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Computers, Materials & Continua
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DOI: 10.32604/cmc.2023.036321

Article

Pythagorean Neutrosophic Planar Graphs with an Application in Decision-Making

P. Chellamani1,2,*, D. Ajay1, Mohammed M. Al-Shamiri3,4 and Rashad Ismail3,4

1Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur, 635601, Tamilnadu, India
2Department of Mathematics, St. Joseph’s College of Engineering, OMR, Chennai, 600119, Tamilnadu, India
3Department of Mathematics, Faculty of Science and Arts, King Khalid University, Muhayl Assir, Saudi Arabia
4Department of Mathematics and Computer, Faculty of Science, Ibb University, Ibb, Yemen
*Corresponding Author: P. Chellamani. Email: joshmani238@gmail.com
Received: 26 September 2022; Accepted: 08 February 2023

Abstract: Graph theory has a significant impact and is crucial in the structure of many real-life situations. To simulate uncertainty and ambiguity, many extensions of graph theoretical notions were created. Planar graphs play a vital role in modelling which has the property of non-crossing edges. Although crossing edges benefit, they have some drawbacks, which paved the way for the introduction of planar graphs. The overall purpose of the study is to contribute to the conceptual development of the Pythagorean Neutrosophic graph. The basic methodology of our research is the incorporation of the analogous concepts of planar graphs in the Pythagorean Neutrosophic graphs. The significant finding of our research is the introduction of Pythagorean Neutrosophic Planar graphs, a conceptual blending of Pythagorean Neutrosophic and Planar graphs. The idea of Pythagorean Neutrosophic multigraphs and dual graphs are also introduced to deal with the ambiguous situations. This paper investigates the Pythagorean Neutrosophic planar values, which form the edges of the Pythagorean neutrosophic graphs. The concept of Pythagorean Neutrosophic dual graphs, isomorphism, co-weak and weak isomorphism have also been explored for Pythagorean Neutrosophic planar graphs. A decision-making algorithm was proposed with a numerical illustration by using the Pythagorean Neutrosophic fuzzy graph.

Keywords: Pythagorean neutrosophic planar graph; planarity value; isomorphism; dual graphs; multigraph

1  Introduction

Graphs are illustrative representations that express the relation between objects and their data. When the relationships are ambiguous, a graph can be implemented as a fuzzy graph model, which has the same structure as a crisp graph but works with ambiguous data. The fuzzy set theory for dealing with incomplete and vague information originated from the work of Zadeh [1]. Following the fuzzy sets, Intuitionistic sets [2], in which the elements possess membership (μ) and non-membership (ρ) grades with the condition that μ+ρ≤1. By considering the vagueness of information, adding some restrictions leads to the extension and development of the neutrosophic set by Smarandache [3], which assigns a truth, indeterminacy, and false membership grade to the elements, with the condition that the sum of the membership grades is within the range of 0 and 3. To expand this concept, Yager [4] proposed the concept of Pythagorean sets, which have an added relaxation in their condition as μ2+ρ2≤1. The fusion of Pythagorean and Neutrosophic sets resulted in the development of the Pythagorean Neutrosophic set, which allows the element to have the membership (μ), indeterminacy (σ) and non-membership grade (γ) with the constraint that μ2+ρ2+γ2≤2.

Kaufmann [5], based on fuzzy relation [6], developed the idea of Fuzzy Graphs (FGs). Later, Rosenfeld [7] defined the basic properties of fuzzy relations, which are generalized with a fuzzy set as a base set, and fuzzy analogs of graphic theoretical concepts like bridges and trees were established with their properties. Bhattacharya [8] introduced the notions of eccentricity, and center explored how a fuzzy group can be associated with a fuzzy graph. Fundamental operations on FGs and their properties were discussed by Mordeson et al. in [9].

Shannon et al. introduced intuitionistic fuzzy relations in [10], and intuitionistic fuzzy graphs (IFG) with their properties were investigated in [11]. The operations of IFGs were described by Parvathi et al. in [12]. In [13,14], Akram et al. established ideas such as strong IFGs and IF hypergraphs. Pythagorean Fuzzy Graphs (PFGs) and their applications were explored by Naz et al. in [15], and the energy of a Pythagorean Fuzzy Graph (PFG) was studied by Akram et al. in [16], followed by the assertion of some PFG operations by Akram et al. in [17]. Akram et al. [18] proposed certain graphs using the base Pythagorean and the abstraction of the fuzzy dual graph with the investigation of its properties in [19].

Yager proposed the concept of fuzzy multiset in [20], and fuzzy planar graphs were developed along with their properties in [21] and [22]. Alshehri et al. [23] established some exciting proofs of Intuitionistic fuzzy planar graphs, and the concept of bipolar fuzzy planar graphs was described in [24]. Strong neutrosophic graphs were introduced in [25], and single-valued neutrosophic graphs were instituted by Broumi et al. [26]. Akram et al. introduced neutrosophic graphs and neutrosophic soft graphs along with their applications [27]. Single-valued neutrosophic hypergraphs and intuitionistic neutrosophic soft graphs were studied in [28,29]. The recent development of planar graphs in this area can be seen in [30–34]. The concept of the Pythagorean Neutrosophic graphs [35] was developed using the Pythagorean Neutrosophic set [36] and further into other graph-theoretical concepts in [37–40].

In this study, the graph-theoretical results are applied in the Pythagorean Neutrosophic fuzzy environment. The concept of planar graphs is more captivating because of their complexity. In designing circuits, circuits or lines are arranged so they do not intersect to avoid circuit problems, and planar graphs can be used to tackle this problem. Though all developments in fuzzy graph theory have advantages, the Pythagorean Neutrosophic graphs have their advantage with more fuzzified inputs. This research article elaborates on the abstraction of Pythagorean Neutrosophic Multi Graphs (PNMGs), Pythagorean Neutrosophic Planar Graphs (PNPGs), and Pythagorean Neutrosophic Dual Graphs (PNDGs). The potential applications of these graphs can be used to assess and design a variety of real-world challenges. Planarity is a crucial feature that is investigated in this work. A significant addition to the literature is the development of the Pythagorean neutrosophic planar and multigraphs with their characterizations.

The following is the order in which this article is organized: Section 2 deals with introducing the Pythagorean Neutrosophic Multi Graphs (PNMGs) and investigating their properties. Section 3 proposes the concept of the Pythagorean Neutrosophic Planar Graphs (PNPGs) along with the results and the investigation of its characteristics. An algorithm for decision-making using Pythagorean Neutrosophic fuzzy graphs has been proposed with the examination of a practical numerical illustration in Section 4 and the work is concluded in Section 5.

