Pythagorean Neutrosophic Planar Graphs with an Application in Decision-Making

P. Chellamani; D. Ajay; Mohammed Al-Shamiri; Rashad Ismail

doi:10.32604/cmc.2023.036321

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Pythagorean Neutrosophic Planar Graphs with an Application in Decision-Making

P. Chellamani^1,2,*, D. Ajay¹, Mohammed M. Al-Shamiri^3,4, Rashad Ismail^3,4

1 Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur, 635601, Tamilnadu, India
2 Department of Mathematics, St. Joseph’s College of Engineering, OMR, Chennai, 600119, Tamilnadu, India
3 Department of Mathematics, Faculty of Science and Arts, King Khalid University, Muhayl Assir, Saudi Arabia
4 Department of Mathematics and Computer, Faculty of Science, Ibb University, Ibb, Yemen

* Corresponding Author: P. Chellamani. Email: email

Computers, Materials & Continua 2023, 75(3), 4935-4953. https://doi.org/10.32604/cmc.2023.036321

Received 26 September 2022; Accepted 08 February 2023; Issue published 29 April 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 $PNMG.$

images

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 $(bc,.4,.6,.4)$ .

images

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 $(ab,.4,.4,.1),(bd,.5,.5,.1),(ad,.4,.4,.1).$

images

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 $PNP$ G 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 $PNDGG$ has 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.

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

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

images

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

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

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

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

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

images

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

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

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

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

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

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

images

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

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

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

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

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

images

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.

images

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

images

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.

images

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.

images

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.

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APA Style

Chellamani, P., Ajay, D., Al-Shamiri, M.M., Ismail, R. (2023). Pythagorean Neutrosophic Planar Graphs with an Application in Decision-Making. Computers, Materials & Continua, 75(3), 4935–4953. https://doi.org/10.32604/cmc.2023.036321

Vancouver Style

Chellamani P, Ajay D, Al-Shamiri MM, Ismail R. Pythagorean Neutrosophic Planar Graphs with an Application in Decision-Making. Comput Mater Contin. 2023;75(3):4935–4953. https://doi.org/10.32604/cmc.2023.036321

IEEE Style

P. Chellamani, D. Ajay, M. M. Al-Shamiri, and R. Ismail, “Pythagorean Neutrosophic Planar Graphs with an Application in Decision-Making,” Comput. Mater. Contin., vol. 75, no. 3, pp. 4935–4953, 2023. https://doi.org/10.32604/cmc.2023.036321

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Copyright © 2023 The Author(s). Published by Tech Science Press.
This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Pythagorean Neutrosophic Planar Graphs with an Application in Decision-Making

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