AdvancesinMathematics:ScientificJournal 9 (2020),no.8,6059–6070 ISSN:1857-8365(printed);1857-8438(electronic) https://doi.org/10.37418/amsj.9.8.74SpecialIssueonICMA-2020
FUZZYMAGICLABELLINGOFNEUTROSOPHICPATHANDSTARGRAPH
D.AJAY1 ANDP.CHELLAMANI
ABSTRACT.GraphLabellingistheassignmentoflabels(integers)totheedges orverticesortoboth.Fuzzylabellingofgraphsismoredetailedandcompatible whencomparedwiththeclassicalmodels.Inthispaper,fuzzymagicandbimagiclabellingofNeutrosophicpathgraphisexamined.Inadditiontothatthe existenceofmagicvalueofIntuitionisticandNeutrosophicstargraphisalso investigated.
1.INTRODUCTION
ThephenomenaofuncertaintyandvaguenessareknownasfuzzyandthelogicalapproachtothisgreatideawasintroducedbyLotfi.AZadeh,in1965[1]. Thisfuzzytheoryhasmorefunctionalrepresentationofvagueconceptsinnaturallanguages.Theconceptoffuzzyhasanextensiveapplicationinmathematicalfields.TheconceptofgraphwaspresentedbyEulerin1936.Graphtheory ismoreusefulinmodellingthefeaturesofsystemwithfinitecomponents.A wayofrepresentinginformationinvolvingrelationshipbetweenobjectsiscalled graph,wheretheverticesandedgesofagraphrepresentstheobjectsandtheir relationrespectively.Thegraphicalmodelsareusedtorepresenttelephonenetwork,railwaynetwork,communicationproblems,trafficnetwork,etc.
Ifthereisvaguenessinthematterofobjectsoritsrelationsorinboth,we areinneedofdesigningafuzzygraphmodel.Fuzzygraphwasfirstdefinedby Kauffmanin1973,andRosenfelddevelopedthetheoryoffuzzygraphtheory
1Correspondingauthor
2010 MathematicsSubjectClassification. 03E72,05C72,05C78.
Keywordsandphrases. Fuzzymagiclabelling,fuzzybi-magiclabelling,Neutrosophicstar graph,Neutrosophicpathgraph,Intuitionisticstargraph. 6059
in1975basedontherelations[2].Crispandfuzzygraphsaresimilarinstructure.Fuzzygraphhasmoreapplicationsindatamining,imagesegmentation, clustering,imagecapturing,networking,planningandscheduling.
GraphlabellingwasintroducedbyRosa[3].Agraphlabellingisthemapping thatcarriesasetofgraphelementsontoasetofnumberscalledlabels.There arenumerouslabellingingraphsandafewofthemaregraceful,cordialand meanlabelling.TheconceptoffuzzylabellingwasintroducedbyANagoorGani etal[4].Thenotionofbi-magiclabellingwasintroducedbyBaskerBabujee,in whichtherearetwomagicvalues.
Inthispaper,takingtheleadofmagiclabelling,wehaveinvestigatedfuzzy magic,bi-magiclabellingofNeutrosophicPathgraphandalsofuzzymagiclabellingofIntuitionisticandNeutrosophicStargraphs.
2.PRELIMINARIES
Afuzzygraph G =(σ,µ) isapairoffunctions σ : V → [0, 1] and µ : V → [0, 1], where ∀ u,v ∈ V , µ (u,v) ≤ σ (u) ∧ σ (v).Alabellingofagraphisan assignmentofvaluestotheverticesandedgesofagraph.ThegraphGissaid tobeafuzzylabellinggraphifitisbijectivesuchthatthemembershipvalueof edgesandverticesaredistinctand µ (u,v) <σ (u) ∧ σ (v) forall u,v ∈ V. Afuzzylabellinggraph G =(σ,µ) issaidtobeafuzzymagiclabellinggraph, ifthereexistanmsuchthat σ (x)+ σ (y)+ µ (xy)= m forall xy ∈ E and x,y ∈ V. AfuzzylabellinggraphGissaidtobeafuzzybi-magiclabellinggraph ifthereexist m1 and m2 suchthat σ (x)+ σ (y)+ µ (xy)= m1 or m2 forall xy ∈ Eandx,y ∈ V. AnIntuitionisticFuzzyGraphisoftheform G =(V,E) where V = {v1,v2,v3, ...,vn} suchthat µ1 : V → [0, 1] and γ1 : V → [0, 1] denotethedegreesof membershipandnon-membershipoftheelement vi ∈ V respectively,and 0 ≤ µ1 (vi)+ γ1 (vi) ≤ 1(1) forevery vi ∈ V (i =1, 2, 3,....n) ,E ⊆ V × V where µ2 : V × V → [0, 1] and γ2 : V × V → [0, 1] aresuchthat µ2 (vi,vj ) ≤ min[µ1 (vi) ,µ1 (vj )] , γ2 (vi,vj ) ≤ max[γ1 (vi) ,γ1 (vj )] 0 ≤ µ2 (vi,vj )+ γ2 (vi,vj ) ≤ 1 forevery (vi,vj ) ∈ E (i,j =1, 2, 3,......,n).