2  Pythagorean Neutrosophic Multigraphs

Definition 2.1. A Pythagorean Neutrosophic Multi Set (PNMS) C of a non-void set H is grouped by functions, `count M', `count NM' and `count I' of C symbolized by CMC, CNMC, and CIC and given as CMC, CNMC, CIC: H → R with R, a collection of all multisets from interval [0,1]. A PNMS C is represented by C={<z,(μC1 (z),μC2 (z),…,μCg (z)),(σC1 (z),σC2 (z),…,σCg (z)),(γC1 (z),γC2(z),…,γCg (z))>|z∈H}. where the M sequence (μC1 (z),μC2 (z),…,μCg (z)), the I  sequence σC1 (z),σC2 (z),…,σCg (z) and the NM sequence (γC1 (z),γC2 (z),…,γCg (z)) may be increasing (or) decreasing order, and sum of μCf (z), σCf (z), γCf(z) ∈ [0, 1] satisfies the criteria 0≤sup μCf(z)+sup σCf(z)+sup γCf(z)≤2 for z∈H and C={<z, μC(z)f, σC(z)f, γC(z)f>/z ∈ H,f=1,2,…,g}.

Definition 2.2. C={<s,μC(s)k,σC(s)k,γC(s)k>/w∈ H,k=1,2,…,q} and O={<s,μO(s)k, σO(s)k,γO(s)k>/s∈ H,k=1−q} be two PNMSs in H. Then,

1.    C⊆O iff μC(s)k≤μO(s)k, σC(s)k≤σO(s)k, γC(s)k≥γO(s)k for k=1 − q and s ∈ H;

2.    C=O  iff  C⊆O and O⊆C.

3.    Cc={<s,γC(s)k,1−σC(s)k,μC(s)k>/s ∈ H,k=1−q}.

4.    C∪O={s,μC(s)k∨μO(s)k,σC(s)k∨ σO(s)k,γC(s)k∧γO(s)k,/s ∈ H,k=1 − q}.

5.    C∩O={s,μC(s)k∧μO(s)k,σC(s)k∧ σO(s)k, γC(s)k∨ γO(s)k,/s ∈ H,k=1 − q}.

Definition 2.3. Let C=(μC,σC,γC) be a Pythagorean Neutrosophic (PN) set on V and O={ds,μO(ds)k,σO(ds)k,γO(ds)k i=1−mds∈ V×V} a PNMS of V×V with μO(ds)k≤min{μO(d),μO(s)},σO(ds)k≤min{σO(d),σO(s)},γO(ds)k≤max{γO(d),γO(s)},∀ k=1−m. G=(C,O) is a PN Multi Graph (PNMG). μO(ds)k,σO(ds)k,γO(ds)k Symbolize the M,I and NM value of ds in G, correspondingly. m represents the count of edges among the vertices. In PNMG G,O represents a PN Multi Edge set (PNME).

Example 2.1. Let the multigraph be G=(V,E) with V={a,b,c,d},E={ab,bc,bc,bc,bd}.

Let C=(μC,σC,γC) be a PN set on V and O=(μO,σO,γO) be a PNME set on V×V defined as

C={<a,.5,.3,.3>,<b,.4,.2,.4>,<c,.5,.4,.3>,<d,.4,.3,.4>},

O={<ab,.3,.2,.3><bc,.3,.2,.3>,<bc,.2,.1,.2>,<bc,.4,.2,.4>,<bd,.3,.2,.2>}.

Definition 2.4. Let O={<ds,μO(ds)k,σO(ds)k,γO(ds)k>,k=1−m|ds ∈ V×V}, a PNME in PNMG G. Degree of a vertex d∈V is

deg(d)=(∑k=1m⁡μO(ds)k,∑k=1m⁡σO(ds)k,∑k=1m⁡γO(ds)k), ∀ s∈V.

Example 2.2. The vertices a,b,c and d in example 2.1, hold the degrees

deg(a)=(.3,.2,.3),deg(b)=(1.2,.7,1.1),deg(c)=(.9,.5,.9),deg(d)=(.3,.2,.2).

Definition 2.5. Let O={(ds,μO(ds)k,σO(ds)k,γO(ds)k,k=1 to m/ds ∈ V×V}  be a PNMS  of PNMG G. Multi edge ds of G is strong if

12 min [μC(d),μC(s)]≤μO(ds)k,12 min [σC(d),σC(s)]≤σO(ds)k,

12 max [γC(d),γC(s)]≥ γO(ds)k,for all k=1 to m.

Definition 2.6. Let O={(ds,μO(ds)k,σO(ds)k,γO(ds)k,k=1 tomds∈ V×V} be a PNMS in PNMG G.

1.    G has order,

O(G)=∑d∈VμC(d),∑d∈VσC(d),∑d∈VγC(d),

2.    G has size,

S(G)=(∑k=1n⁡μO(ds)k,∑k=1n⁡σO(ds)k,∑k=1n⁡γO(ds)k),∀ds∈V×V.

3.    The total degree of d ∈ V is,

tdG(d)=(∑k=1n⁡μO(ds)k,∑k=1n⁡σO(ds)k,∑k=1n⁡γO(ds)k), ∀d ∈ V.

Definition 2.7. Let O={(ds,μO(ds)k,σO(ds)k,γO(ds)k,k=1 to m/ds ∈ V×V}, a PNME in PNMG G. G is complete if min[μC(d),μC(s)]=μO(ds)k, min[σC(d),σC(s)]=σO(ds)k, max[γC(d),γC(s)]=γO(ds)k,∀ k=1− m and ∀ d,s ∈ V.

Example 2.3. Consider PNMG G in Fig. 1. Using the calculation as defined, it is verified that G in Fig. 1 is a complete PNMG.

Figure 1: Pythagorean neutrosophic multigraph

Definition 2.8. If each node of G has the same degree of M, I and NM values, then G is a regular  PNMG.

Definition 2.9. Let G be a PNMG such that O={ds,μO(ds)i,σO(ds)i,γO(ds)i,i=1 to m/ds ∈ V×V}.

1.    The edge ds has degree

DG(ds)=((degμ)G(d)+(degμ)G(s)−2μO(ds)i),((degσ)G(d)+(degσ)G(s)−2σO(ds)i),((degσ)G(d)+(degσ)G(s)−2σO(ds)i).

