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Interval-valuedneutrosophiccompetitiongraphs

MuhammadAkramandMaryamNasir

Received19April2017; Revised2May2017; Accepted10May2017

Abstract. Weﬁrstintroducetheconceptofinterval-valuedneutrosophiccompetitiongraphs.Wethendiscusscertaintypes,including kcompetitioninterval-valuedneutrosophicgraphs, p-competitionintervalvaluedneutrosophicgraphsand m-stepinterval-valuedneutrosophiccompetitiongraphs.Moreover,wepresenttheconceptof m-stepintervalvaluedneutrosophicneighbourhoodgraphs.

2010AMSClassiﬁcation: 03E72,05C72,05C78,05C99

Keywords: Interval-valuedneutrosophicdigraphs,Interval-valuedneutrosophic competitiongraphs.

CorrespondingAuthor: MuhammadAkram(m.akram@pucit.edu.pk)

1. Introduction

In1975,Zadeh[35]introducedthenotionofinterval-valuedfuzzysetsasan extensionoffuzzysets[34]inwhichthevaluesofthemembershipdegreesareintervalsofnumbersinsteadofthenumbers.Interval-valuedfuzzysetsprovideamore adequatedescriptionofuncertaintythantraditionalfuzzysets.Itisthereforeimportanttouseinterval-valuedfuzzysetsinapplications,suchasfuzzycontrol.One ofthecomputationallymostintensivepartoffuzzycontrolisdefuzziﬁcation[19]. Atanassov[12]proposedtheextendedformoffuzzysettheorybyaddinganew component,called,intuitionisticfuzzysets.Smarandache[26, 27]introducedtheconcept ofneutrosophicsetsbycombiningthenon-standardanalysis.Inneutrosophicset, themembershipvalueisassociatedwiththreecomponents:truth-membership(t), indeterminacy-membership(i)andfalsity-membership(f ),inwhicheachmembershipvalueisarealstandardornon-standardsubsetofthenon-standardunitinterval ]0 , 1+[andthereisnorestrictionontheirsum.Smarandache[28]andWangetal. [29]presentedthenotionofsingle-valuedneutrosophicsetstoapply neutrosophic setsinreallifeproblemsmoreconveniently.Insingle-valuedneutrosophicsets,three componentsareindependentandtheirvaluesaretakenfromthestandardunitinterval[0, 1].Wangetal.[30]presentedtheconceptofinterval-valuedneutrosophic

MuhammadAkrametal./Ann.FuzzyMath.Inform. 14 (2017),No.1,99–120 sets,whichismorepreciseandmoreﬂexiblethanthesingle-valuedneutrosophicset. Aninterval-valuedneutrosophicsetisageneralizationoftheconceptofsingle-valued neutrosophicset,inwhichthreemembership(t,i,f )functionsareindependent,and theirvaluesbelongtotheunitinterval[0, 1].

Kauffman[18]gavethedefinitionofafuzzygraph.Fuzzygraphswerenarrated byRosenfeld[22].Afterthat,someremarksonfuzzygraphswererepresentedby Bhattacharya[13].Heshowedthatalltheconceptsoncrispgraphtheorydonot havesimilaritiesinfuzzygraphs.Wu[32]discussedfuzzydigraphs.Theconceptof fuzzy k-competitiongraphsand p-competitionfuzzygraphswasfirstdevelopedby SamantaandPalin[23],itwasfurtherstudiedin[11, 21, 25].Samantaetal.[24] introducedthegeneralizationoffuzzycompetitiongraphs,called m-stepfuzzycompetitiongraphs.Samantaetal.[24]alsointroducedtheconceptsoffuzzy m-step neighbourhoodgraphs,fuzzyeconomiccompetitiongraphs,and m-stepeconomic competitiongraphs.Theconceptsofbipolarfuzzycompetitiongraphsandintuitionisticfuzzycompetitiongraphsarediscussedin[21, 25].HongmeiandLianhua [16],gavedefinitionofinterval-valuedfuzzygraphs.Akrametal.[1, 2, 3, 4]have introducedseveralconceptsoninterval-valuedfuzzygraphsandinterval-valuedneutrosophicgraphs.AkramandShahzadi[6]introducedthenotionofneutrosophic softgraphswithapplications.Akram[7]introducedthenotionofsingle-valuedneutrosophicplanargraphs.AkramandShahzadi[8]studiedpropertiesofsingle-valued neutrosophicgraphsbylevelgraphs.Recently,AkramandNasir[5]havediscussed someconceptsofinterval-valuedneutrosophicgraphs.Inthispaper,wefirstintroducetheconceptofinterval-valuedneutrosophiccompetition graphs.Wethen discusscertaintypes,including k-competitioninterval-valuedneutrosophicgraphs, p-competitioninterval-valuedneutrosophicgraphsand m-stepinterval-valuedneutrosophiccompetitiongraphs.Moreover,wepresenttheconceptof m-stepintervalvaluedneutrosophicneighbourhoodgraphs.

Wehaveusedstandarddeﬁnitionsandterminologiesinthispaper.Forother notations,terminologiesandapplicationsnotmentionedinthepaper,thereaders arereferredto[6, 9, 10, 13, 14, 15, 17, 20, 26, 33, 36].

2. Interval-ValuedNeutrosophicCompetitionGraphs

Deﬁnition2.1 ([35]). Theinterval-valuedfuzzyset A in X isdeﬁnedby A = {(s, [tl A(s),tu A(s)]): s ∈ X}, where, tl A(s)and tu A(s)arefuzzysubsetsof X suchthat tl A(s) ≤ tu A(s)forall x ∈ X Aninterval-valuedfuzzyrelationon X isaninterval-valuedfuzzyset B in X × X

Deﬁnition2.2 ([30, 31]). Theinterval-valuedneutrosophicset(IVN-set) A in X isdeﬁnedby A = {(s, [tl A(s),tu A(s)], [il A(s),iu A(s)], [f l A(s),f u A(s)]): s ∈ X}, where, tl A(s), tu A(s), il A(s), iu A(s), f l A(s),and f u A(s)areneutrosophicsubsetsof X suchthat tl A(s) ≤ tu A(s), il A(s) ≤ iu A(s)and f l A(s) ≤ f u A(s)forall s ∈ X.Anintervalvaluedneutrosophicrelation(IVN-relation)on X isaninterval-valuedneutrosophic set B in X × X 100

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Deﬁnition2.3 ([5]). Aninterval-valuedneutrosophicdigraph(IVN-digraph)ona non-emptyset X isapair G =(A, −→ B ),(inshort, G),where A =([tl A,tu A], [il A,iu A],[f l A, f u A])isanIVN-seton X and B =([tl B ,tu B ], [il B,iu B ],[f l B ,f u B ])isanIVN-relationon X, suchthat:

(i) tl B −−−→ (s,w) ≤ tl A(s) ∧ tl A(w), tu B −−−→ (s,w) ≤ tu A(s) ∧ tu A(w), (ii) il B −−−→ (s,w) ≤ il A(s) ∧ il A(w), iu B −−−→ (s,w) ≤ iu A(s) ∧ iu A(w), (iii) f l B −−−→ (s,w) ≤ f l A(s) ∧ f l A(w), f u B −−−→ (s,w) ≤ f u A(s) ∧ f u A(w),forall s,w ∈ X.

Example2.4. WeconstructanIVN-digraph G =(A, −→ B )on X = {a,b,c} asshown inFig. 1 a ([0 . 2, 0. 4], [0 . 3, 0. 5], [0 . 6, 0. 7])

b([0 .6 ,08] ,[0 .3 ,0 .8] ,[02 ,0 .9])

([0.1,02],[0.2,03],[01,06]) ([0 .1 ,0 . 2] ,[0 .1 ,0 . 3] ,[02 ,0 . 6]) ([0 . 1, 0. 2] , [0 2, 0 3] , [0 . 2, 0. 5])Figure1. IVN-digraph

c([0 1, 0 2], [0 2, 0 4], [0 3, 0 7])

Deﬁnition2.5. Let −→ G beanIVN-digraphtheninterval-valuedneutrosophicoutneighbourhoods(IVN-out-neighbourhoods)ofavertex s isanIVN-set

N+(s)=(X + s ,[t(l)+ s , t(u)+ s ],[i(l)+ s , i(u)+ s ],[f (l)+ s , t(u)+ s ]), where X + s = {w|[tl B −−−→ (s,w) > 0, tu B −−−→ (s,w) > 0],[il B −−−→ (s,w) > 0,iu B −−−→ (s,w) > 0],[f l B −−−→ (s,w) > 0, f u B −−−→ (s,w) > 0]}, suchthat t(l)+ s : X + s → [0, 1],definedby t(l)+ s (w)= tl B −−−→ (s,w), t(u)+ s : X + s → [0, 1], definedby t(u)+ s (w)= tu B −−−→ (s,w), i(l)+ s : X + s → [0, 1],definedby i(l)+ s (w)= il B −−−→ (s,w), i(u)+ s : X + s → [0, 1],definedby i(u)+ s (w)= iu B −−−→ (s,w), f (l)+ s : X + s → [0, 1],definedby f (l)+ s (w)= f l B −−−→ (s,w), f (u)+ s : X + s → [0, 1],definedby f (u)+ s (w)= f u B −−−→ (s,w).