AnIntuitionisticfuzzylabellinggraphisanIntuitionisticfuzzymagiclabelling graphifthereexistsanMsuchthat M = {µ1 (vi)+ µ1 (vj )+ µ2 (vi,vj ) ,γ1 (vi)+ γ1 (vj )+ γ2 (vi,vj )}
AnIntuitionisticfuzzylabellinggraphisaIntuitionisticfuzzymagiclabelling graph ∃ M1,M2 suchthat M1 or M2 = {µ1 (vi)+ µ1 (vj )+ µ2 (vi,vj ) , γ1 (vi)+ γ1 (vj )+ γ2 (vi,vj )}. AneutrosophicFuzzyGraphis G =(V,E) where V = {v1,v2,v3,.......,vn} suchthat T1 : V → [0, 1], I1 : V → [0, 1] and F1 : V → [0, 1] denotethedegree oftruth-membership,indeterminacy-membershipandfalsity-membershipofthe element vi ∈ V respectively,and 0 ≤ T1 (vi)+ I1 (vi)+ F1 (vi) ≤ 3 forevery vi ∈ V (i =1,...n), E ⊆ V × V where T2 : V × V → [0, 1], I2 : V × V → [0, 1] and F2 : V × V → [0, 1] aresuchthat
T2 (vi,vj ) ≤ min[T1 (vi) ,T1 (vj )] , I2 (vi,vj ) ≤ min[I1 (vi) ,I1 (vj )] , F2 (vi,vj ) ≤ min[F1 (vi) ,F1 (vj )] 0 ≤ T2 (vi,vj )+ I2 (vi,vj )+ F2 (vi,vj ) ≤ 3 forevery (vi,vj ) ∈ E (i,j =1,...,n). ANeutrosophicfuzzylabellinggraphisaNeutrosophicfuzzymagiclabelling graphifthereexistsanMsuchthatMequals
{T1 (vi)+ T1 (vj )+ T2 (vi,vj ) , I1 (vi)+ I1 (vj )+ I2 (vi,vj ) , F1 (vi)+ F1 (vj )+ F2 (vi,vj )}.
3.FUZZY MAGICAND BI-MAGIC LABELLINGOF NEUTROSOPHIC PATH GRAPH
ANeutrosophicpathgraphsatisfiestheconditionsdefinedforaneutrosophic graph.Andinthissectionwearegoingtoinvestigatethemagicandbi-magic valueoftheneutrosophicpathgraph.
Theorem3.1. AnyNeutrosophicPathgraph Pn,n ≥ 2(n ∈ N ),admitsfuzzy magiclabelling.
Proof. Let n ∈ N,n ≥ 2, V = {v1,v2,v3,...,vn} and E = {ej = vj v(j+1) :1 ≤ j ≤ n 1} andPathgraphbe Pn =(V,E).
Forall j, (1 ≤ j ≤ n),thetruth,indeterminacyandfalsemembershipfunctions σ : V → [0, 1], ρ : V → [0, 1] µ : V → [0, 1] respectivelyfortheverticesare
definedasfollows: σ (vj )= 4n (j+1) 2(2n 1) ifjisodd 3n (j+1) 2(2n 1) ifjiseven
ρ (vj )= 4n (j+(n 1)) 2(2n 1) ifjisodd 3n (j+(n 1)) 2(2n 1) ifjiseven µ (vj )= 4n (j+(n 1)) 10(n 1) ifjisodd 3n (j+(n 1)) 10(n 1) ifjiseven
Themembershipofedges ∀j, (1 ≤ j ≤ n 1),aredefinedbythefunctions
α : E → [0, 1] by α (ej )= j 2n 1 , β : E → [0, 1] by β (ej )= j 2n 1 , γ : E → [0, 1] by γ (ej )= j 5(n 1) where ej = vj vj+1 ∈ E.Afuzzylabellinggraph,satisfies thecondition µ (ej ) <σ (vj ) ∧ σ (vj+1).
