3.14 and 3.6, and Lemma 3.17, we have
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 thatZ = {q1, q2, … , qn} is the universe of discourse. LetM = {(qi, tM(qi), fM(qi): qi ∈ Z} andN = {(qi, tN (qi), fN (qi): qi ∈ Z} be two VSs. The new distance measure is defined asClearly,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 ofnnumber of patients. Suppose thatis the symptoms of the diseasesEi, andis the symptoms of patientTjgiven in VSs. So, we havewherei = 1, 2, … , mandj = 1, 2, … , n. The distance between each pair of diseases and patients can be expressed as the following matrix:
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
| → Disease | Cholecystitis (CH) | Migraine (MI) | Dyspepsia (DY) | Diverticulitis (DI) | Inflammatory |
|---|---|---|---|---|---|
| ↓ Symptoms | bowel disease (IBD) | ||||
| Jaundice (JA) | (0.7, 0.2) | (0.2, 0.2) | (0.2, 0.5) | (0.6, 0.2) | (0.3, 0.5) |
| Nausea (NA) | (0.1, 0.4) | (0.7, 0.3) | (0.2, 0.4) | (0.3, 0.5) | (0.3, 0.2) |
| Heartburn (HB) | (0.2, 0.3) | (0.3, 0.4) | (0.7, 0.1) | (0.3, 0.5) | (0.5, 0.4) |
| Constipation (CO) | (0.6, 0.3) | (0.2, 0.4) | (0.3, 0.4) | (0.7, 0.2) | (0.2, 0.6) |
| Chronic diarrhea (CD) | (0.2, 0.3) | (0.3, 0.5) | (0.2, 0.6) | (0.4, 0.5) | (0.7, 0.2) |
Symptoms–diseases VR.
TABLE 2
| Jaundice (JA) | Nausea (NA) | Heartburn (HB) | Constipation (CO) | Chronic diarrhea (CD) | |
|---|---|---|---|---|---|
| Safari | (0.3, 0.6) | (0.7, 0.2) | (0.4, 0.5) | (0.3, 0.2) | (0.2, 0.4) |
| Najafi | (0.3, 0.4) | (0.2, 0.5) | (0.4, 0.4) | (0.3, 0.5) | (0.7, 0.1) |
| Ahmadi | (0.8, 0.1) | (0.4, 0.3) | (0.5, 0.2) | (0.6, 0.3) | (0.3, 0.4) |
| Rahmani | (0.2, 0.3) | (0.3, 0.5) | (0.8, 0.2) | (0.3, 0.4) | (0.3, 0.5) |
Patient–symptoms VR.
Now, we show the patients and symptoms as VSs as follows:
Here, we calculate the vague distance between the disease and the patients based on their symptoms.
In the same way, we have
The distance matrix for the aforementioned values is as follows:
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.
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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.
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Summary
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
Volume
11 - 2023
Edited by
Yilun Shang, Northumbria University, United Kingdom
Reviewed by
Sovan Samanta, Tamralipta Mahavidyalaya, India
Madhumangal Pal, Vidyasagar University, India
Updates
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
This article was submitted to Statistical and Computational Physics, a section of the journal Frontiers in Physics
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