Deﬁnition2.6. Let −→ G beanIVN-digraphtheninterval-valuedneutrosophicinneighbourhoods(IVN-in-neighbourhoods)ofavertex s isanIVN-set

N (s)=(Xs ,[t(l) s , t(u) s ],[i(l) s , i(u) s ],[f (l) s , t(u) s ]), where Xs = {w|[tl B −−−→ (w,s) > 0, tu B −−−→ (w,s) > 0],[il B −−−→ (w,s) > 0,iu B −−−→ (w,s) > 0],[f l B −−−→ (w,s) > 0, f u B −−−→ (w,s) > 0]}, 101

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suchthat t(l) s : Xs → [0, 1],definedby t(l) s (w)= tl B −−−→ (w,s), t(u) s : Xs → [0, 1], definedby t(u) s (w)= tu B −−−→ (w,s), i(l) s : Xs → [0, 1],definedby i(l) s (w)= il B −−−→ (w,s), i(u) s : Xs → [0, 1],definedby i(u) s (w)= iu B −−−→ (w,s), f (l) s : Xs → [0, 1],definedby f (l) s (w)= f l B −−−→ (w,s), f (u) s : Xs → [0, 1],definedby f (u) s (w)= f u B −−−→ (w,s).

Example2.7. ConsideranIVN-digraph G =(A, −→ B )on X = {a,b,c} asshownin Fig. 2. a([0 .2 ,0 .4] ,[0 .3 ,0 .5] ,[0 .6 ,0 .7]) b ([0 6 , 0 8] , [0 3 , 0 8] , [0 . 2 , 0 9])

c ([0 1 , 0 2] , [0 2 , 0 4] , [0 3 , 0 7])

([0 .1 ,0 . 2] ,[0 .2 ,0 . 3] ,[0 .1 ,0 . 6]) ([0 . 1, 0. 2], [0 . 1, 0. 3], [0 . 2, 0. 6]) ([0 1, 0 2], [0 2, 0 3], [0 2, 0 5])

Figure2. IVN-digraph

WehaveTable 1 andTable 2 representinginterval-valuedneutrosophicoutand in-neighbourhoods,respectively.

Table1. IVN-out-neighbourhoods s N+(s) a {(b,[0.1,0.2],[0.2,0.3],[0.1,0.6]),(c,[0.1,0.2],[0.1,0.3],[0.2,0.6])} b ∅ c {(b,[0.1,0.2],[0.2,0.3],[0.2,0.5])}

Table2. IVN-in-neighbourhoods s N (s) a ∅ b {(a,[0.1,0.2],[0.2,0.3],[0.1,0.6]),(c,[0.1,0.2],[0.2,0.3],[0.2,0.5])} c {(a,[0.1,0.2],[0.1,0.3],[0.2,0.6])}

Deﬁnition2.8. TheheightofIVN-set A =(s, [tl A,tu A], [il A,iu A], [f l A,f u A])inuniverse ofdiscourse X isdeﬁnedas:forall s ∈ X, h(A)=([hl 1(A),hu 1 (A)], [hl 2(A),hu 2 (A)], [hl 3(A),hu 3 (A)]), =([sup s∈X tl A(s), sup s∈X tu A(s)], [sup s∈X il A(s), sup s∈X iu A(s)], [inf s∈X f l A(s), inf s∈X f u A(s)]) 102

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Deﬁnition2.9. Aninterval-valuedneutrosophiccompetitiongraph(IVNC-graph) ofaninterval-valuedneutrosophicgraph(IVN-graph) −→ G =(A, −→ B )isanundirected IVN-graph C −−→ (G)=(A,W )whichhasthesamevertexsetasin −→ G andthereisan edgebetweentwovertices s and w ifandonlyif N+(s) ∩ N+(w) = ∅.Thetruthmembership,indeterminacy-membershipandfalsity-membershipvaluesoftheedge (s,w)aredeﬁnedas:forall s,w ∈ X,

(i) tl W (s,w)=(tl A(s) ∧ tl A(w))hl 1 (N+(s) ∩ N+(w), tu W (s,w)=(tu A(s) ∧ tu A(w))hu 1 (N+(s) ∩ N+(w),

(ii) il W (s,w)=(il A(s) ∧ il A(w))hl 2(N+(s) ∩ N+(w), iu W (s,w)=(iu A(s) ∧ iu A(w))hu 2 (N+(s) ∩ N+(w), (iii) f l W (s,w)=(f l A(s) ∧ f l A(w))hl 3 (N+(s) ∩ N+(w), f u W (s,w)=(f u A(s) ∧ f u A(w))hu 3 (N+(s) ∩ N+(w).

Example2.10. ConsideranIVN-digraph G =(A, −→ B )on X = {a,b,c} asshown inFig. 3

([0 .1 ,0 .2] ,[0 .2 ,0 .3] ,[0 .1 ,0 .6]) ([0 . 1, 0. 2], [0 . 1, 0. 3], [0 . 2, 0. 6]) ([0 1,0 2],[0 2,0 3],[0.2,0 5])

a([0 .2,04] ,[0 .3,05] ,[0 .6,07]) b ([0 6, 0 . 8] , [0 . 3, 0 8] , [0 2, 0 9])

c([0 1, 0 2], [0 2, 0 4], [0 3, 0 7])

Figure3. IVN-digraph

WehaveTable 3 andTable 4 representinginterval-valuedneutrosophicoutand in-neighbourhoods,respectively.

Table3. IVN-out-neighbourhoods s N+(s) a {(b,[0.1,0.2],[0.2,0.3],[0.1,0.6]),(c,[0.1,0.2],[0.1,0.3],[0.2,0.6])} b ∅ c {(b,[0.1,0.2],[0.2,0.3],[0.2,0.5])}

Table4. IVN-in-neighbourhoods

s N (s) a ∅ b {(a,[0.1,0.2],[0.2,0.3],[0.1,0.6]),(c,[0.1,0.2],[0.2,0.3],[0.2,0.5])} c {(a,[0.1,0.2],[0.1,0.3],[0.2,0.6])}

103

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ThenIVNC-graphofFig. 3 isshowninFig. 4. a([0 . 2, 0. 4], [0 . 3, 0. 5], [0 . 6, 0. 7])

([0 01, 0 04], [0 04, 0 12], [0 06, 0 42])

b([0 6, 0 8], [0 3, 0 8], [0 2, 0 9]) c([0 .1 ,0 . 2] ,[0 .2 ,0 . 4] ,[0 .3 ,0 . 7])

Figure4. IVNC-graph

Deﬁnition2.11. ConsideranIVN-graph G =(A, B),where A =([Al 1, Au 1 ],[Al 2, Au 2 ],[Al 3, Au 3 )]and B =([Bl 1, Bu 1 ],[Bl 2, Bu 2 ],[Bl 3, Bu 3 )].then,anedge(s, w), s, w ∈ X iscalledindependentstrong,if 1 2 [Al 1(s) ∧ Al 1(w)] <Bl 1(s,w), 1 2 [Au 1 (s) ∧ Au 1 (w)] <Bu 1 (s,w), 1 2 [Al 2(s) ∧ Al 2(w)] <Bl 2(s,w), 1 2 [Au 2 (s) ∧ Au 2 (w)] <Bu 2 (s,w), 1 2 [Al 3(s) ∧ Al 3(w)] >Bl 3(s,w), 1 2 [Au 3 (s) ∧ Au 3 (w)] >Bu 3 (s,w)

Otherwise,itiscalledweak.

Westatethefollowingtheoremswithouttheirproofs.

Theorem2.12. Suppose −→ G isanIVN-digraph.If N+(s) ∩ N+(w) containsonly oneelementof −→ G ,thentheedge (s, w) of C(−→ G ) isindependentstrongifandonlyif

|[N+(s) ∩ N+(w)]|tl > 0.5, |[N+(s) ∩ N+(w)]|tu > 0.5, |[N+(s) ∩ N+(w)]|il > 0.5, |[N+(s) ∩ N+(w)]|iu > 0.5, |[N+(s) ∩ N+(w)]|f l < 0 5, |[N+(s) ∩ N+(w)]|f u < 0 5

Theorem2.13. IfalltheedgesofanIVN-digraph −→ G areindependentstrong,then Bl 1(s,w)

(Al 1(s) ∧ Al 1(w))2 > 0.5, Bu 1 (s,w) (Au 1 (s) ∧ Au 1 (w))2 > 0.5, Bl 2(s,w) (Al 2(s) ∧ Al 2(w))2 > 0 5, Bu 2 (s,w) (Au 2 (s) ∧ Au 2 (w))2 > 0 5, Bl 3(s,w) (Al 3(s) ∧ Al 3(w))2 < 0 5, Bu 3 (s,w) (Au 3 (s) ∧ Au 3 (w))2 < 0 5,

foralledges (s, w) in C(−→ G )

Deﬁnition2.14. Theinterval-valuedneutrosophicopen-neighbourhood(IVN-openneighbourhood)ofavertex s ofanIVN-graph G =(A,B)isIVN-set N(s)=(Xs, [tl s, tu s ],[il s, iu s ],[f l s, f u s ]),where 104