TruthValue:If j isoddthen σ (vj )= 4n (j+1) 2(2n 1) and j = j +1,becomeseven,then σ (vj+1)= 3n (j+1) 2(2n 1) Weknow 3n< 4n,andalso 3n (j +1) < 4n (j +1), ⇒ 3n (j +1) 2(2n 1) < 4n (j +1) 2(2n 1) , ⇒ σ (vj ) ∧ σ (vj+1)= 3n (j +1) 2(2n 1)
Themembershipofedgeis α (ej )= j 2n 1 and j 2n 1 < 3n (j +1) 2(2n 1) ⇒ α (ej ) <σ (vj ) ∧ σ (vj+1)
Themembershipvalueofneutrosophicgraphsatisfiesfuzzylabellingcondition. Wechecktheexistenceofmagicvalueofneutrosophicpathgraph.
ExistenceofMagicvalue: Wehave σ (vj )= 4n (j+1) 2(2n 1) and σ (vj+1)= 3n (j+1) 2(2n 1) .
Thus, M = 4n (j+1) 2(2n 1) + 3n (j+1) 2(2n 1) + j 2n 1 = 7n 2 2(2n 1)
IndeterminacyValue: Ifjisodd,wehave ρ (vj )= 4n (j+(n 1)) 2(2n 1) and ρ (vj+1)= 3n (j+(n 1)) 2(2n 1) . Weknow 3n< 4n,andalso 3n (j +(n 1)) < 4n (j +(n 1))
⇒
3n (j +(n 1)) 2(2n 1) < 4n (j +(n 1)) 2(2n 1)
⇒ ρ (vj ) ∧ ρ (vj+1)= 3n (j +(n 1)) 2(2n 1)
Themembershipofedgeis
β (ej )= j 2n 1 and j 2n 1 < 3n (j +(n 1)) 2(2n 1)
⇒ β (ej ) <ρ (vj ) ∧ ρ (vj+1) .
Themembershipvalueischeckedanditisverifiedthatneutrosophicgraphsatisfiesfuzzylabellingcondition.Nowwechecktheexistenceofmagicvalueof neutrosophicpathgraph.
ExistenceofMagicvalue: Weget ρ (vj )= 4n (j+(n 1)) 2(2n 1) and ρ (vj+1)= 3n (j+(n 1)) 2(2n 1) .
Thus, M = 4n (j+(n 1)) 2(2n 1) + 3n (j+(n 1)) 2(2n 1) + j 2n 1 = 5n+2 2(2n 1) .
FalseValue: Ifjisodd, µ (vj )= 4n (j+(n 1)) 10(n 1) and µ (vj+1)= 3n (j+(n 1)) 10(n 1) .
Weknow 3n< 4n,andalso 3n (j +(n 1)) < 4n (j +(n 1))
⇒ 3n (j +(n 1)) 10(n 1) < 4n (j +(n 1)) 10(n 1)
⇒ µ (vj ) ∧ µ (vj+1)= 3n (j +(n 1)) 10(n 1) .
Themembershipofedgeis
γ (ej )= j 5(n 1) and j 5(n 1) < 3n (j +(n 1)) 10(n 1)
⇒ γ (ej ) <µ (vj ) ∧ µ (vj+1)
Thustheneutrosophicgraphsatisfiesfuzzylabellingcondition.Nextwecheck theexistenceofmagicvalueofneutrosophicpathgraph.
ExistenceofMagicvalue: Wehave
µ (vj )= 4n (j +(n 1)) 10(n 1) and
µ (vj+1)= 3n (j +(n 1)) 10(n 1)
Thus, M = 4n (j+(n 1)) 10(n 1) + 3n (j+(n 1)) 10(n 1) + j 5(n 1) = 5n+2 10(n 1) .
ThereforetheMagicvalueofNeutrosophicpathgraphis
M = 7n 2 2(2n 1) , 5n+2 2(2n 1) , 5n+2 10(n 1) .
Example3.2.
ThemagicvalueoftheaboveNeutrosophicPathgraph P8 is (1 8, 1 4, 0 6) Theorem3.3. AnyNeutrosophicPathgraph Pn,n ≥ 3(n ∈ N ),admitsfuzzy bi-magiclabelling.