2.    The edge ds has degree

tDG(ds)=((degμ)G(d)+(degμ)G(s)−2μO(ds)i), ((degσ)G(d)+(degσ)G(s)−2σO(ds)i),

  ((degσ)G(d)+(degσ)G(s)−2σO(ds)i),

where  ((ds)i) is the ith edge between d & s.

Definition 2.10. If the degree of M,I and NM of every edge in PNMG G are equal, then G is Edge Regular (ER).

Example 2.4. Degree of the edges in Example 1 are DG(ab)=(1.2,.7,1.1),DG(bd)=(1.2,.7,1.2),DG(bc)=(1.2,.7,1.1), DG(bc)=(1.1,.7,1), DG(bc)=(1.3,.8,1.2) and total degree of edges are tDG(ab)=(1.5,0.9,1.4), tDG(bd)=(1.5,0.9,1.4), tDG(bc)=(1.5,0.9,1.4), tDG(bc)=(1.5,0.9,1.4), tDG(bc)=(1.5,0.9,1.4).

Theorem 2.1. Let G=(C,O) be a PNMG. If G is regular and edge regular PNMG, then the M μ(ds)k,I σ(ds)k,NM γ(ds)k for every line (ds)∈V×V are constants.

Proof. Consider, G=(C,O), a PNMG; G is regular and edge regular PNMG. There exists constants p1,p2,p3 and q1,q2,q3 for regular and edge regular correspondingly so that for every node,

deg G(d)=((degμ)G(d),(degσ)G(d),(degσ)G(d))= p1,p 2,p3 for every edge ds∈V×V,

DG(ds)=((Dμ)G(ds),(Dσ)G(ds),(Dγ)G(ds)),=((degμ)G(d)+(degμ)G(s)−2μO(ds)k),((degσ)G(d)+(degσ)G(s)−2σO(ds)k),((degσ)G(d)+(degσ)G(s)−2γO(ds)k).=(q1,q2,q3).

Thus, for the M,I and NM values, p1+p1−2μO(ds)k=2q1, p1+p1−μO(ds)k=2q1, 2p1−2q1=2μO(ds)k, p1−q1=μO(ds)k, Similarly, p2−q2=σO(ds)k and p3−q3=γO(ds)k.

Thus, the M,I and NM values of a regular PNMG with edge regular are constant.

Theorem 2.2. Let G=(C,O) be a PNMG on G∗=(V,E). If G∗ is y–regular multigraph, μO(ds)k,σO(ds)k and γO(ds)k are constants for every edge ds∈V×V, then G is a regular and edge regular  PNMG.

Proof. Consider G∗=(V,E) as a y− regular multigraph. Consider μO(ds)k=q1, σO(ds)k=q2 and γO(ds)k=q3. For every vertex d∈V,

degG(d)=((degμ)G(d),(degσ)G(d),(degσ)G(d)),

=(∑d≠sμO(ds)k,∑d≠sσO(ds)k,∑d≠sγO(ds)k),

=(y×q1,y×q2,y×q3 ),

=(∑d≠sμO(yx)k,∑d≠sσO(yx)k,∑d≠sγO(yx)k),

=((degμ)G(s),(degσ)G(s),(degσ)G(s))=degG(s).

For every edge, ds∈V×V,

DG(ds)=((Dμ)G(ds),(Dσ)G(ds),(Dγ)G(ds))=((degμ)G(d)+(degμ)G(s)−2μO(ds)k),((degσ)G(d)+(degσ)G(s)−2σO(ds)k),((degσ)G(d)+(degσ)G(s)−2γO(ds)k).=((y×q1)+(y×q1) – 2(q1),(y×q2)+(y×q2)−2(q2),(y×q3)+(y ×;q3) − 2(q3))=(2q1 (y − 1),2q2 (y − 1),2q3 (y − 1)).

Thus, G is regular and edge regular PNMG.

Definition 2.11. The strength of PN edge fj is determined by the value

Sfi=((Sμ)fi,(Sσ)fi,(Sγ)fi)=(μO(fj)imin(μ C(f),μ C(j)),σO(fj)imin(σ C(f),σ C(j)),γO(fj)imin(γ C(f),γC(j))).

An edge fj of a PNMG is PN strong if (Sμ)fj≥0.5,(Sσ)fj≥0.5,(Sγ)fj≥0.5.

Definition 2.12. Let G  be a PNMG such that O has 2 edges (ab,μo(ab)k,σo(ab)k,γo(ab)k) and (cd,μo(ab)y,σo(ab)y,γo(ab)y), which intersect at P,k and y are fixed integers. At P, the intersecting value is defined by,

S P=((Sμ)P,(Sσ)P,(Sγ)P)=((Sμ)ab+(Sμ)cd2+(Sσ)ab+(Sσ)cd2+(Sγ)ab+(Sγ)cd2).

The planarity decreases when the number of points of intersection in PNMG increases.

SP is inversely proportional to the planarity for PNMG.

3  Pythagorean Neutrosophic Planar Graphs

The concept of the Pythagorean Neutrosophic planar graphs has been discussed.

Definition 3.1. Let G be a PNMG and the point of intersections among the edges be P1,P2,…,P m,G is a PN  Planar Graph (PNPG) with PN Planarity Value (PNPV) f=(fμ,fσ,fγ), where

f=(fμ,fσ,fγ)=(11+{(Sμ)P1+(Sμ)P2+…+(Sμ)Pm},11+{(Sσ)P1+(Sσ)P2+…+(Sσ)Pm},11+{(Sγ)P1+(Sγ)P2+…+(Sγ)Pm}).

0≤fμ≤1,0≤fσ≤1,0≤fγ≤1. The PNPV is (1,1,1) for a geometrical representation of PNPG  if it has no intersecting point.

Example 3.1. Take a multigraph G∗=(V,E) such that V={a,b,c,d,e}, E={ab,ac,ad,ad,bc,bd,cd,ce,ae,de,be}. Let C=(μC,σC,γC) be a PN set on  V and O =(μO,σO,γO) be a PNME set on V×V as are described as,

C={<a,.5,.5,.2>,<b,.6,.7,.3>,<c,.4,.6,.4>,<d,.7,.5,.3>,<e,.8,.6,.5>},

O={<ab,.5,.4,.2>,<ac,.4,.5,.3>,<ad,.5,.5,.3>,<ad,.4,.4,.2>,<bc,.4,.6,.4>,<bd,.6,.5,.2>,<cd,.3,.5,.3>,<ae,.5,.5,.4>,<ce,.3,.6,.5>,<de,.7,.5,.4>,<be,.6,.6,.4>}.