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Xs = {w|[Bl 1(s,w) > 0, Bu 1 (s,w) > 0],[Bl 2(s,w) > 0, Bu 2 (s,w) > 0],[Bl 3(s,w) > 0, Bu 3 (s,w) > 0]}, and tl s : Xs → [0, 1]definedby tl s(w)= Bl 1(s, w), tu s : Xs → [0, 1]definedby tu s (w)= Bu 1 (s, w), il s : Xs → [0, 1]definedby il s(w)= Bl 2(s, w), iu s : Xs → [0, 1] definedby iu s (w)= Bu 2 (s, w), f l s : Xs → [0, 1]definedby f l s(w)= Bl 3(s, w), f u s : Xs → [0, 1]definedby f u s (w)= Bu 3 (s, w).Foreveryvertex s ∈ X,theintervalvaluedneutrosophicsingletonset, ˘ As =(s,[Al′ 1 , Au′ 1 ],[Al′ 2 , Au′ 2 ],[Al′ 3 , Au′ 3 )suchthat: Al′ 1 : {s}→ [0, 1], Au′ 1 : {s}→ [0, 1], Al′ 2 : {s}→ [0, 1], Au′ 2 : {s}→ [0, 1], Al′ 3 : {s}→ [0, 1], Au′ 3 : {s}→ [0, 1],definedby Al′ 1 (s)= Al 1(s), Au′ 1 (s)= Au 1 (s), Al′ 2 (s)= Al 2(s), Au′ 2 (s)= Au 2 (s), Al′ 3 (s)= Al 3(s)and Au′ 3 (s)= Au 3 (s),respectively.Theintervalvaluedneutrosophicclosed-neighbourhood(IVN-closed-neighbourhood)ofavertex s is N[s]= N(s) ∪ As

Deﬁnition2.15. Suppose G =(A, B)isanIVN-graph.Interval-valuedneutrosophicopen-neighbourhoodgraph(IVN-open-neighbourhood-graph)of G isan IVN-graph N(G)=(A, B′)whichhasthesameIVN-setofverticesin G andhasan interval-valuedneutrosophicedgebetweentwovertices s, w ∈ X in N(G)ifandonly if N(s) ∩ N(w)isanon-emptyIVN-setin G.Thetruth-membership,indeterminacymembership,falsity-membershipvaluesoftheedge(s, w)aregivenby:

Bl′ 1 (s,w)=[Al 1(s) ∧ Al 1(w)]hl 1(N(s) ∩ N(w)),

Bl′ 2 (s,w)=[Al 2(s) ∧ Al 2(w)]hl 2(N(s) ∩ N(w)),

Bl′ 3 (s,w)=[Al 3(s) ∧ Al 3(w)]hl 3(N(s) ∩ N(w)),

Bu′ 1 (s,w)=[Au 1 (s) ∧ Au 1 (w)]hu 1 (N(s) ∩ N(w)),

Bu′ 2 (s,w)=[Au 2 (s) ∧ Au 2 (w)]hu 2 (N(s) ∩ N(w)),

Bu′ 3 (s,w)=[Au 3 (s) ∧ Au 3 (w)]hu 3 (N(s) ∩ N(w)), respectively

Deﬁnition2.16. Suppose G =(A, B)isanIVN-graph.Interval-valuedneutrosophicclosed-neighbourhoodgraph(IVN-closed-neighbourhood-graph)of G isan IVN-graph N(G)=(A, B′)whichhasthesameIVN-setofverticesin G andhasan interval-valuedneutrosophicedgebetweentwovertices s, w ∈ X in N[G]ifandonly if N[s] ∩ N[w]isanon-emptyIVN-setin G.Thetruth-membership,indeterminacymembership,falsity-membershipvaluesoftheedge(s, w)aregivenby:

Bl′ 1 (s,w)=[Al 1(s) ∧ Al 1(w)]hl 1(N[s] ∩ N[w]),

Bl′ 2 (s,w)=[Al 2(s) ∧ Al 2(w)]hl 2(N[s] ∩ N[w]),

Bl′ 3 (s,w)=[Al 3(s) ∧ Al 3(w)]hl 3(N[s] ∩ N[w]),

Bu′ 1 (s,w)=[Au 1 (s) ∧ Au 1 (w)]hu 1 (N[s] ∩ N[w]),

Bu′ 2 (s,w)=[Au 2 (s) ∧ Au 2 (w)]hu 2 (N[s] ∩ N[w]),

Bu′ 3 (s,w)=[Au 3 (s) ∧ Au 3 (w)]hu 3 (N[s] ∩ N[w]), respectively

Wenowdiscussthemethodofconstructionofinterval-valuedneutrospohiccompetitiongraphoftheCartesianproductofIVN-digraphinfollowingtheoremwhich canbeproofusingsimilarmethodasusedin[21],henceweomititsproof. 105

MuhammadAkrametal./Ann.FuzzyMath.Inform. 14 (2017),No.1,99–120 Theorem2.17. Let C(−→ G1)=(A1, B1) and C(−→ G2)=(A2, B2) betwoIVNC-graphs ofIVN-digraphs −→ G1 =(A1, −→ L1) and −→ G2 =(A2, −→ L2),respectively.Then C(−→ G1✷ −→ G2)= GC(−→ G1)∗ ✷C(−→ G2)∗ ∪ G✷ ,where GC(−→ G1)∗ ✷C(−→ G2)∗ isanIVN-graphonthecrispgraph (X1 × X2,EC(−→ G1)∗ ✷EC(−→ G2)∗ ), C(−→ G1)∗ and C(−→ G2)∗ arethecrispcompetitiongraphs of −→ G1 and −→ G2,respectively. G✷ isanIVN-graphon (X1 × X2,E ✷) suchthat: (1) E ✷ = {(s1,s2)(w1,w2): w1 ∈ N (s1)∗,w2 ∈ N+(s2)∗} EC(−→ G1 )∗ ✷EC(−→ G2 )∗ = {(s1,s2)(s1,w2): s1 ∈ X1,s2w2 ∈ EC(−→ G2)∗ } ∪{(s1,s2)(w1,s2): s2 ∈ X2,s1w1 ∈ EC(−→ G1 )∗ } (2) tl A1 ✷A2 = tl A1 (s1) ∧ tl A2 (s2), il A1✷A2 = il A1 (s1) ∧ il A2 (s2), f l A1✷A2 = f l A1 (s1) ∧ f l A2 (s2), tu A1 ✷A2 = tu A1 (s1) ∧ tu A2 (s2), iu A1✷A2 = iu A1 (s1) ∧ iu A2 (s2), f u A1✷A2 = f u A1 (s1) ∧ f u A2 (s2).

(3) tl B ((s1,s2)(s1,w2))=[tl A1 (s1)∧tl A2 (s2)∧tl A2 (w2)]×∨a2 {tl A1 (s1)∧tl −→ L2 (s2a2)∧ tl −→ L2 (w2a2)}, (s1,s2)(s1,w2) ∈ EC(−→ G1)∗ ✷EC(−→ G2)∗ ,a2 ∈ (N+(s2) ∩ N+(w2))∗

(4) il B ((s1,s2)(s1,w2))=[il A1 (s1)∧il A2 (s2)∧il A2 (w2)] ×∨a2 {il A1 (s1)∧il −→ L2 (s2a2)∧ il −→ L2 (w2a2)}, (s1,s2)(s1,w2) ∈ EC(−→ G1)∗ ✷EC(−→ G2)∗ ,a2 ∈ (N+(s2) ∩ N+(w2))∗ .

(5) f l B ((s1,s2)(s1,w2))=[f l A1 (s1)∧f l A2 (s2)∧f l A2 (w2)]×∨a2 {f l A1 (s1)∧f l −→ L2 (s2a2)∧ f l −→ L2 (w2a2)}, (s1,s2)(s1,w2) ∈ EC(−→ G1)∗ ✷EC(−→ G2)∗ ,a2 ∈ (N+(s2) ∩ N+(w2))∗

(6) tu B ((s1,s2)(s1,w2))=[tu A1 (s1)∧tu A2 (s2)∧tu A2 (w2)]×∨a2 {tu A1 (s1)∧tu −→ L2 (s2a2)∧ tu −→ L2 (w2a2)}, (s1,s2)(s1,w2) ∈ EC(−→ G1)∗ ✷EC(−→ G2)∗ ,a2 ∈ (N+(s2) ∩ N+(w2))∗

(7) iu B ((s1,s2)(s1,w2))=[iu A1 (s1)∧iu A2 (s2)∧iu A2 (w2)] ×∨a2 {iu A1 (s1)∧iu −→ L2 (s2a2)∧ iu −→ L2 (w2a2)}, (s1,s2)(s1,w2) ∈ EC(−→ G1)∗ ✷EC(−→ G2)∗ ,a2 ∈ (N+(s2) ∩ N+(w2))∗

(8) f u B ((s1,s2)(s1,w2))=[f u A1 (s1)∧f u A2 (s2)∧f u A2 (w2)]×∨a2 {f u A1 (s1)∧f u −→ L2 (s2a2)∧ f u −→ L2 (w2a2)}, (s1,s2)(s1,w2) ∈ EC(−→ G1)∗ ✷EC(−→ G2)∗ ,a2 ∈ (N+(s2) ∩ N+(w2))∗

(9) tl B ((s1,s2)(w1 ,s2))=[tl A1 (s1)∧tl A1 (w1)∧tl A2 (s2)]×∨a1 {tl A2 (s2)∧tl −→ L1 (s1a1)∧ tl −→ L1 (w1a1)}, (s1,s2)(w1,s2) ∈ EC(−→ G1)∗ ✷EC(−→ G2)∗ ,a1 ∈ (N+(s1) ∩ N+(w1))∗ (10) il B ((s1,s2)(w1,s2))=[il A1 (s1)∧il A1 (w1)∧il A2 (s2)] ×∨a1 {il A2 (s2)∧il −→ L1 (s1a1)∧ il −→ L1 (w1a1)}, (s1,s2)(w1,s2) ∈ EC(−→ G1)∗ ✷EC(−→ G2)∗ ,a1 ∈ (N+(s1) ∩ N+(w1))∗ . 106

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(11) f l B ((s1,s2)(w1,s2))=[f l A1 (s1)∧f l A1 (w1)∧f l A2 (s2)]×∨a1 {tl A2 (s2)∧f l −→ L1 (s1a1)∧ f l −→ L1 (w1a1)}, (s1,s2)(w1,s2) ∈ EC(−→ G1)∗ ✷EC(−→ G2)∗ ,a1 ∈ (N+(s1) ∩ N+(w1))∗

(12) tu B ((s1,s2)(w1 ,s2))=[tu A1 (s1)∧tu A1 (w1)∧tu A2 (s2)]×∨a1 {tu A2 (s2)∧tu −→ L1 (s1a1)∧ tu −→ L1 (w1a1)}, (s1,s2)(w1,s2) ∈ EC(−→ G1)∗ ✷EC(−→ G2)∗ ,a1 ∈ (N+(s1) ∩ N+(w1))∗

(13) iu B ((s1,s2)(w1,s2))=[iu A1 (s1)∧iu A1 (w1)∧iu A2 (s2)] ×∨a1 {iu A2 (s2)∧iu −→ L1 (s1a1)∧ iu −→ L1 (w1a1)}, (s1,s2)(w1,s2) ∈ EC(−→ G1)∗ ✷EC(−→ G2)∗ ,a1 ∈ (N+(s1) ∩ N+(w1))∗

(14) f u B ((s1,s2)(w1,s2))=[f u A1 (s1)∧f u A1 (w1)∧f u A2 (s2)]×∨a1 {tu A2 (s2)∧f u −→ L1 (s1a1)∧ f u −→ L1 (w1a1)}, (s1,s2)(w1,s2) ∈ EC(−→ G1)∗ ✷EC(−→ G2)∗ ,a1 ∈ (N+(s1) ∩ N+(w1))∗ (15) tl B ((s1,s2)(w1 ,w2))=[tl A1 (s1) ∧ tl A1 (w1) ∧ tl A2 (s2) ∧ tl A2 (w2)] × [tl A1 (s1) ∧ tl −→ L1 (w1s1) ∧ tl A2 (w2) ∧ tl −→ L2 (s2w2)], (s1,w1)(s2,w2) ∈ E ✷ (16) il B ((s1,s2)(w1,w2))=[il A1 (s1) ∧ il A1 (w1) ∧ il A2 (s2) ∧ il A2 (w2)] × [il A1 (s1) ∧ il −→ L1 (w1s1) ∧ il A2 (w2) ∧ il −→ L2 (s2w2)], (s1,w1)(s2,w2) ∈ E ✷ . (17) f l B ((s1,s2)(w1,w2))=[f l A1 (s1) ∧ f l A1 (w1) ∧ f l A2 (s2) ∧ f l A2 (w2)] × [f l A1 (s1) ∧ f l −→ L1 (w1s1) ∧ f l A2 (w2) ∧ f l −→ L2 (s2w2)], (s1,w1)(s2,w2) ∈ E ✷ (18) tu B ((s1,s2)(w1 ,w2))=[tu A1 (s1) ∧ tu A1 (w1) ∧ tu A2 (s2) ∧ tu A2 (w2)] × [tu A1 (s1) ∧ tu −→ L1 (w1s1) ∧ tu A2 (w2) ∧ tu −→ L2 (s2w2)], (s1,w1)(s2,w2) ∈ E ✷ (19) iu B ((s1,s2)(w1,w2))=[iu A1 (s1) ∧ iu A1 (w1) ∧ iu A2 (s2) ∧ iu A2 (w2)] × [iu A1 (s1) ∧ iu −→ L1 (w1s1) ∧ iu A2 (w2) ∧ iu −→ L2 (s2w2)], (s1,w1)(s2,w2) ∈ E ✷ (20) f u B ((s1,s2)(w1,w2))=[f u A1 (s1) ∧ f u A1 (w1) ∧ f u A2 (s2) ∧ f u A2 (w2)] × [f u A1 (s1) ∧ f u −→ L1 (w1s1) ∧ f u A2 (w2) ∧ f u −→ L2 (s2w2)], (s1,w1)(s2,w2) ∈ E ✷

A. k-competitioninterval-valuedneutrosophicgraphs

WenowdiscussanextensionofIVNC-graphs,called k-competitionIVN-graphs. Deﬁnition2.18. ThecardinalityofanIVN-set A isdenotedby |A| = |A|tl , |A|tu , |A|il , |A|iu , |A|f l , |A|f u . Where |A|tl , |A|tu , |A|il , |A|iu and |A|f l , |A|f u representthesumoftruthmembershipvalues,indeterminacy-membershipvaluesandfalsity-membershipvalues,respectively,ofalltheelementsof A 107

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Example2.19. ThecardinalityofanIVN-set A = {(a,[0.5,0.7],[0.2,0.8], [0.1, 0.3]),(b,[0.1,0.2],[0.1,0.5],[0.7,0.9]),(c,[0.3,0.5],[0.3,0.8],[0.6,0.9])} in X = {a, b, c} is |A| = |A|tl , |A|tu , |A|il , |A|iu , |A|f l , |A|f u =([0 9, 1 4], [0 6, 2 1], [1 4, 2 1]).

Wenowdiscuss k-competitionIVN-graphs. Deﬁnition2.20. Let k beanon-negativenumber.Then k-competitionIVN-graph Ck(−→ G )ofanIVN-digraph −→ G =(A, −→ B )isanundirectedIVN-graph G =(A, B) whichhassameIVN-setofverticesasin −→ G andhasaninterval-valuedneutrosophicedgebetweentwovertices s, w ∈ X in Ck(−→ G )ifandonlyif |(N+(s) ∩ N+(w))|tl >k, |(N+(s) ∩ N+(w))|tu >k, |(N+(s) ∩ N+(w))|il >k, |(N+(s) ∩ N+(w))|iu >k, |(N+(s) ∩ N+(w))|f l >k and |(N+(s) ∩ N+(w))|f u >k.The interval-valuedtruth-membershipvalueofedge(s, w)in Ck(−→ G )is tl B (s, w)= kl 1 k kl 1 [tl A(s) ∧ tl A(w)]hl 1 (N+(s) ∩ N+(w)),where kl 1 = |(N+(s) ∩ N+(w))|tl and tu B (s, w)= ku 1 k ku 1 [tu A(s) ∧ tu A(w)]hu 1 (N+(s) ∩ N+(w)),where ku 1 = |(N+(s) ∩ N+(w))|tu ,the interval-valuedindeterminacy-membershipvalueofedge(s, w)in Ck(−→ G )is il B (s, w)= kl 2 k kl 2 [il A(s) ∧ il A(w)]hl 2 (N+(s) ∩ N+(w)),where kl 2 = |(N+(s) ∩ N+(w))|il ,and iu B(s, w)= ku 2 k ku 2 [iu A(s) ∧ iu A(w)]hu 2 (N+(s) ∩ N+ (w)),where ku 2 = |(N+(s) ∩ N+ (w))|iu , theinterval-valuedfalsity-membershipvalueofedge(s, w)in Ck(−→ G )is f l B(s, w)= kl 3 k kl 3 [f l A(s) ∧ f l A(w)]hl 3(N+(s) ∩ N+(w)),where kl 3 = |(N+(s) ∩ N+(w))|f l ,and f u B (s, w)= ku 3 k ku 3 [f u A(s) ∧ f u A(w)]hu 3 (N+(s) ∩ N+(w)),where ku 3 = |(N+(s) ∩ N+(w))|f u .

Example2.21. ConsideranIVN-digraph G =(A, −→ B )on X = {s,w,a,b,c},such that A = {(s,[0.4, 0.5],[0.5, 0.7],[0.8, 0.9]),(w,[0.6, 0.7],[0.4, 0.6],[0.2, 0.3]),(a, [0 2, 0 6],[0 3, 0 6],[0 2, 0 6]),(b,[0 2, 0 6],[0 1, 0 6],[0 2, 0 6]),(c,[0 2, 0 7],[0 3, 0 5], [0 2, 0 6])},and B = {( −−−→ (s,a),[0 1, 0 4],[0 3, 0 6],[0 2, 0 6]),( −−−→ (s,b),[0 2, 0 4],[0 1, 0 5], [0.2, 0.6]),( −−−→ (s,c),[0.2, 0.5],[0.3, 0.5],[0.2, 0.6]),( −−−→ (w,a),[0.2, 0.5],[0.2, 0.5],[0.2, 0.3]), ( −−−→ (w,b),[0 2, 0 6],[0 1, 0 6],[0 2, 0 3]),( −−−→ (w,c),[0 2, 0 7],[0 3, 0 5],[0 2, 0 3])},asshown inFig. 5

Wecalculate N+(s)= {(a,[0 1, 0 4],[0 3, 0 6],[0 2, 0 6]),(b,[0 2, 0 4],[0 1, 0 5], [0.2, 0.6]),(c,[0.2, 0.5],[0.3, 0.5],[0.2, 0.6])} and N+(w)= {(a,[0.2, 0.5],[0.2, 0.5], [0.2, 0.3]),(b,[0.2, 0.6],[0.1, 0.6],[0.2, 0.3]),(c,[0.2, 0.7],[0.3, 0.5],[0.2, 0.3])}.Therefore, N+(s) ∩ N+(w)= {(a,[0.1, 0.4],[0.2, 0.5],[0.2, 0.3]),(b,[0.2, 0.4],[0.1, 0.5], [0.2, 0.3]),(c,[0.2, 0.5],[0.3, 0.5],[0.2, 0.3)}.So, kl 1 =0.5, ku 1 =1.3, kl 2 =0.6, ku 2 =1.5, kl 3 =0.6and ku 3 =0.9.Let k =0.4,then, tl B (s, w)=0.02, tu B (s, w)=0 56, il B (s, w)=0 06, iu B(s, w)=0 82, f l B(s, w)=0 02and f u B (s, w)=0 11. ThisgraphisdepictedinFig. 6

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s([0 4, 0 5], [0 5, 0 7], [0 8, 0 9])

([01,0 4],[03,0 6],[02,0 6]) ([02, 0.4],[0.1, 05],[02, 0.6]) ([0 .2 ,0 .5] ,[0 .3 ,0 .5] ,[0 .2 ,0 .6])

([0.2,06],[01,0.6],[02,0.3])

w([0 6, 0 7], [0 4, 0 6], [0 2, 0 3])

([0 2, 07],[0.3, 05],[0 2, 03])

a([0 2, 0 6], [0 3, 0 6], [0 2, 0 6])

([02,05],[02,05],[02,03])

b([0 2, 0 6], [0 1, 0 6], [0 2, 0 6])

c([0 2, 0 7], [0 3, 0 5], [0 2, 0 6])

Figure5. IVN-digraph

a([0 .2 ,0 .6] ,[0 .3 ,06] ,[0 .2 ,06])

b([0 .2 ,0 .6] ,[0 .1 ,0 .6] ,[0 .2 ,06])

c([0 .2 ,0 .7] ,[0 .3 ,0 .5] ,[0 .2 ,06])

([0 02, 0 56], [0 06, 0 82], [0 02, 0 11])

s([0 .4 ,0 .5] ,[0 .5 ,07] ,[08 ,0 .9]) w([0. 6, 0. 7], [0. 4, 0. 6], [02, 0. 3])

Figure6. 0 4-CompetitionIVN-graph

Proof. Let −→ G =(A, −→ B )beanIVN-digraph.Let Ck(−→ G )bethecorresponding k-competitionIVN-graph.

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If hl 1(N+(s) ∩ N+(w))=1and |(N+(s) ∩ N+(w))|tl > 2k,then kl 1 > 2k.Thus, tl B (s,w)= kl 1 k kl 1 [tl A(s) ∧ tl A(w)]hl 1(N+(s) ∩ N+(w))

or,tl B (s,w)= kl 1 k kl 1 [tl A(s) ∧ tl A(w)]

tl B (s,w) [tl A(s) ∧ tl A(w)] = kl 1 k kl 1 > 0 5

If hu 1 (N+(s) ∩ N+(w))=1and |(N+(s) ∩ N+(w))|tu > 2k,then ku 1 > 2k.Thus, tu B (s,w)= ku 1 k ku 1 [tu A(s) ∧ tu A(w)]hu 1 (N+(s) ∩ N+(w))

or,tu B (s,w)= ku 1 k ku 1 [tu A(s) ∧ tu A(w)] tu B (s,w) [tu A(s) ∧ tu A(w)] = ku 1 k ku 1 > 0.5.

If hl 2(N+(s) ∩ N+(w))=1and |(N+(s) ∩ N+(w))|il > 2k,then kl 2 > 2k.Thus, il B (s,w)= kl 2 k kl 2 [il A(s) ∧ il A(w)]hl 2(N+(s) ∩ N+(w))

or,il B (s,w)= kl 2 k kl 2 [il A(s) ∧ il A(w)] il B(s,w) [il A(s) ∧ il A(w)] = kl 2 k kl 2 > 0 5

If hu 2 (N+(s) ∩ N+(w))=1and |(N+(s) ∩ N+(w))|iu > 2k,then ku 2 > 2k.Thus, iu B (s,w)= ku 2 k ku 2 [iu A(s) ∧ iu A(w)]hu 2 (N+(s) ∩ N+(w))

or,iu B (s,w)= ku 2 k ku 2 [iu A(s) ∧ iu A(w)] iu B (s,w) [iu A(s) ∧ iu A(w)] = ku 2 k ku 2 > 0 5

If hl 3(N+(s) ∩ N+(w))=1and |(N+(s) ∩ N+(w))|f l < 2k,then kl 3 < 2k.Thus, f l B (s,w)= kl 3 k kl 3 [f l A(s) ∧ f l A(w)]hl 3(N+(s) ∩ N+(w))

or,f l B (s,w)= kl 3 k kl 3 [f l A(s) ∧ f l A(w)]

f l B(s,w) [f l A(s) ∧ f l A(w)] = kl 3 k kl 3 < 0 5 110

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If hu 3 (N+(s) ∩ N+(w))=1and |(N+(s) ∩ N+(w))|f u < 2k,then ku 3 < 2k.Thus, f u B (s,w)= ku 3 k ku 3 [f u A(s) ∧ f u A(w)]hu 3 (N+(s) ∩ N+(w)) or,f u B (s,w)= ku 3 k ku 3 [f u A(s) ∧ f u A(w)] f u B (s,w) [f u A(s) ∧ f u A(w)] = ku 3 k ku 3 < 0 5

So,theedge(s,w)isindependentstrongin Ck(−→ G ). B. p-competitioninterval-valuedneutrosophicgraphs

WenowdefineanotherextensionofIVNC-graphs,called p-competitionIVN-graphs. Definition2.23. ThesupportofanIVN-set A =(s,[tl A, tu A],[il A, iu A],[f l A, f u A])in X isthesubsetof X definedby supp(A)= {s ∈ X :[tl A(s) =0, tu A(s) =0],[il A(s) =0, iu A(s) =0],[f l A(s) =1, f u A(s) =1]} and |supp(A)| isthenumberofelementsintheset. Example2.24. ThesupportofanIVN-set A = {(a,[0 5,0 7],[0 2,0 8], [0 1,0 3]), (b,[0 1,0 2],[0 1,0 5],[0 7,0 9]),(c,[0 3,0 5],[0 3,0 8],[0 6,0 9]),(d,[0,0],[0,0], [1,1])} in X = {a, b, c, d} is supp(A)= {a,b,c} and |supp(A)| =3. Wenowdefine p-competitionIVN-graphs. Definition2.25. Let p beapositiveinteger.Then p-competitionIVN-graph Cp(−→ G ) oftheIVN-digraph −→ G =(A, −→ B )isanundirectedIVN-graph G =(A, B)which hassameIVN-setofverticesasin −→ G andhasaninterval-valuedneutrosophicedge betweentwovertices s, w ∈ X in Cp(−→ G )ifandonlyif |supp(N+(s) ∩ N+(w))|≥ p.Theinterval-valuedtruth-membershipvalueofedge(s, w)in Cp(−→ G )is tl B (s, w)= (i p)+1 i [tl A(s) ∧ tl A(w)]hl 1(N+(s) ∩ N+(w)),and tu B (s, w)= (i p)+1 i [tu A(s) ∧ tu A(w)]hu 1 (N+(s) ∩ N+(w)),theinterval-valuedindeterminacy-membershipvalueof edge(s, w)in Cp(−→ G )is il B (s, w)= (i p)+1 i [il A(s) ∧ il A(w)]hl 2(N+(s) ∩ N+(w)),and iu B(s, w)= (i p)+1 i [iu A(s) ∧ iu A(w)]hu 2 (N+(s) ∩ N+(w)),theinterval-valuedfalsitymembershipvalueofedge(s, w)in Cp(−→ G )is f l B (s, w)= (i p)+1 i [f l A(s) ∧ f l A(w)]hl 3 (N+(s) ∩N+(w)),and f u B (s, w)= (i p)+1 i [f u A(s) ∧ f u A(w)]hu 3 (N+(s) ∩ N+(w)),where i = |supp(N+(s) ∩ N+(w))|.

Example2.26. ConsideranIVN-digraph G =(A, −→ B )on X = {s,w,a,b,c},such that A = {(s,[0 4, 0 5],[0 5, 0 7],[0 8, 0 9]),(w,[0 6, 0 7],[0 4, 0 6],[0 2, 0 3]),(a, [0 2, 0 6],[0 3, 0 6],[0 2, 0 6]),(b,[0 2, 0 6],[0 1, 0 6],[0 2, 0 6]),(c,[0 2, 0 7],[0 3, 0 5], [0.2, 0.6])},and B = {( −−−→ (s,a),[0.1, 0.4],[0.3, 0.6],[0.2, 0.6]),( −−−→ (s,b),[0.2, 0.4],[0.1, 0.5], [0 2, 0 6]),( −−−→ (s,c),[0 2, 0 5],[0 3, 0 5],[0 2, 0 6]),( −−−→ (w,a),[0 2, 0 5],[0 2, 0 5],[0 2, 0 3]), ( −−−→ (w,b),[0 2, 0 6],[0 1, 0 6],[0 2, 0 3]),( −−−→ (w,c),[0 2, 0 7],[0 3, 0 5],[0 2, 0 3])},asshown inFig. 7

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s([04,0.5],[05,07],[0.8,09]) w ([0 6 , 0 7] , [0 4 , 0 6] , [0 2 , 0 3])

([01 ,04] ,[03 ,0.6] ,[0 .2 ,0.6])

([0 .2 ,0 . 4] ,[0 .1 ,0 . 5] ,[0 .2 ,0 . 6])

([0 2 , 0 . 5] ,[0 .3 , 0 5] ,[0 2 , 0 . 6])

([02, 0.6],[01, 06],[0.2, 03]) ([0 .2 ,0 . 7] ,[0 .3 ,0 . 5] ,[0 .2 ,0 . 3])

([0 .2 ,0 .5] ,[0 .2 ,05] ,[0 .2 ,0 .3])

b ([0 .2 ,0 . 6] ,[0 .1 ,0 . 6] ,[0 2 ,0 6])

a ([0 .2 ,0 . 6] ,[0 .3 ,0 . 6] ,[0 .2 ,0 . 6]) c([0.2,07],[03,0.5],[0.2,06])

Figure7. IVN-digraph

Wecalculate N+(s)= {(a,[0 1, 0 4],[0 3, 0 6],[0 2, 0 6]),(b,[0 2, 0 4],[0 1, 0 5], [0 2, 0 6]),(c,[0 2, 0 5],[0 3, 0 5],[0 2, 0 6])} and N+(w)= {(a,[0 2, 0 5],[0 2, 0 5], [0 2, 0 3]),(b,[0 2, 0 6],[0 1, 0 6],[0 2, 0 3]),(c,[0 2, 0 7],[0 3, 0 5],[0 2, 0 3])}.Therefore, N+(s) ∩ N+(w)= {(a,[0 1, 0 4],[0 2, 0 5],[0 2, 0 3]),(b,[0 2, 0 4],[0 1, 0 5], [0 2, 0 3]),(c,[0 2, 0 5],[0 3, 0 5],[0 2, 0 3)}.Now, i = |supp(N+(s) ∩ N+(w))| =3. For p =3,wehave, tl B (s, w)=0.02, tu B (s, w)=0.08, il B (s, w)=0.04, iu B (s, w)=0 1, f l B (s, w)=0 01and f u B (s, w)=0 03.ThisgraphisdepictedinFig. 8

a([0 .2 ,0 .6] ,[0 .3 ,0 .6] ,[0 .2 ,0 .6])

b([0 .2 ,06] ,[0 .1 ,0 .6] ,[0 .2 ,0 .6]) ([0 02, 0 08], [0 04, 0 1], [0 01, 0 03])

c([0 .2 ,0 .7] ,[0 .3 ,0 .5] ,[0 .2 ,0 .6])

s([0 .4 ,05] ,[05 ,0 .7] ,[0 .8 ,0 .9]) w([0. 6, 07], [0. 4, 0. 6], [0. 2, 0. 3])

Figure8. 3-CompetitionIVN-graph Westatethefollowingtheoremwithoutitsproof. Theorem2.27. Let −→ G =(A, −→ B ) beanIVN-digraph.If hl 1(N+(s) ∩ N+(w))=1,hl 2(N+(s) ∩ N+(w))=1,hl 3(N+(s) ∩ N+(w))=0, hu 1 (N+(s) ∩ N+(w))=1,hu 2 (N+(s) ∩ N+(w))=1,hu 3 (N+(s) ∩ N+(w))=0, 112

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in C[ i 2 ](−→ G ),thentheedge (s,w) isstrong,where i = |supp(N+(s) ∩ N+(w))|.(Note thatforanyrealnumber s, [s]=greatestintegernotesceeding s.)

C. m-stepinterval-valuedneutrosophiccompetitiongraphs

WenowdeﬁneanotherextensionofIVNC-graphknownas m-stepIVNC-graph.We willusethefollowingnotations:

P m s,w :Aninterval-valuedneutrosophicpathoflength m from s to w −→ P m s,w :Adirectedinterval-valuedneutrosophicpathoflength m from s to w. N+ m(s): m-stepinterval-valuedneutrosophicout-neighbourhoodofvertex s Nm(s): m-stepinterval-valuedneutrosophicin-neighbourhoodofvertex s Nm(s): m-stepinterval-valuedneutrosophicneighbourhoodofvertex s Nm(G): m-stepinterval-valuedneutrosophicneighbourhoodgraphoftheIVN-graph G Cm −−→ (G): m-stepIVNC-graphoftheIVN-digraph −→ G

Deﬁnition2.28. Suppose −→ G =(A, −→ B )isanIVN-digraph.The m-stepIVNdigraphof −→ G isdenotedby −→ G m =(A, B),whereIVN-setofverticesof −→ G issame withIVN-setofverticesof −→ G m andhasanedgebetween s and w in −→ G m ifandonly ifthereexistsaninterval-valuedneutrosophicdirectedpath −→ P m s,w in −→ G

Deﬁnition2.29. The m-stepinterval-valuedneutrosophicout-neighbourhood(IVNout-neighbourhood)ofvertex s ofanIVN-digraph −→ G =(A, −→ B )isIVN-set N+ m(s)=(X + s ,[t(l)+ s , t(u)+ s ],[i(l)+ s , i(u)+ s ],[f (l)+ s , f (u)+ s ]),where X + s = {w| thereexistsadirectedinterval-valuedneutrosophicpathoflength m from s to w, −→ P m s,w}, t(l)+ s : X + s → [0,1], t(u)+ s : X + s → [0,1], i(l)+ s : X + s → [0, 1], i(u)+ s : X + s → [0,1], f (l)+ s : X + s → [0,1] f (u)+ s : X + s → [0,1]aredeﬁned by t(l)+ s =min{tl−−−−−→ (s1, s2),(s1, s2)isanedgeof −→ P m s,w}, t(u)+ s =min{tu −−−−−→ (s1, s2), (s1, s2)isanedgeof −→ P m s,w}, i(l)+ s =min{il−−−−−→ (s1, s2),(s1, s2)isanedgeof −→ P m s,w}, i(u)+ s =min{iu −−−−−→ (s1, s2),(s1, s2)isanedgeof −→ P m s,w}, f (l)+ s =min{f l−−−−−→ (s1, s2),(s1, s2)isanedgeof −→ P m s,w}, f (u)+ s =min{f u −−−−−→ (s1, s2),(s1, s2)isanedgeof −→ P m s,w}, respectively.

Example2.30. ConsideranIVN-digraph G =(A, −→ B )on X = {s,w,a,b,c,d},such that A = {(s,[0 4, 0 5],[0 5, 0 7],[0 8, 0 9]),(w,[0 6, 0 7],[0 4, 0 6],[0 2, 0 3]),(a, [0.2, 0.6],[0.3, 0.6],[0.2, 0.6]),(b,[0.2, 0.6],[0.1, 0.6],[0.2, 0.6]),(c,[0.2, 0.7],[0.3, 0.5], [0 2, 0 6]), d([0 2, 0 6], [0 3, 0 6], [0 2, 0 6])},and B = {( −−−→ (s,a),[0 1, 0 4],[0 3, 0 6],[0 2, 0 6]),( −−−→ (a,c),[0 2, 0 6],[0 3, 0 5],[0 2, 0 6]),( −−−→ (a,d),[0 2, 0 6],[0 3, 0 5],[0 2, 0 4]), ( −−−→ (w,b),[0 2, 0 6],[0 1, 0 6],[0 2, 0 3]),( −−→ (b,c),[0 2, 0 4],[0 1, 0 2],[0 1, 0 3]),( −−−→ (b,d), [0 1, 0 3],[0 1, 0 2],[0 2, 0 4])},asshowninFig. 9 113

MuhammadAkrametal./Ann.FuzzyMath.Inform. 14 (2017),No.1,99–120

s([0 4, 0 5], [0 5, 0 7], [0 8, 0 9]) w([0 6, 0 7], [0 4, 0 6], [0 2, 0 3])

([0. 1, 0. 4], [0. 3, 0. 6], [0. 2, 0. 6]) ([0 .2 ,0 . 6] ,[0 .1 ,0 . 6] ,[0 .2 ,0 . 3])

([0 .2 ,06] ,[0 .3 ,0 .5] ,[0 .2 ,0 .4])

([0 .2 ,0 . 6] ,[0 .3 ,0 . 5] ,[0 .2 ,0 . 6])

a ([0 2 ,0 . 6] ,[0 .3 ,0 6] ,[0 .2 ,0 . 6]) c([0 2, 0 7], [0 3, 0 5], [0 2, 0 6])

([0. 2, 0. 4], [0. 1, 0. 2], [0. 1, 0. 3]) ([0 . 1, 0. 3], [0 . 1, 0. 2], [0 . 2, 0. 4])

b ([0 . 2, 0. 6] , [0 . 1, 0 6] , [0 . 2, 0. 6]) d([0 2, 0 6], [0 3, 0 6], [0 2, 0 6])

Figure9. IVN-digraph

Wecalculate2-stepIVN-out-neighbourhoodsas, N+ 2 (s)= {(c,[0 1, 0 4],[0 3, 0 5], [0.2, 0.6]),(d,[0.1, 0.4],[0.3, 0.5],[0.2, 0.4])} and N+ 2 (w)= {(c,[0.2, 0.4],[0.1, 0.2], [0.1, 0.3]),(d,[0.1, 0.3],[0.1, 0.2],[0.2, 0.3])}.

Deﬁnition2.31. The m-stepinterval-valuedneutrosophicin-neighbourhood(IVNin-neighbourhood)ofvertex s ofanIVN-digraph −→ G =(A, −→ B )isIVN-set Nm(s)=(Xs ,[t(l) s , t(u) s ],[i(l) s , i(u) s ],[f (l) s , f (u) s ]),where

Xs = {w| thereexistsadirectedinterval-valuedneutrosophicpathoflength m from w to s, −→ P m w,s}, t(l) s : Xs → [0,1], t(u) s : Xs → [0,1], i(l) s : Xs → [0, 1], i(u) s : Xs → [0,1], f (l) s : Xs → [0,1] f (u) s : Xs → [0,1]aredeﬁned by t(l) s =min{tl−−−−−→ (s1, s2),(s1, s2)isanedgeof −→ P m w,s}, t(u) s =min{tu −−−−−→ (s1, s2), (s1, s2)isanedgeof −→ P m w,s}, i(l) s =min{il−−−−−→ (s1, s2),(s1, s2)isanedgeof −→ P m w,s}, i(u) s =min{iu −−−−−→ (s1, s2),(s1, s2)isanedgeof −→ P m w,s}, f (l) s =min{f l−−−−−→ (s1, s2),(s1, s2)isanedgeof −→ P m w,s}, f (u) s =min{f u −−−−−→ (s1, s2),(s1, s2)isanedgeof −→ P m w,s}, respectively.

Example2.32. ConsideranIVN-digraph G =(A, −→ B )on X = {s,w,a,b,c,d},such that A = {(s,[0 4, 0 5],[0 5, 0 7],[0 8, 0 9]),(w,[0 6, 0 7],[0 4, 0 6],[0 2, 0 3]),(a, [0.2, 0.6],[0.3, 0.6],[0.2, 0.6]),(b,[0.2, 0.6],[0.1, 0.6],[0.2, 0.6]),(c,[0.2, 0.7],[0.3, 0.5], [0 2, 0 6]), d([0 2, 0 6], [0 3, 0 6], [0 2, 0 6])},and B = {( −−−→ (s,a),[0 1, 0 4],[0 3, 0 6],[0 2, 0 6]),( −−−→ (a,c),[0 2, 0 6],[0 3, 0 5],[0 2, 0 6]),( −−−→ (a,d),[0 2, 0 6],[0 3, 0 5],[0 2, 0 4]), ( −−−→ (w,b),[0 2, 0 6],[0 1, 0 6],[0 2, 0 3]),( −−→ (b,c),[0 2, 0 4],[0 1, 0 2],[0 1, 0 3]),( −−−→ (b,d), [0 1, 0 3],[0 1, 0 2],[0 2, 0 4])},asshowninFig. 10 114

MuhammadAkrametal./Ann.FuzzyMath.Inform. 14 (2017),No.1,99–120

s([0 4, 0 5], [0 5, 0 7], [0 8, 0 9]) w([0 6, 0 7], [0 4, 0 6], [0 2, 0 3])

([01,0. 4],[03,0. 6],[02,0. 6]) ([0 .2 ,0 .6] ,[0 .1 ,06] ,[02 ,0 .3])

a ([0 .2 ,0 6] ,[0 .3 ,0 . 6] ,[0 .2 ,0 . 6]) c([0 2, 0 7], [0 3, 0 5], [0 2, 0 6])

([0 .2 ,0 .6] ,[03 ,0 .5] ,[0 .2 ,0 .4])

([0 .2 ,0 . 6] ,[0 .3 ,0 . 5] ,[0 .2 ,0 . 6])

([02, 0. 4], [01, 0. 2], [01, 0. 3]) ([0 . 1, 0. 3], [0 . 1, 0. 2], [0 . 2, 0. 4])

b ([0 . 2, 0. 6] , [0 . 1, 0 6] , [0 2, 0. 6]) d([0 2, 0 6], [0 3, 0 6], [0 2, 0 6])

Figure10. IVN-digraph

Wecalculate2-stepIVN-in-neighbourhoodsas, N2 (s)= {(c,[0.1,0.4],[0.3,0.5], [0 2,0 6]),(d,[0 1,0 4],[0 3,0 5],[0 2,0 4])} and N2 (w)= {(c,[0 2,0 4],[0 1,0 2], [0 1,0 3]),(d,[0 1,0 3],[0 1,0 2],[0 2,0 3])}

Deﬁnition2.33. Suppose −→ G =(A, −→ B )isanIVN-digraph.The m-stepIVNCgraphofIVN-digraph −→ G isdenotedby Cm(−→ G )=(A, B)whichhassameIVN-set ofverticesasin −→ G andhasanedgebetweentwovertices s, w ∈ X in Cm(−→ G )ifand onlyif(N+ m(s) ∩ N+ m(w))isanon-emptyIVN-setin −→ G .Theinterval-valuedtruthmembershipvalueofedge(s, w)in Cm(−→ G )is tl B (s, w)=[tl A(s) ∧ tl A(w)]hl 1 (N+ m(s) ∩ N+ m(w)),and tu B (s, w)=[tu A(s) ∧ tu A(w)]hu 1 (N+ m(s) ∩ N+ m(w)),theinterval-valued indeterminacy-membershipvalueofedge(s, w)in Cm(−→ G )is il B (s, w)=[il A(s) ∧ il A(w)]hl 2 (N+ m(s) ∩ N+ m(w)),and iu B (s, w)=[iu A(s) ∧ iu A(w)]hu 2 (N+ m(s) ∩ N+ m(w)),the interval-valuedfalsity-membershipvalueofedge(s, w)in Cm(−→ G )is f l B(s, w)= [f l A(s) ∧ f l A(w)]hl 3(N+ m(s) ∩ N+ m(w)),and f u B (s, w)=[f u A(s) ∧ f u A(w)]hu 3 (N+ m(s) ∩ N+ m(w)).

The2 stepIVNC-graphisillustratedbythefollowingexample.

Example2.34. ConsideranIVN-digraph G =(A, −→ B )on X = {s,w,a,b,c,d},such that A = {(s,[0 4, 0 5],[0 5, 0 7],[0 8, 0 9]),(w,[0 6, 0 7],[0 4, 0 6],[0 2, 0 3]),(a, [0.2, 0.6],[0.3, 0.6],[0.2, 0.6]),(b,[0.2, 0.6],[0.1, 0.6],[0.2, 0.6]),(c,[0.2, 0.7],[0.3, 0.5], [0 2, 0 6]), d([0 2, 0 6], [0 3, 0 6], [0 2, 0 6])},and B = {( −−−→ (s,a),[0 1, 0 4],[0 3, 0 6],[0 2, 0 6]),( −−−→ (a,c),[0 2, 0 6],[0 3, 0 5],[0 2, 0 6]),( −−−→ (a,d),[0 2, 0 6],[0 3, 0 5],[0 2, 0 4]), ( −−−→ (w,b),[0 2, 0 6],[0 1, 0 6],[0 2, 0 3]),( −−→ (b,c),[0 2, 0 4],[0 1, 0 2],[0 1, 0 3]),( −−−→ (b,d), [0 1, 0 3],[0 1, 0 2],[0 2, 0 4])},asshowninFig. 11 115

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s([0 4, 0 5], [0 5, 0 7], [0 8, 0 9]) w([0 6, 0 7], [0 4, 0 6], [0 2, 0 3])

([01, 04],[0. 3, 0. 6],[0. 2, 0. 6]) ([0 .2 ,06] ,[0 .1 ,0 .6] ,[02 ,03])

a ([0 .2 ,0 . 6] ,[0 3 ,0 6] ,[0 .2 ,0 6]) c([0 2, 0 7], [0 3, 0 5], [0 2, 0 6])

([0 .2 ,0 .6] ,[03 ,0 .5] ,[02 ,0 .4])

([0 .2 ,0 . 6] ,[0 .3 ,0 . 5] ,[0 .2 ,0 . 6])

([0. 2, 0. 4], [0. 1, 0. 2], [0. 1, 0. 3]) ([0 . 1, 0. 3], [0 . 1, 0. 2], [0 . 2, 0. 4])

b ([0 . 2, 0. 6] , [0 . 1, 0 6] , [0 2, 0. 6]) d([0 2, 0 6], [0 3, 0 6], [0 2, 0 6])

Figure11. IVN-digraph

Wecalculate N+ 2 (s)= {(c,[0.1, 0.4],[0.3, 0.5],[0.2, 0.6]),(d,[0.1, 0.4],[0.3, 0.5], [0 2, 0 4])} and N+ 2 (w)= {(c,[0 2, 0 4],[0 1, 0 2],[0 1, 0 3]),(d,[0 1, 0 3],[0 1, 0 2], [0 2, 0 3])}.Therefore, N+ 2 (s) ∩ N+ 2 (w)= {(c,[0 1, 0 4],[0 1, 0 2],[0 2, 0 6]),(d, [0.1, 0.3],[0.1, 0.2],[0.2, 0.4])}.Thus, tl B (s, w)=0.04, tu B (s, w)=0.20, il B (s, w)=0 04, iu B (s, w)=0 12, f l B(s, w)=0 04and f u B (s, w)=0 12.Thisgraph isdepictedinFig. 12

s([0 4, 0 5], [0 5, 0 7], [0 8, 0 9]) w([0 6, 0 7], [0 4, 0 6], [0 2, 0 3])

([0 04, 0 20], [0 04, 0 12], [0 04, 0 12])

a([0 2, 0 6], [0 3, 0 6], [0 2, 0 6])

b([0 2, 0 6], [0 1, 0 6], [0 2, 0 6])

c([0 2, 0 7], [0 3, 0 5], [0 2, 0 6])

d([0 2, 0 6], [0 3, 0 6], [0 2, 0 6])

Figure12. 2-StepIVNC-graph

Ifapredator s attacksoneprey w,thenthelinkageisshownbyanedge −−−→ (s,w) inanIVN-digraph.But,ifpredatorneedshelpofmanyothermediators s1, s2,..., sm 1,thenlinkageamongthemisshownbyinterval-valuedneutrosophicdirected path −→ P m s,w inanIVN-digraph.So, m-steppreyinanIVN-digraphisrepresentedby avertexwhichisthe m-stepout-neighbourhoodofsomevertices.Now,thestrength ofanIVNC-graphsisdeﬁnedbelow.

Deﬁnition2.35. Let −→ G =(A, −→ B )beanIVN-digraph.Let w beacommonvertexof m-stepout-neighbourhoodsofvertices s1, s2,..., sl.Also,let −→ Bl 1(u1,v1), −→ Bl 1(u2,v2),..., −→ Bl 1(ur,vr)and −→ Bu 1 (u1,v1), −→ Bu 1 (u2,v2),..., −→ Bu 1 (ur,vr)betheminimuminterval-valuedtruth-membershipvalues, −→ Bl 2(u1,v1), −→ Bl 2(u2,v2),..., −→ Bl 2(ur,vr) and −→ Bu 2 (u1,v1), −→ Bu 2 (u2,v2),..., −→ Bu 2 (ur,vr)betheminimumindeterminacy-membership 116

MuhammadAkrametal./Ann.FuzzyMath.Inform. 14 (2017),No.1,99–120 values, −→ Bl 3(u1,v1), −→ Bl 3(u2,v2),..., −→ Bl 3(ur,vr)and −→ Bu 3 (u1,v1), −→ Bu 3 (u2,v2),..., −→ Bu 3 (ur, vr)bethemaximumfalse-membershipvalues,ofedgesofthepaths −→ P m s1,w, −→ P m s2 ,w, ..., −→ P m sr ,w,respectively.The m-stepprey w ∈ X isstrongpreyif −→ Bl 1(ui,vi) > 0 5, −→ Bl 2(ui,vi) > 0 5, −→ Bl 3(ui,vi) < 0 5, −→ Bu 1 (ui,vi) > 0 5, −→ Bu 2 (ui,vi) > 0 5, −→ Bu 3 (ui,vi) < 0 5, forall i =1, 2,...,r.

Thestrengthoftheprey w canbemeasuredbythemapping S : X → [0, 1],such that: S(w)= 1 r

r i=1 [ −→ Bl 1(ui,vi)]+ r i=1 [ −→ Bu 1 (ui,vi)]+ r i=1 [ −→ Bl 2(ui,vi)] + r i=1 [ −→ Bu 2 (ui,vi)] r i=1 [ −→ Bl 3(ui,vi)] r i=1 [ −→ Bu 3 (ui,vi)]

Example2.36. ConsideranIVN-digraph −→ G =(A, −→ B )asshowninFig. 11,the strengthoftheprey c isequalto (0 2+0 2)+(0 6+0 4)+(0 1+0 1)+(0 6+0 2) (0 2+0 1) (0 3+0 3) 2 =1 5 > 0 5 Hence, c isstrong2-stepprey.

Westatethefollowingtheoremwithoutitsproof.

Theorem2.37. Ifaprey w of −→ G =(A, −→ B ) isstrong,thenthestrengthof w, S(w) > 0 5

Remark2.38. Theconverseoftheabovetheoremisnottrue,i.e.if S(w) > 0 5, thenallpreysmaynotbestrong.Thiscanbeexplainedas: Let S(w) > 0.5foraprey w in −→ G .So, S(w)= 1 r

r i=1 [ −→ Bl 1(ui,vi)]+ r i=1 [ −→ Bu 1 (ui,vi)]+ r i=1 [ −→ Bl 2(ui,vi)] + r i=1 [ −→ Bu 2 (ui,vi)] r i=1 [ −→ Bl 3(ui,vi)] r i=1 [ −→ Bu 3 (ui,vi)] . Hence, r i=1 [ −→ Bl 1(ui,vi)]+ r i=1 [ −→ Bu 1 (ui,vi)]+ r i=1 [ −→ Bl 2(ui,vi)] + r i=1 [ −→ Bu 2 (ui,vi)] r i=1 [ −→ Bl 3(ui,vi)] r i=1 [ −→ Bu 3 (ui,vi)] > r 2 117

MuhammadAkrametal./Ann.FuzzyMath.Inform. 14 (2017),No.1,99–120

Thisresultdoesnotnecessarilyimplythat −→ Bl 1(ui,vi) > 0.5, −→ Bl 2(ui,vi) > 0.5, −→ Bl 3(ui,vi) < 0.5, −→ Bu 1 (ui,vi) > 0 5, −→ Bu 2 (ui,vi) > 0 5, −→ Bu 3 (ui,vi) < 0 5, forall i =1, 2,...,r. Since,alledgesofthedirectedpaths −→ P m s1 ,w, −→ P m s2 ,w,..., −→ P m sr ,w,arenotstrong.So, theconverseoftheabovestatementisnottruei.e.,if S(w) > 0 5,theprey w of −→ G maynotbestrong.Now, m-stepinterval-valuedneutrosophicneighbouhoodgraphs aredeﬁnesbelow.

Deﬁnition2.39. The m-stepIVN-out-neighbourhoodofvertex s ofanIVN-digraph −→ G =(A, −→ B )isIVN-set Nm(s)=(Xs,[tl s, tu s ],[il s, iu s ],[f l s, f u s ]),where Xs = {w| thereexistsadirectedinterval-valuedneutrosophicpathoflength m from s to w, Pm s,w}, tl s : Xs → [0,1], tu s : Xs → [0,1], il s : Xs → [0,1], iu s : Xs → [0,1], f l s : Xs → [0,1], f u s : Xs → [0,1],aredeﬁnedby tl s =min{tl(s1, s2),(s1, s2)is anedgeof Pm s,w}, tu s =min{tu(s1, s2),(s1, s2)isanedgeof Pm s,w}, il s =min{il(s1, s2),(s1, s2)isanedgeof Pm s,w}, iu s =min{iu(s1, s2),(s1, s2)isanedgeof Pm s,w}, f l s =min{f l(s1, s2),(s1, s2)isanedgeof Pm s,w}, f u s =min{f u(s1, s2),(s1, s2)isan edgeof Pm s,w},respectively.

Deﬁnition2.40. Suppose G =(A, B)isanIVN-graph.Then m-stepintervalvaluedneutrosophicneighbouhoodgraph Nm(G)isdeﬁnedby Nm(G)=(A, ´ B) where A =([Al 1, Au 1 ],[Al 2, Au 2 ],[Al 3, Au 3 ]), ´ B =([ ´ Bl 1, ´ Bu 1 ],[ ´ Bl 2, ´ Bu 2 ],[ ´ Bl 3, ´ Bu 3 ]), ´ Bl 1 : X × X → [0,1], ´ Bu 1 : X × X → [0,1], ´ Bl 2 : X × X → [0, 1], ´ Bu 2 : X × X → [0, 1], ´ Bl 3 : X × X → [0,1],and ´ Bu 3 : X × X → [0, 1]aresuchthat: ´ Bl 1(s,w)= Al 1(s) ∧ Al 1(w)hl 1(Nm(s) ∩ Nm(w)), ´ Bl 2(s,w)= Al 2(s) ∧ Al 2(w)hl 2(Nm(s) ∩ Nm(w)), ´ Bl 3(s,w)= Al 3(s) ∧ Al 3(w)hl 3(Nm(s) ∩ Nm(w)), ´ Bu 1 (s,w)= Au 1 (s) ∧ Au 1 (w)hu 1 (Nm(s) ∩ Nm(w)), ´ Bu 2 (s,w)= Au 2 (s) ∧ Au 2 (w)hu 2 (Nm(s) ∩ Nm(w)), ´ Bu 3 (s,w)= Au 3 (s) ∧ Au 3 (w)hu 3 (Nm(s) ∩ Nm(w)), respectively

Westatethefollowingtheoremswithoutthierproofs.

Theorem2.41. Ifallpreysof −→ G =(A, −→ B ) arestrong,thenalledgesof Cm(−→ G )= (A, B) arestrong.

Arelationisestablishedbetween m-stepIVNC-graphofanIVN-digraphand IVNC-graphof m-stepIVN-digraph.

Theorem2.42. If −→ G isanIVN-digraphand −→ Gm isthe m-stepIVN-digraphof −→ G , then C(−→ G m)= Cm(−→ G )

Theorem2.43. Let −→ G =(A, −→ B ) beanIVN-digraph.If m> |X| then Cm(−→ G )= (A,B) hasnoedge.

118

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Theorem2.44. IfalltheedgesofIVN-digraph −→ G =(A, −→ B ) areindependentstrong, thenalltheedgesof Cm(−→ G ) areindependentstrong.

3. Conclusions

Graphtheoryisanenjoyableplaygroundfortheresearchofprooftechniquesin discretemathematics.Therearemanyapplicationsofgraphtheoryindiﬀerentﬁelds. WehaveintroducedIVNC-graphsand k-competitionIVN-graphs, p-competition IVN-graphsand m-stepIVNC-graphsasthegeneralizedstructuresofIVNC-graphs. Wehavedescribedinterval-valuedneutrosophicopenandclosed-neighbourhood. Alsowehaveestablishedsomeresultsrelatedtothem.Weaimtoextendour researchworkto(1)Interval-valuedfuzzyroughgraphs;(2)Interval-valuedfuzzy roughhypergraphs,(3)Interval-valuedfuzzyroughneutrosophicgraphs,and(4) DecisionsupportsystemsbasedonIVN-graphs.

Acknowledgment: TheauthorsarethankfultoEditor-in-Chiefandthereferees fortheirvaluablecommentsandsuggestions.

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MuhammadAkram (makrammath@yahoo.com)

DepartmentofMathematics,UniversityofthePunjab,NewCampus,Lahore,Pakistan

MaryamNasir (maryamnasir912@gmail.com)

DepartmentofMathematics,UniversityofthePunjab,NewCampus,Lahore,Pakistan

120

Interval-valued neutrosophic competition graphs

@FMI

Interval-valuedneutrosophiccompetitiongraphs

1. Introduction

2. Interval-ValuedNeutrosophicCompetitionGraphs

Figure4. IVNC-graph

Figure10. IVN-digraph

3. Conclusions

References