Proof. Let n ∈ N,n ≥ 3,V = {v1,v2,v3,...,vn} and E = {ej = vj vj+1 :1 ≤ j ≤ n 1} andpathgraphbe Pn =(V,E).
Forall j (1 ≤ j ≤ n),thetruth,indeterminacyandfalsemembershipforthe vertices σ : V → [0, 1], ρ : V → [0, 1], µ : V → [0, 1] aredefinedrespectivelyas follows:
σ (vj )= 4n (j+1) 2(2n 1) ifjisodd 3n (j+1) 2(2n 1) ifjiseven
ρ (vj )= 4n (j+(n 1)) 2(2n 1) ifjisodd 3n (j+(n 1)) 2(2n 1) ifjiseven
µ (vj )= 4n (j+(n 1)) 10(n 1) ifjisodd 3n (j+(n 1)) 10(n 1) ifjiseven
Thetruth,indeterminacyandfalsemembershipofedges ∀j, (1 ≤ j ≤ n 1), aredefinedbythefunction
α : E → [0, 1] byα (ej )= j 2n 4 j odd j 2n+5 j even
β : E → [0, 1] byβ (ej )= j 2n 4 j odd j 2n+3 j even
γ : E → [0, 1] byγ (ej )= j 5(n 4) j odd j 5(n+5) j even where ej = vj vj+1 ∈ E.
Afuzzylabellinggraph,satisfiesthecondition α (ej ) <σ (vj ) ∧ σ (vj+1).
TruthValue:If j isoddthen σ (vj )= 4n (j+1) 2(2n 1) and σ (vj+1)= 3n (j+1) 2(2n 1) .
Weknow 3n< 4n,andalso 3n (j +1) < 4n (j +1) ⇒ 3n (j +1) 2(2n 1) < 4n (j +1) 2(2n 1) ⇒ σ (vj ) ∧ σ (vj+1)= 3n (j +1) 2(2n 1)
Themembershipofedgeis α (ej )= j 2n 4 j 2n+5 , j 2n 4 < 3n (j+1) 2(2n 1) ⇒ j 2n+5 < 3n (j+1) 2(2n 1) ⇒ α (ej ) <σ (vj ) ∧ σ (vj+1).
Thusthetruthvalueofneutrosophicpathgraphsatisfiesfuzzylabellingcondition.Nowwechecktheexistenceofmagicvalueoftruthmembershipvalue.
ExistenceofBi-Magicvalue: Wehave σ (vj )= 4n (j+1) 2(2n 1) and σ (vj+1)= 3n (j+1) 2(2n 1)
Subcasea:whenjisodd, α (ej )= j 2n 4 .Thus, M = 4n (j+1) 2(2n 1) + 3n (j+1) 2(2n 1) + j 2n 4 M = 7n 4 2(2n 1) + 1 2n 4
Subcaseb:whenjiseven, α (ej )= j 2n+5 .Thus, M = 4n (j+1) 2(2n 1) + 3n (j+1) 2(2n 1) + j 2n+5 M = 7n 6 2(2n 1) + 1 2n+5 .
IndeterminacyValue: Ifjisodd,wehave ρ (vj )= 4n (j+(n 1)) 2(2n 1) and ρ (vj+1)= 3n (j+(n 1)) 2(2n 1) .Weknow 3n< 4n,andalso 3n (j +(n 1)) < 4n (j +(n 1)), ⇒ 3n (j+(n 1)) 2(2n 1) < 4n (j+(n 1)) 2(2n 1) , ⇒ ρ (vj ) ∧ ρ (vj+1)= 3n (j+(n 1)) 2(2n 1) .
Themembershipofedgeis β (ej )= j 2n 4 j 2n+3 , j 2n 4 < 3n (j+(n 1)) 2(2n 1) ⇒ j 2n+3 < 3n (j+(n 1)) 2(2n 1) ⇒ β (ej ) <ρ (vj ) ∧ ρ (vj+1).
Itisverifiedthatneutrosophicgraphsatisfiesfuzzylabellingcondition.Nowwe
checktheexistenceofmagicvalueofneutrosophicpathgraph.
ExistenceofBi-Magicvalue:
Wehave ρ (vj )= 4n (j+(n 1)) 2(2n 1) and ρ (vj+1)= 3n (j+(n 1)) 2(2n 1)
Subcasea:whenjisodd, β (ej )= j 2n 4 .Thus, M = 4n (j+(n 1)) 2(2n 1) + 3n (j+(n 1)) 2(2n 1) + j 2n 4 . M = 5n 2(2n 1) + 1 2n 4 .