The PNMG has two points of intersection in Fig. 2 (P1 and P2).P1 is a point among the lines (ad,.5,.5,.3) and (bc,.4,.6,.4) and P2 is a point among the edges (ad,.4,.4,.2) and (bc,.4,.6,.4).

Figure 2: Pythagorean neutrosophic planar graph

The strength for the edges ab,ad and bc are Sad=(.5.5,.5.5,.3.3)=(1,1,1), Sad=(.4.5,.4.5,.2.3)=(.8,.8,.67), Sbc=(.4.4,.6.6,.4.4)=(1,1,1). For P1, intersecting value SP1 is (1,1,1) and for P2, SP2 is (.9,.9,.835).

Therefore PNPV for the  PNMG given in Fig. 2 is (.345,.345,.353).

Theorem 3.1. Let G be a complete PNMG. The PNPV,f=(fμ,fσ,fγ) of G is given by fμ=11+np,fσ=11+np and fγ=11+np such that fμ+fσ+fγ≤3, where np is the count of point of intersection among the lines in G.

Definition 3.2. A PNPG G is called strong (𝒮PNPG) if the PNPV f=(fμ,fσ,fγ) of the graph is fμ≥0.5,fσ≥0.5,fγ≥0.5.

Theorem 3.2. Let G be a 𝒮 PNPG. The number of points of intersections among 𝒮 lines in G is utmost one.

Proof. Let G be a 𝒮 PNPG. Consider G has at least 2 points of intersections P1 and P2 between 2 S lines in G. For any 𝒮 edge (wq,μO(wq)i,σO(wq)i,γO(wq)i), μO(wq)i≥12min{μC(w),μC(q)}, σO(wq)i≥12min{σC(w),σC(q)}, γO(wq)i≤12max{γC(w),γC(q)}.

Thus, (Sμ)wq,(Sμσ)wq,(Sγ)wq≥.5. Thus, for two intersecting 𝒮 edges (wq,μO(wq)k,σO(wq)k, γO(wq)k) and (cd,μO(cd)j,σO(cd)j,γO(cd)j),

(Sμ)wq+(Sμ)cd2+(Sσ)wq+(Sσ)cd2+(Sγ)wq+(Sγ)cd2≥.5,

(i.e.,) (Sμ)P1,(Sσ)P1≥.5,(Sγ)P1≤.5, Likewise, (Sμ)P2,(Sσ)P2≥ .5,(Sγ)P2≤.5.

⇒1+(Sμ)P1+(Sμ)P2≥ 2,1+(Sσ)P1+(Sσ)P2≥ 2,1+(Sγ)P1+(Sγ)P2≥2.

fμ=11+(Sμ)P1+(Sμ)P2≤.5 fσ=11+(Sσ)P1+(Sσ)P2≤.5 fγ=11+(Sγ)P1+(Sγ)P2≥.5.

This becomes a contradiction to the fact PN graph is a 𝒮PNPG. Thus the number of points of intersections between 𝒮 edges cannot be two. If the count of point of intersections of PN edges increases, the PNPV decreases. When the count of the point of intersection of 𝒮 edges is 1, then the PNPV fμ≥0.5,fσ≥0.5,fγ≥0.5. A 𝒮PNPG is a PNPG without any crossing between edges. Thus, the largest number of points of intersections among the 𝒮 edges in G is 1 parameter. The region bounded by PN edges is a face of a PN graph. Every PN Face (PNF) in its boundary is characterized by PN edges. If every edge in the boundary of a PNF have μO,σO,γO values (1,1,1) and (0,0,0), then it is a crisp face. When one among those edges is removed or has μO,σO,γO values (0,0,0) and (1,1,1) correspondingly, the PNF does not exist. The existence of a PNF depends on the minimal strength of PN edges in its boundary. A PNF and its μO,σO,γO Values of PNG are expressed below.

Definition 3.3. Let G be a PNPG and O= {(ds,μO(ds)k,σO(ds)k,γO(ds)k,k=1 to m/ds ∈V×V}. A PNF of G is a region and is bounded by the set of PN lines E′⊂E, of a pictorial demonstration of G. The M,I and NM of PNF are:

min{μO(ds)kmin{μC(d),μC(s)},k=1,2,…mds∈ E′},

min{σO(ds)kmin{σC(d),σC(s)},k=1,2,…mds∈ E′},

max{γO(ds)kmax{γC(d),γC(s)},k=1,2,…mds∈ E′}.

Definition 3.4. A PNF is (𝒮) PNF if the value of M,I is larger than 0.5, NM is below 0.5, and weak otherwise. The infinite region in every PNPG is termed as an outer PNF, and other faces are called inner PNFs.

Example 3.2. The PNPG as in Fig. 3, has the following faces: PNF F1 is bounded by the edges (ab,.4,.4,.1),(bc,.5,.5,.1),(ac,.4,.4,.1). Outer PNF F2 surrounded by edges (ac,.4,.4,.1),(ad,.4,.4,.1),(bd,.5,.5,.1),(bc,.5,.5,.1).PNF F3 is bounded by lines (ab,.4,.4,.1),(bd,.5,.5,.1),(ad,.4,.4,.1).

Figure 3: Faces in pythagorean neutrosophic planar graph

Clearly, the M,I and NM value of a PNF F1 is  (.8,.8,.5). Thus, F1 is a 𝒮PNF.

Definition 3.5. Let G be a PNPG and let O= {(ds,μO(ds)k,σO(ds)k,γO(ds)k,k=1 − m/ds ∈ V×V}. Let F1,F2,…,Fk be the 𝒮PNFs  of G. The PN Dual Graph (PNDG)  of G is a PNPG G′=(V′,C′,O′) with V= {xk,k=1 − k}, and the vertex  xk of G′ is for Fk of G. The M,I,NM values of vertices are C′=(μC′,σC′,γC′): V′→[0,1]3 such that

μC′(xk)=max{μO′(ua)k,k=1 to p ua is in the boundary of 𝒮PNF Fk},

σC′(xk)=max{σO′(ua)k,k=1 to p ua  is in the boundary of 𝒮PNF Fk},

γC′(xk)=min{γO′(ua)k,k=1 to p ua is in the boundary of 𝒮PNF Fk}.

Two common faces Fk and F b  of G might exist between one common line. There may be more than 1 edge among 2 vertices xk and xb in PNDG G′. μO′ (xkxb) represent the M value of the lth edge among xk and xb and γO′(xkxb) represent the NM value of the lth edge amidst xk and xb. M,I and NM values of PN  edges of the PNDG are presented by μO′(xkxb)l=μO′(ua)b, σO′(xkxb)l=σO′(ua)b, γO′(xkxb)l=γO′(ua)b, with (ua)b is an edge in the boundary between 2 𝒮PNF Fk and F b and l=1  to S, where 𝒮 is the count of lines among xk and xb. PNDG  of  PNPG does not hold point of intersection of edges for a some representation, so it is  PNPG with PNPV (1,1,1). The PNF  of PNDG can be similarly expressed as in PNPG.

Theorem 3.3. Let G be a PNPG  whose count of vertices, total of PN edges, count of 𝒮PNF  are symbolized by m,p,n correspondingly. G′  be the PNDG of G, then count of vertices, edges, PNF of  G′ equals m,p,n correspondingly.

Theorem 3.4. Let G=(V,C,O) be a PNPG without weak lines and the PNPG of G  be  G′=(V′,C′,O′). The M,I and NM values of PN lines of G′  equals values of G.

Definition 3.6. Let G=(C,O) be a PNPG where O={(ds,μO(ds)k,σO(ds)k,γO(ds)k,b)= 1to nab∈ V×V}. Let F1,F2 ,…,Fk be 𝒮PNFs of G. Then PNDG of  G is a PNPG G′=(C′,O′), where  V′={rb,b=1  to K} and the vertex rb of  G′ is taken for Fb of G.

The M,I,NM by mapping G′=(V′,C′,O′):V′→[0,1]3 such that

μC′(rb)=max {μO′(ua)b,b=1 to mab is in the neighbourhood of 𝒮PNF Fb},

σC′(rb)=max {σO′(ua)b,b=1 to m ab is in the neighbourhood of 𝒮PNF Fb},

γC′(rb)=min {γO′(ua)b,b=1 to mab is in the neighbourhood of 𝒮PNF Fb}.

Between Fk and Fb of G, at least one common edge may occur. Among two vertices, there may exist beyond a single edge rkrb in PNDG G′. M,I and NM values of PN edges of PNDG are μC′(rkrb)S=μOS(ua)k, σC′(rkrb)S=σOS(ua)k, γC′(rkrb)S=γOS(ua)k where (ab)S is in the surrounding among 𝒮PN faces Fk and Fb and S=1 – l, is the count of common edges in the neighborhood of Fk and Fb. The PNDGG′of PNDGGhas no crossing among lines for some definite geometric representation, PNPG of PNPV (1,1,1).

Example 3.3. Take a PNG G=(V,C,O) as displayed in Fig. 4 with V={a1,a2,a3,a4,a5}. Let C  and O be PN vertex set and PN edge set.

C=<(a1,.7,.5,.3),(a1,.69,.55,.4),(a1,.35,.45,.5),(a4,.76,.8,.3)>.

O=<(a1a2,.6,.48,.2),(a2a3,.3,.4,.43),(a3a4,.3,.4,.45),(a4a1,.65,.4,.25),(a1a3,.3,.4,.4)>.

Figure 4: Pythagorean neutrosophic dual graph

PNF F1 is enclosed by the edges (a1a3,.3,.4,.4), (a3a4,.3,.4,.45), (a4a1,.65,.4,.25).

PNFF2 is enclosed by the edges  (a1a2,.6,.48,.2), (a2a3,.3,.4,.43), (a1a3,.3,.4,.4).

PNFF3 is enclosed by the edges  (a1a2,.6,.48,.2),  (a2a3,.3,.4,.43),  (a3a4,.3,.4,.45), (a4a1,.65,.4,.25).

We symbolize the vertices of PN  Dual Graph (PNDG)  by a dot and edges by dashed lines. We take a vertex for each face of PNDG  with V′={r1,r2,r3,r4}.

μC′(r1)=max {.3,.3,.65}=.65,σC′(r1)=max {.4,.4,.4}=.4,γC′(r1)=min {.4,.45,.25}=.25,

μC′(r2)=max {.6,.3,.3}=.6,σC′(r2)=max {.48,.4,.4}=.48,γC′(r2)=min {.2,.43,.4}=.2,

μC′(r3)=max {.6,.3,.3,.65}=.65,σC′(r3)=max {.48,.4,.4,.4}=.48,γC′(r3)=min {.2,.43,.45,.25}=.2

The vertex set V′ has the vertices <r1,(.65,.4,.25)>, <r2,(.6,.48,.2)>,<r3,(.65,.48,.2)>.

There is one common edge a1a3 amidst F1,F2 in G. Thus, there exists a single line among vertices r1 and r2 in PNDG  of G. The edges for the PNDG are constructed as in Fig. 4.

Definition 3.7. An isomorphism of two PNPGsG1 and G2, y :G1→G2 is a bijective mapping y :V1→V2 that holds the following

1.    μC1(r)=μC2(y(r)), σC1(r)=σC2(y(r)), γC1(r)=γC2(y(r)).

2.    μO1(rs)=μO2(y(r) y(s)), σO1(rs)=σO2(y(r) y(s)), γO1(rs)=γO2(y(r) y(s)), ∀ r∈V1,∀ rs∈E1.

Example 3.4. Consider two PNPGsG1=(C1,O1) and G2=(C2,O2) as in Fig. 5 such that

C1={<a1,.8,.7,.2>,<a2,.7,.6,.4>,<a3,.6,.5,.5>,<a4,.5,.4,.4>},

O1={<a1a2,.6,.5,.3>,<a2a3,.5,.4,.4>,<a3a4,.4,.3,.3>,<a4a1,.3,.2,.3>},

C2={<r1,.5,.4,.4>,<r3,.7,.6,.4>,<r2,.6,.5,.5>,<r4,.8,.7,.2>},

O2={<r1r2,.4,.3,.3>,<r2r3,.5,.4,.4>,<r3r4,.6,.5,.3>,<r4r1,.3,.2,.3>}.

Figure 5: Pythagorean neutrosophic planar graph

y : V1→ V2 given by y(a1)=r4,y(a2)=r3,y(a3)=r2,y(a4)=r1 satisfies

μC1(rk)=μC2(y(rk)), σC1(rk)=σC2(y(rk)), γC1(rk)=γC2(y(rk)), μO1(rkrb)=μO2(y(rk)y(rb)),σO1(rkrb)=σO2(y(rk)y(rb)),  γO1(rkrb)=γO2(y(rk)y(rb)) for all rk∈V1,rkrb∈E1, where i,j=1 to 4. Thus G1 is isomorphic to G2.

The M,I and NM of the edges of PNDG are μO′(r1r2)=μO(a1a3)=.3, σO′(r1r2)=σO(a1a3)=.4, γO′(r1r2)=γO(a1a3)=.4, μO′(r2r3)=μO(a1a2)=.6, σO′(r2r3)=σO(a1a2)=.48, γO′(r2r3)=γO(a1a2)=.2, μO′(r3r1)=μO(a3a4)=.65, σO′(r3r1)=σO(a3a4)=.4, γO′(r3r1)=γO(a3a4)=.45, μO′(r1r3)=μO(a4a1)=.65, σO′(r1r3)=σO(a4a1)=.4, γO′(r1r3)=γO(a4a1)=.25, μO′(r2r3)=μO(a2a3)=.3, σO′(r2r3)=σO(a2a3)=.4, γO′(r2r3)=γO(a2a3)=.43.

Thus, the PNDG edge set is,

O′=<(r1r2,.3,.4,.4),(r2r3,.6,.48,.2),(r3r1,.65,.4,.45),(r1r3,.65,.4,.25),(r2r3,.3,.4,.43)>. Thus G1 is a PNDG of G2.

Definition 3.8. A weak isomorphism of two PNPGsG1 and G2, y: G1→G2  is a bijective mapping y: V1→V2 that holds the following:

1.    y is a homomorphism.

2.    μC1(r)=μC2(y(r)), σC1(r)=σC2(y(r)), γC1(r)=γC2(y(r))∀ r ∈V1.

Example 3.5. Consider two PNPG, G1=(C1,O1) and G2=(C2,O2) as in Fig. 6 such that

C1={<a1,.9,.5,.3>,<a2,.8,.6,.2>,<a3,.7,.4,.4>,<a4,.6,.3,.4>,<a5,.5,.4,.3>},

O1={<a1a2,.7,.4,.2>,<a2a3,.6,.4,.3>,<a3a4,.4,.3,.3>,<a4a1,.5,.3,.2>,

<a4a5,.5,.3,.2>,<a5a1,.4,.3,.1>,<a2a5,.4,.3,.15>},

C2={<r1,.7,.4,.4>,<r2,.6,.3,.4>,<r3,.8,.6,.2>,<r4,.5,.4,.3>,<r5,.9,.5,.3>},

O2={<r1r2,.4,.2,.1>,<r2r3,.5,.3,.2>,<r3r5,.5,.2,.1>,<r3r4,.3,.2,.1>,<r2r4,.4,.2,.1><r4r5,.2,.2,.5>}.

Figure 6: Pythagorean neutrosophic planar graphs

A mapping y: V1→ V2 given by y(a1)=r5,y(a2)=r3,y(a3)=r1,y(a4)=r2,y(a5)=r4 satisfies μC1(rk)=μ2(y(rk), σC1(rk)=σ2(y(rk)),γC1(rk)=γ2(y(rk)) for all rk ∈V1, where k,b=1,2,3,4,5.

But μC1(rkrb)≠μC2(y(rk) y(rb)), σC1(rkrb)≠σC2(y(rk) y(rb)), γC1(rkrb)≠γC2(y(rk) y(rb)).

Thus G1 is a weak isomorphic to G2.

Definition 3.9. A co-weak isomorphism of two PNPGsG1 and G2, y:G1→G2 is a bijective mapping y:V1→ V2 that holds

1.    y is a homomorphism.

2.    μC1(rs)=μC2(y(r) y(s)), σC1(rs)=σC2(y(r) y(s)), γC1(rs)=γC2(y(r) y(s)), ∀ rs∈E1.

Example 3.6. Take PNPG, G1=(C1,O1) and G2=(C2,O2) as in Fig. 7 such that

C1={<a1,.8,.7,.6>,<a2,.7,.6,.5>,<a3,.9,.5,.4>,<a4,.8,.6,.2>,<a5,.7,.5,.3>}

O1={<a1a2,.6,.5,.4>,<a2a3,.5,.4,.35>,<a1a4,.65,.5,.3>,<a2a5,.55,.4,.25>,<a4a5,.6,.4,.1>,<a2a5,.6,.4,.35>}

C2={<p1,.75,.65,.5>,<p2,.6,.5,.45>,<p3,.8,.4,.3>,<p4,.7,.5,.1>,<p5,.65,.45,.25>}

O2={<p1p2,.6,.5,.4>,<p2p3,.5,.4,.35>,<p1p4,.65,.5,.3>,<p2p5,.55,.4,.25>,<p2p4,.5,.4,.2>,<p3p5,.6,.4,.2>,<p4p5,.6,.4,.1>,<p2p5,.6,.4,.35>}.

Figure 7: Pythagorean neutrosophic planar graphs

A mapping y: V1→ V2 illustrated by y(a1)=p1,y(a2)=p2,y(a3)=p3,y(a4)=p4,y(a5)=p5 satisfies μO1(rkrb)=μO2(y(rk)y(rb)), σO1(rkrb)=σO2(y(rk)y(rb)), γO1(rkrb)=γO2(y(rk)y(rb)).

forall rkrb ∈E1 , where k,b=1,2,3,4,5 but μC1(rk)≠μC2(y(rk)), σC1(rk)≠σC2(y(rk)),γC1(rk)≠γC2(y(rk)). Thus G1 is a weak isomorphic to G2.

4  Application in Decision Making Problem

4.1 Algorithm

The following algorithm is our proposed technique for multi-criteria decision making.

Step 1: Input the alternatives B=(B1,B2,…,Bn) and set of criteria’s C=(C1,C2,…,Cm) and create the PNF relation (M(k)=mlp(k))nxn according to each criteria.

Step 2: Aggregate all mlp(k)=(μlp(k),βlp(k),σlp(k)) (l,p=1,2,…,n) regarding criteria Cj and derive M(k)=(mlp)nxn where mlp is the value assigned for alternate ml over mp according to criteria Cj by PNF averaging (PNFA) operator. pi(k)=PNFA (mi1(k),mi2(k),…,min(k)), (k = 1, 2,…, m)

=(1−(∏j=1n(1−μij2))1/n,1−(∏j=1n(1−βij2))1/n ,(∏j=1n(1−σij2))1/n),i=1,2,…,m

Step 3: Calculate the aggregated value of each criteria Cm and compute the aggregated matrix.

Step 4: Use the score function,

S(Bi(j))=1+μ+β−σ3,(i=1 to n,j=1 to m).

calculate the score matrix for the problem.

Step 5: Calculate the choice matrix by the function

S(Bi)=∑j=1n⁡Bijn,(i=1 to m).

Step 6: Now arrange the alternatives in an order and choose the maximum as the optimal decision.

4.2 Numerical Approach

In this competitive world, time is the most precious asset for everyone. In the given 24 h a day, saving time and using it for multiple duties, and chores is an important quality. Even though time management is in our hands, travelling from one point destination for the people who doesn’t drive is hard in recent times, cabs and travelling applications is one of the trending and useful facility in our cities. By the existing trends and techniques, the fastest and money-saving possibility is vital in day-to-day life. Consider the following scenario: a person wants to travel from a point to his destination and is given a set of mobile booking applications to choose a ride. Let there be these 5 cabs booking applications namely Bi (i=1 to 5) that are effective nowadays. The decision-makers provide their priors by comparing these applications concerning criteria’s Cj (j=1,2,3,4,5).

C1=Availability,C2=Travelling speed,C3=Safety,C4=Cost,C5=User friendly

Step 1: The alternatives are B=(B1,B2,B3,B4,B5) and the criteria’s are C=(C1,C2,C3,C4,C5). The Pythagorean neutrosophic fuzzy relation according to criteria Ci=(i=1,2,…,5) is given in structure as in Fig. 8 and values are detailed in Tables 1 to 5.

Figure 8: Pythagorean neutrosophic fuzzy directed graph for M(k) (k  = 1, 2, 3, 4, 5)

Step 2 and 3: Using PNFA operator, the aggregated values are calculated as follows:

C1:B1(1)=(.5269,.3984,.0423),B2(1)=(.4687,.3543,.08662),B3(1)=(.521,.325,.0321),B4(1)=(.7275,.3791,.04975),B5(1)=(.69123,.61604,.0321).

C2:B1(2)=(.5967,.2851,.0234),B2(2)=(.7223,.26435,.0184),B3(2)=(.48703,.3332,.04975),B4(2)=(.2121,.2851,.0321),B5(2)=(.6953,.5701,.0321).

C3:B1(3)=(.6088,.3248,.0321),B2(3)=(.51595,.4729,.0558),B3(3)=(.7763,.3248,.0243),B4(3)=(.5847,.3922,.0443),B5(3)=(.6214,.3543,.0321).

C4:B1(4)=(.6247,.3407,.0321),B2(4)=(.5523,.3332,.0321),B3(4)=(.5376,.3734,.0377),B4(4)=(.70114,.3017,.0321),B5(4)=(.6955,.3248,.0377).

C5:B1(5)=(.5316,.347,.0497),B2(5)=(.7029,.3161,.0243),B3(5)=(.5523,.3922,.0423),B4(5)=(.5931,.3108,.0423),B5(5)=(.6807,.2766,.0243).

Step 4: By using the score function, the score matrix is calculated in Table 6.

Step 5: Deriving the choice values of alternatives using the function, we get

S(B1)=.62702,S(B2)=.6324,S(B3)=.629,S(B4)=.6192,S(B5)=.691.

Step 6: The order of ranking obtained for the problem is

B5>B2>B3>B1>B4.

Thus, the alternate with maximum value B5 is chosen to be the optimal decision.

4.3 Comparative Analysis

The proposed model is compared with the decision-making method in [39] and is verified that the same ranking is obtained. Table 7 provides a comparison of both algorithms, showing the optimal alternative and results. Both algorithms provide the same optimum decision, as can be seen in the comparison table.

5  Discussion

Graph theory is known for its vast applications in various fields most vitally in designing networking problems. In particular, numerous graph theoretical concepts have been introduced to model vagueness in networking problems. PN graphs, an extension of FGs, and a fusion of Pythagorean and Neutrosophic graphs have better flexibility to be applied in real-world problems. The article has initiated the concept of PN Multi Graph and PN Planar Graph employing the concept of PNGs. The concept of PN Dual Graph, isomorphism, weak and co-weak isomorphism has been explored for PN Planar Graphs and their results have been examined. An algorithm has been proposed using Pythagorean Neutrosophic fuzzy graphs with a numerical example for a real-life problem. The limitation of the set and study is that it is limited when it is compared with the newly proposed sets, but the advancements pave for the new concept of planar graphs in Pythagorean neutrosophic environment. The advantage of this proposed study is this set is more fuzzifying than the previous studies because of the set and their properties. This research can be extended further to investigate Interval- valued PNGs, bipolar PNGs, and their implementation in real-life situations.

Acknowledgement: The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through the Large Group Research Project under grant number (R.G.P.2/181/44).

Funding Statement: This research was supported by Deanship of Scientific Research at King Khalid University.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

References

1. L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, pp. 338–353, 1965.

2. K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, pp. 87–96, 1986.

3. F. Smarandache, “A unifying field in logics,” in Neutrosophy: Neutrosophic Probability, Set and Logic, Rehoboth: American Research Press, 1999.

4. R. R. Yager, “Pythagorean fuzzy subsets,” in Proc. of the Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, AB, Canada, pp. 57–61, 2013.

5. A. Kaufmann, “Introduction a la Theorie des Sour-Ensembles Flous; Masson etCie 1,” Compléments et Nouvelles Applications, Masson, Paris, vol. 4, 1977.

6. L. A. Zadeh, “Similarity relations and fuzzy orderings,” Information Sciences, vol. 3, pp. 177–200, 2020.

7. A. Rosenfeld, “Fuzzy graphs,” In: L. A. Zadeh, K. S. Fu and M. Shimura (Eds.Fuzzy Sets and Their Applications to Cognitive and Decision Processes, pp. 77–95, Cambridge, MA, USA: Academic Press, 1975.

8. P. Bhattacharya, “Some remarks on fuzzy graphs,” Pattern Recognition Letters, vol. 5, pp. 297–302, 1987.

9. J. N. Mordeson and C. S. Peng, “Operations on fuzzy graphs,” Information Sciences, vol. 79, pp. 159–170, 1994.

10. A. Shannon and K. T. Atanassov, “A first step to a theory of intuitionistic fuzzy graphs,” In: D.Lakov, (Ed.Proc. of the Fuzzy Based Expert Systems, Sofia, Bulgaria, pp. 59–61, 1994.

11. A. Shannon and K. T. Atanassov, “Intuitionistic fuzzy graphs from α−,β−,(α,β)-levels,” Notes Intuitionistic Fuzzy Sets, vol. 1, pp. 32–35, 1995.

12. R. Parvathi, M. G. Karunambigai and K. T. Atanassov, “Operations on intuitionistic fuzzy graphs,” in Proc. of the IEEE Int. Conf. on Fuzzy Systems, Jeju Island, Korea, pp. 1396–1401, 2009.

13. M. Akram and W. A. Dudek, “Intuitionistic fuzzy hypergraphs with applications,” Information Sciences, vol. 218, pp. 182–193, 2013.

14. M. Akram and B. Davvaz, “Strong intuitionistic fuzzy graphs,” Filomat, vol. 26, pp. 177–196, 2012.

15. S. Naz, S. Ashraf and M. Akram, “A novel approach to decision-making with pythagorean fuzzy information,” Mathematics, vol. 6, pp. 1–28, 2018.

16. M. Akram and S. Naz, “Energy of pythagorean fuzzy graphs with applications,” Mathematics, vol. 6, pp. 136, 2018.

17. M. Akram, A. Habib, F. Ilyas and J. M. Dar, “Specific types of pythagorean fuzzy graphs and application to decision-making,” Mathematical and Computational Applications, vol. 23, pp. 42, 2018.

18. M. Akram, J. M. Dar and S. Naz, “Certain graphs under pythagorean fuzzy environment,” Complex & Intelligent Systems, vol. 5, pp. 127–144, 2019.

19. N. Abdul-Jabbar, J. H. Naoom and E. H. Ouda, “Fuzzy dual graphs,” Al-Nahrain Journal of Science, vol. 12, pp. 168–171, 2009.

20. R. R. Yager, “On the theory of bags,” International Journal of General Systems, vol. 13, pp. 23–37, 1986.

21. A. Pal, S. Samanta and M. Pal, “Concept of fuzzy planar graphs,” in Proc. of the Science and Information Conf., London, UK, vol. 14, pp. 557–563, 2013.

22. A. Pal, S. Samanta and M. Pal, “New concept of fuzzy planar graphs,” International Journal of Advanced Research in Artificial Intelligence, vol. 3, pp. 52–59, 2014.

23. N. Alshehri and M. Akram, “Intuitionistic fuzzy planar graphs,” Discrete Dynamics in Nature and Society, vol. 2014, pp. 1–9, 2014.

24. M. Akram, S. Samanta and M. Pal, “Application of bipolar fuzzy sets in planar graphs,” International Journal of Applied and Computational Mathematics, vol. 3, pp. 773–785, 2017.

25. R. Dhavaseelan, R. Vikramaprasad and V. Krishnaraj, “Certain types of neutrosophic graphs,” International Journal of Mathematical Sciences and Applications, vol. 5, pp. 333–339, 2015.

26. S. Broumi, M. Talea, A. Bakali and F. Smarandache, “Single-valued neutrosophic graphs,” Journal of New Theory, vol. 10, pp. 86–101, 2016.

27. M. Akram and S. Shahzadi, “Neutrosophic soft graphs with application,” Journal of Intelligent & Fuzzy Systems, vol. 32, pp. 841–858, 2017.

28. M. Akram and S. Shahzadi, “Representation of graphs using intuitionistic neutrosophic soft sets,” Journal of Mathematical Analysis, vol. 7, pp. 1–23, 2016.

29. M. Akram, S. Shahzadi and A. B. Saeid, “Single-valued neutrosophic hypergraphs,” TWMS Journal of Applied and Engineering Mathematics, vol. 8, pp. 122–135, 2018.

30. G. Muhiuddin, S. Hameed, A. Rasheed and U. Ahmad, “Cubic planar graph and its application to road network,” Mathematical Problems in Engineering, vol. 2022, pp. 1–12, 2022.

31. T. Mahapatra, S. Sahoo, G. Ghorai and M. Pal, “Interval valued m-polar fuzzy planar graph and its application,” Artificial Intelligence Review, vol. 54, no. 3, pp. 1649–1675, 2021.

32. R. Mahapatra, S. Samanta and M. Pal, “Generalized neutrosophic planar graphs and its application,” Journal of Applied Mathematics and Computing, vol. 65, no. 1, pp. 693–712, 2021.

33. G. Sur and Ö. Erkan, “Surface quality optimization of CFRP plates drilled with standard and step drill bits using TAGUCHI, TOPSIS and AHP method,” Engineering Computations, vol. 38, no. 5, pp. 2163–2187, 2021.

34. Ö. Erkan, D. Mustafa, I. Birahan and N. T. Ibrahim, “Selection of optimal machining conditions for the composite materials by using taguchi and GONNs,” Measurement, vol. 48, pp. 306–313, 2014.

35. D. Ajay and P. Chellamani, “Pythagorean neutrosophic fuzzy graphs,” International Journal of Neutrosophic Science, vol. 11, pp. 108–114, 2020.

36. R. Jansi, K. Mohana and F. Smarandache, “Correlation measure for pythagorean neutrosophic fuzzy sets with T and F as dependent neutrosophic components,” Neutrosophic Sets and Systems, vol. 30, pp. 1–16, 2019.

37. D. Ajay, P. Chellamani, G. Rajchakit, N. Boonsatit and P. Hammachukiattikul, “Regularity of pythagorean neutrosophic graphs with an illustration in MCDM,” AIMS Mathematics, vol. 7, no. 5, pp. 9424–9442, 2022.

38. D. Ajay, S. John Borg and P. Chellamani, “Domination in pythagorean neutrosophic graphs with an application in fuzzy intelligent decision making,” in Int. Conf. on Intelligent and Fuzzy Systems, Cham,Turkey, Springer, pp. 667–675, 2022.

39. P. Chellamani and D. Ajay, “Pythagorean neutrosophic dombi fuzzy graphs with an application to MCDM,” Neutrosophic Sets and Systems, vol. 47, pp. 411–431, 2021.

40. D. Ajay and P. Chellamani, “Operations on pythagorean neutrosophic graphs,” AIP Conference Proceedings, vol. 2516, no. 1, pp. 200028, 2022.

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