Subcaseb:whenjiseven, β (ej )= j 2n+3 .Thus, M = 4n (j+(n 1)) 2(2n 1) + 3n (j+(n 1)) 2(2n 1) + j 2n+3 . M = 5n 2 2(2n 1) + 1 2n+3 .
FalseValue: Ifjisodd, µ (vj )= 4n (j+(n 1)) 10(n 1) and µ (vj+1)= 3n (j+(n 1)) 10(n 1) Weknow 3n< 4n,andalso 3n (j +(n 1)) < 4n (j +(n 1)) ⇒ 3n (j+(n 1)) 10(n 1) < 4n (j+(n 1)) 10(n 1) ⇒ µ (vj ) ∧ µ (vj+1)= 3n (j+(n 1)) 10(n 1) .
Themembershipofedgeis γ (ej )= j 5(n 4) j 5(n+5) j 5(n 4) < 3n (j +(n 1)) 10(n 1) ⇒ j 5(n +5) < 3n (j +(n 1)) 10(n 1)
⇒ γ (ej ) <µ (vj ) ∧ µ (vj+1).Itisverifiedthatneutrosophicgraphsatisfiesfuzzy labellingcondition.Nowwechecktheexistenceofmagicvalueofneutrosophic pathgraph.
ExistenceofBi-Magicvalue:
Wehave µ (vj )= 4n (j+(n 1)) 10(n 1) and µ (vj+1)= 3n (j+(n 1)) 10(n 1)
Subcasea:whenjisodd, γ (ej )= j 5(n 4) .Thus, M = 4n (j+(n 1)) 10(n 1) + 3n (j+(n 1)) 10(n 1) + j 5(n 4) . M = 5n 10(n 1) + 1 5(n 4) .
Subcaseb:whenjiseven, γ (ej )= j 5(n+5) .Thus, M = 4n (j+(n 1)) 10(n 1) + 3n (j+(n 1)) 10(n 1) + j 5(n+5) . M = 5n 2 10(n 1) + 1 5(n+5) .
Thusthebi-magicvalueofNeutrosophicpathgraphis
M1,M2 = 7n 4 2(2n 1) + 1 2n 4 , 5n 2(2n 1) + 1 2n 4 , 5n 10(n 1) + 1 5(n 4) ifjisodd 7n 6 2(2n 1) + 1 2n+5 , 5n 2 2(2n 1) + 1 2n+3 , 5n 2 10(n 1) + 1 5(n+5) ifjiseven
Example3.4.
Thebi-magicvalueoftheaboveNeutrosophicPathgraph P11 is (1 8, 1 4, 0 6) and (1 7, 1 3, 0 5).
4.FUZZY MAGIC LABELLINGOF INTUITIONISTIC &NEUTROSOPHICSTAR GRAPH
Nowwearegoingtofindtheexistenceofmagicvalueforintuitionisticand neutrosophicstargraph.
Theorem4.1. AnyIntuitionisticStargraph Sn, n ≥ 2 admitsfuzzymagiclabelling. Proof. Let n ∈ N,n ≥ 2, V = {v1,v2,v3,...,vn} and E = {ej = vj vj+1 :1 ≤ j ≤ n 1} andPathgraphbe Sn =(V,E).
Forall j, (1 ≤ j ≤ n),themembershipvalueandthenon-membershipvalue forthevertices σ : V → [0, 1], ρ : V → [0, 1] aredefinedrespectivelyasfollows: σ (vj )= 2n (j 1) 2(2n 1) and ρ (vj )= 2n (j 1) 3(2n 1) .
Themembershipandnon-membershipvaluesofedges ∀j, (1 ≤ j ≤ n 1),are definedbythefunctions α : E → [0, 1] by α (ej )= j 2(2n 1) and β : E → [0, 1] by β (ej )= j 3(2n 1) where ej = vj vj+1 ∈ E
Clearly,thefuzzylabellingconditionissatisfiedforthedefinedmembership andnon-membershipvalues.Thusnowthecruxofthetheoremistheexistence ofMagicvalueofIntuitionisticstargraph Sn.
ExistenceofMagicvalueformembershipvalue:
Instargraph,alltheverticesareincidentedwiththevertex v1 andthuswehave σ (v1)= 2n 2(2n 1) and σ (vj )= 2n (j 1) 2(2n 1) .Thus, M = 2n 2(2n 1) + 2n (j 1) 2(2n 1) + j 2(2n 1) = 4n+1 2(2n 1) .
ExistenceofMagicvaluefornon-membershipvalue:
Instargraphalltheverticesareincidentedwiththevertex v1 andthuswe have σ (v1)= 2n 3(2n 1) and σ (vj )= 2n (j 1) 3(2n 1) .Thus, M = 2n 3(2n 1) + 2n (j 1) 3(2n 1) + j 3(2n 1) = 4n+1 3(2n 1) .ThereforethemagicvalueoftheIntuitionisticstargraphis M = 4n+1 2(2n 1) , 4n+1 3(2n 1) .
Example4.2. ThemagicvalueoftheaboveIntuitionisticStargraph S9 is (1.1, 0.7). Theorem4.3. AnyNeutrosophicstargraph Sn, n ≥ 2 hasfuzzymagiclabelling. Proof. Let n ∈ N,n ≥ 2, V = {v1,v2,v3,...,vn} and E = {ej = vj vj+1 :1 ≤ j ≤ n 1} andPathgraphbe Sn =(V,E)
Forall j (1 ≤ j ≤ n),thetruth,indeterminacyandfalsemembershipforthe vertices σ : V → [0, 1], ρ : V → [0, 1], µ : V → [0, 1] aredefinedasfollows: σ (vj )= 4n (j+(n 1)) 2(2n+1) ,ρ (vj )= 2n (j 1) 2(2n 1) ,µ (vj )= 2n (j 1) 3(2n 1)
Thetruth,indeterminacyandfalsemembershipofedges ∀j, (1 ≤ j ≤ n 1), aredefinedbythefunctions α : E → [0, 1] by α (ej )= j 2(2n+1) , β : E → [0, 1] by β (ej )= j 2(2n 1) , γ : E → [0, 1] by γ (ej )= j 3(2n 1) respectively,where ej = vj vj+1 ∈ E.
Clearly,thefuzzylabellingconditionissatisfiedforthetruth,indeterminacy andfalsemembershipdefined.ThusnowthecruxofthetheoremistheexistenceofMagicvalueofNeutrosophicstargraph Sn ExistenceofMagicvaluefortruthvalue: Instargraph,allverticesareincidentedwiththevertex v1 andthuswehave σ (v1)= 3n 2(2n+1) and σ (vj )=
4n (j+(n 1)) 2(2n+1) .Thus, M = 3n 2(2n+1) + 4n (j+(n 1)) 2(2n+1) + j 2(2n+1) = 6n+1 2(2n+1)
ExistenceofMagicvalueforindeterminacyvalue: Alltheverticesareincidentedwiththevertex v1 andthuswehave σ (v1)= 2n 2(2n 1) and σ (vj )= 2n (j 1) 2(2n 1) . Thus, M = 2n 2(2n 1) + 2n (j 1) 2(2n 1) + j 2(2n 1) = 4n+1 2(2n 1)
ExistenceofMagicvalueforfalsevalue: Alltheverticesareincidentedwith thevertex v1,thuswehave σ (v1)= 2n 3(2n 1) and σ (vj )= 2n (j 1) 3(2n 1) .Thus, M = 2n 3(2n 1) + 2n (j 1) 3(2n 1) + j 3(2n 1) = 4n+1 3(2n 1) .ThereforethemagicvalueofNeutrosophic stargraphis M = 6n+1 2(2n+1) , 4n+1 2(2n 1) , 4n+1 3(2n 1) .
Example4.4.
ThemagicvalueoftheaboveNeutrosophicStargraph S8 is (1 4, 1 1, 0 7)
5.CONCLUSION
Inthispaper,theexistenceofmagicvalueforNeutrosophicpathgraphand thebi-magicvalueofNeutrosophicpathgraphareinvestigated.Inaddition tothatmagicvaluesofIntuitionisticandNeutrosophicstargraphshavebeen found.
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DEPARTMENTOF MATHEMATICS,
SACRED HEART COLLEGE (AUTONOMOUS), TIRUPATTUR ,VELLORE,TAMIL NADU,INDIA Emailaddress:
DEPARTMENTOF MATHEMATICS,
SACRED HEART COLLEGE (AUTONOMOUS), TIRUPATTUR ,VELLORE,TAMIL NADU,INDIA Emailaddress: