Single-ValuedNeutrosophicGraphStructures
MuhammadAkramandMuzzamalSitara
DepartmentofMathematics,UniversityofthePunjab,NewCampus,Lahore,Pakistan, E-mail:makrammath@yahoo.com,m.akram@pucit.edu.pk,muzzamal.sitara@gmail.com
Abstract
Agraphstructureisageneralizationofundirectedgraphwhichisquiteusefulinstudyingsomestructures, includinggraphsandsignedgraphs.Inthisresearchpaper, weapplytheideaofsingle-valuedneutrosophic setstographstructure,andexploresomeinterestingpropertiesofsingle-valuedneutrosophicgraphstructure. Wealsodiscusstheconceptof φ-complementofsingle-valuedneutrosophicgraphstructure.
Key-words:Graphstructure,Single-valuedNeutrosophicgraphstructure, φ-complement. MathematicsSubjectClassification2000:03E72,05C72,05C78,05C99
1Introduction
FuzzysettheorywasintroducedbyZadeh[14]tosolveproblemswithuncertainties.Atpresent,inmodelingand controllingunsuresystemsinindustry,societyandnature,fuzzy setsandfuzzylogicareplayingavitalrole.In decisionmaking,theycanbeusedaspowerfullmathematicaltoolswhichfacilitateforapproximatereasoning. Theyplayasignificantroleincomplexphenomenawhichisnoteasilydescribedbyclassicalmathematics. Atanassov[3]illustratedtheextensionoffuzzysetsbyaddinganewcomponent,called,intuitionisticfuzzy sets.Theintuitionisticfuzzysetshaveessentiallyhigherdescribing possibilitiesthanfuzzysets.Theideaof intuitionisticfuzzysetismoremeaningfulaswellasinventiveduetothepresenceofdegreeoftruth,degreeof falseandthehesitationmargin.Thehesitationmarginofintuitionistic fuzzysetisitsindeterminacyvalueby default.Samrandache[10]submittedtheideaofneutrosophicset (NS)bycombiningthenon-standardanalysis, atricomponentlogic/set/probabilitytheoryandphilosophy.“Itisa branchofphilosophywhichstudiesthe origin,natureandscopeofneutralitiesaswellastheirinteractions withdifferentideationalspectra”[11].A NShasthreecomponents:truthmembership,indeterminacymembershipandfalsitymembership,inwhich eachmembershipvalueisarealstandardornon-standardsubset ofthenonstandardunitinterval]0 , 1+[ ([10]).ToapplyNSsinreal-lifeproblemsmoreconveniently,Wangetal. [12]definedsingle-valuedneutrosophic sets(SVNSs).ASVNSisageneralizationofintuitionisticfuzzysets[3].InSVNSthreecomponentsarenot dependentandtheirvaluesarecontainedinthestandardunitinterval[0,1].
FuzzygraphswerenarratedbyRosenfeld[8]in1975.DineshandRamakrishnan[5]introducedthenotionof afuzzygraphstructureanddiscussedsomerelatedproperties. AkramandAkmal[1]introducedtheconcept ofbipolarfuzzygraphstructures.Broumetal.[4]portrayedsingle-valuedneutrosophicgraphs.Akramand Shahzadi[2]introducedthenotionofneutrosophicsoftgraphswithapplications.Inthisresearchpaper,we applytheideaofsingle-valuedneutrosophicsetstographstructure,andexploresomeinterestingpropertiesof single-valuedneutrosophicgraphs.Wealsodiscusstheconceptof φ-complementofsingle-valuedneutrosophic graphstructure.
2Single-ValuedNeutrosophicGraphStructures
Definition2.1. Gn =(Q,Q1,Q2,...,Qn)iscalledasingle-valuedneutrosophicgraphstructure(SVNSGS)ofa graphstucture G =(S,S1,S2...,Sn)if Q =< (m,n),T (m,n),I(m,n),F (m,n) > isasingle-valuedneutrosophic (SVNS)seton S and Qi =<n,Ti(n),Ii(n),Fi(n) > isasingle-valuedneutrosophicseton Si suchthat Ti(m,n) ≤ min{T (m),T (n)}, Ii(m,n) ≤ min{I(m),I(n)}, Fi(m,n) ≤ max{F (m),F (n)}, ∀m,n ∈ S.
Notethat Ti(m,n)=0= Ii(m,n)= Fi(m,n)forall(m,n) ∈ S × S Si and 0 ≤ Ti(m,n)+ Ii(m,n)+ Fi(m,n) ≤ 3 ∀(m,n) ∈ Si ,where S and Si (i=1,2,...,n)areunderlyingvertexand underlyingi-edgesetsof ˇ Gn respectively.
Definition2.2. Let Gn =(Q,Q1,Q2,...,Qn)beaSVNSGSof G.If Hn =(Q′,Q′ 1,Q′ 2,...,Q′ n)isaSVNSGS of G suchthat
T ′(n) ≤ T (n),I ′(n) ≤ I(n),F ′(n) ≥ F (n) ∀n ∈ S,
T ′ i (m,n) ≤ Ti(m,n), I ′ i (m,n) ≤ Ii(m,n), F ′ i (m,n) ≥ Fi(m,n), ∀mn ∈ Si, i =1, 2,...,n
Then ˇ Hn iscalledaSVNSsubgraphstructureofSVNSGS ˇ Gn.
Definition2.3. ASVNSGS Hn =(Q′,Q′ 1,Q′ 2,...,Q′ n)iscalledaSVNSinducedsubgraphstructureof Gn by asubset R of S if
T ′(n)= T (n),I ′(n)= I(n),F ′(n)= F (n) ∀n ∈ R,
T ′ i (m,n)= Ti(m,n), I ′ i (m,n)= Ii(m,n), F ′ i (m,n)= Fi(m,n), ∀m,n ∈ R, i =1, 2,...,n
Definition2.4. ASVNSGS Hn =(Q′,Q′ 1,Q′ 2,...,Q′ n)iscalledaSVNSspanningsubgraphstructureof Gn if Q′ = Q and T ′ i (m,n) ≤ Ti(m,n), I ′ i (m,n) ≤ Ii(m,n), F ′ i (m,n) ≥ Fi(m,n),i =1, 2,...,n.
Example2.5. ConsideraGS ˇ G =(S,S1 ,S2)and Q, Q1,Q2 beSVNSsubsetsof S,S1 ,S2 respectively,suchthat Q = {(n1,.5,.2,.3), (n2,.7,.3,.4), (n3,.4,.3,.5), (n4,.7,.3,.6)}, Q1 = {(n1n2,.5,.2,.4), (n2n4,.7,.3,.6)}, Q2 = {(n3n4,.4,.3,.6), (n1n4,.5,.2,.6)}.
Directcalculationsshowthat Gn =(Q,Q1,Q2)isaSVNSGSof G asshowninFig.2.1.
Q1( 5,.2,.4)
Q 2 ( 5 ,. 2 ,. 6) Q1 (7,.3,.6) Q2( 4,.3,.6)
n1( 5,.2,.3) n2( 7 3 4) n3( 4,.3,.5) n4( 7,.3,.6)
Figure2.1:Asingle-valuedneutrosophicgraphstructure
Example2.6. SVNSGS Kn =(Q′,Q11,Q12)isaSVNSsubgraphstructureof Gn asshowninFig.2.2.
n1( 4,.1,.4) n2( 6,.2,.5) n3( 3,.2,.6) n4( 6,.2,.7)
Q11( 4,.1,.5) Q 12 ( 4 ,. 1 ,. 7) Q11(. 5,.2,.7) Q12( 3,.2,.8)
Figure2.2:ASVNSsubgraphstructure ˇ Kn
Definition2.7. Let ˇ Gn =(Q,Q1,Q2,...,Qn)beaSVNSGSof ˇ G.Then mn ∈ Si iscalledaSVNS Qi-edge orsimply Qi-edge,if Ti(m,n) > 0or Ii(m,n) > 0or Fi(m,n) > 0orallthreeconditionshold.Consequently, supportof Qi;i=1,2,...,nis: supp(Qi)= {mn ∈ Qi : Ti(m,n) > 0}∪{mn ∈ Qi : Ii(m,n) > 0}∪{mn ∈ Qi : Fi(m,n) > 0}
Definition2.8. Qi-pathinaSVNSGS ˇ Gn =(Q,Q1,Q2,...,Qn)isasequenceofdistinctvertices n1,n2,...,nm (exceptchoicethat nm = n1)in S,suchthat nj inj isaSVNS Qi-edge ∀j =1, 2,...,m
Definition2.9. ASVNSGS ˇ Gn =(Q,Q1,Q2,...,Qn)iscalled Qi-strongforsome i ∈{1, 2,...,n} if Ti(m,n)=min{T (m),T (n)}, Ii(m,n)=min{I(m),I(n)}, Fi(m,n)=max{F (m),F (n)}, ∀mn ∈ supp(Qi). SVNSGS ˇ Gn iscalledstrongifitis Qi-strongforall i ∈{1, 2,...,n}.
Example2.10. ConsiderSVNSGS Gn =(Q,Q1,Q2)asshowninFig.2.3.Then Gn isastrongSVNSGS sinceitisboth Q1 and Q2 strong.
n4( 4,.4,.5)
Q2( 3,.6,.6) Q1(. 3,.4,.5) Q 1 ( 1 ,. 6 ,. 7) Q 2 ( 2 ,. 4 ,. 6) Q2(2,.6,.6) Q1( 2,.4,.5)
n1( 3,.6,.5) n2( 3,.7,.6) n3( 2,.6,.5)
Q 2 ( 3 ,. 4 ,. 5) Q 1 ( 2 ,. 6 ,. 5)
n5( 3,.5,.4)
n8( 1,.8,.7)
Q2(1,.5,.7) Q2( 3,.5,.5) Q1(.2,.4,.5) Q1( 1,.4,.7)
n6( 6,.6,.5)
n7( 2,.4,.3)
Figure2.3:AstrongSVNSGS Gn =(Q,Q1,Q2)
Definition2.11. ASVNSGS ˇ Gn =(Q,Q1,Q2,...,Qn)iscalledcompleteor Q1Q2...Qn-complete,if ˇ Gn isa strongSVNSGS, supp(Qi) = φ foralli=1,2,...,nandforeverypairofvertices m,n ∈ S, mn isan Qi edge for somei.
Example2.12. Let Gn =(Q,Q1,Q2)beaSVNSGSofgraphstructure G =(S,S1 ,S2)suchthat S = {n1,n2,n3}, S1 = {n1n2}, S2 = {n2n3,n1n3} asshowninFig.2 4.bysimplecalculations,itcanbeseenthat Gn isastrongSVNSGS. n1 ( 5,.4,.5) n2( 4,.7,.8) n3( 5 7 6) Q2( 4,.7,.8)
Q 1(4 ,.4 ,.8) Q2 ( . 5,. 4,. 6)
Figure2.4:AcomplteSVNSGS
Moreover, supp(Q1) = φ, supp(Q2) = φ andeachpairofverticesin S,iseithera Q1-edgeoran Q2-edge. So Gn isacomplete,i.e., Q1Q2-completeSVNSGS. Definition2.13. Let ˇ Gn =(Q,Q1,Q2,...,Qn)beaSVNSGS.Thentruthstrength,indeterminacystrength andfalsitystrengthofa Qi-path PQi =n1,n2,...,nm aredenotedby T.PQi , I.PQi and F.PQi respectivelyand definedas
T.PQi = m j=2 [T P Qi (nj 1 nj )], I.PQi = m j=2 [I P Qi (nj 1 nj )], F.PQi = m j=2 [F P Qi (nj 1 nj )].
Example2.14. ConsideraSVNSGS ˇ Gn =(Q,Q1,Q2)asshowninFig.2 4.Wefoundthat PQ2 = n2,n1,n3 isa Q2-path.So T.PQ2 =0.4, I.PQ2 =0.4and F.PQ2 =0.8. Definition2.15. Let ˇ Gn =(Q,Q1,Q2,...,Qn)beaSVNSGS.Then Qi-truthstrengthofconnectednessbetween m and n isdefinedby T ∞ Qi (mn)= j≥1 {T j Qi (mn)},suchthat T j Qi (mn) =(T j 1 Qi ◦ T 1 Qi )(mn)for j ≥ 2and T 2 Qi (mn)=(T 1 Qi ◦ T 1 Qi )(mn)= z (T 1 Qi (mz) ∧ T 1 Qi )(zn). Qi-indeterminacystrengthofconnectednessbetween m and n isdefinedby I ∞ Qi (mn)= j≥1 {I j Qi (mn)},suchthat I j Qi (mn)=(I j 1 Qi ◦ I 1 Qi )(mn)for j ≥ 2and I 2 Qi (mn)=(I 1 Qi ◦ I 1 Qi )(mn)= z (I 1 Qi (mz) ∧ I 1 Qi )(zn).
Qi-Falsitystrengthofconnectednessbetween m and n isdefinedby F ∞ Qi (mn)= j≥1 {F j Qi (mn)},suchthat F j Qi (mn)=(F j 1 Qi ◦ F 1 Qi )(mn)for j ≥ 2and F 2 Qi (mn)=(F 1 Qi ◦ F 1 Qi )(mn)= z (F 1 Qi (mz) ∨ F 1 Qi )(zn).
Definition2.16. ASVNSGS Gn =(Q,Q1,Q2,...,Qn)isa Qi-cycleif(supp(Q),supp(Q1),supp(Q2),...,supp(Qn)) isa Qi cycle
Definition2.17. ASVNSGS Gn =(Q,Q1,Q2,...,Qn)isaSVNSfuzzy Qi-cycle(forsomei)if Gn isa Qi-cycle, nounique Qi-edge mn isin Gn suchthat TQi (mn)=min{TQi (rs): rs ∈ Si = supp(Qi)} orIQi (mn)=min{IQi (rs): rs ∈ Si = supp(Qi)} or FQi (mn)=max{FQi (rs): rs ∈ Si = supp(Qi)}
Example2.18. ConsideraSVNSGS ˇ Gn =(Q,Q1,Q2)asshowninFig.2 3.Then ˇ Gn isa Q1-cycleandSVNS fuzzy Q1 cycle,since(supp(Q),supp(Q1),supp(Q2))isa Q1-cycleandthereisnounique Q1-edgesatisfying abovecondition.
Definition2.19. Let ˇ Gn =(Q,Q1,Q2,...,Qn)beaSVNSGSand p beavertexin ˇ Gn.Let(Q′,Q′ 1,Q′ 2,...,Q′ n) beaSVNSGSinducedby S \{p} suchthat ∀v = p,w = p TQ′ (p)=0= IQ′ (p)= FQ′ (p), TQ′ i (pv)=0= IQ′ i (pv)= FQ′ i (pv) ∀ edges pv ∈ ˇ Gn. TQ′ (v)= TQ(v), IQ′ (v)= IQ(v), FQ′ (v)= FQ(v), ∀v = p. TQ′ i (vw)= TQi (vw), IQ′ i (vw)= IQi (vw), FQ′ i (vw)= FQi (vw), Then p isSVNSfuzzy Qi-cutvertexforanyi,if T ∞ Qi (vw) >T ∞ Q′ i (vw), I ∞ Qi (vw) >I ∞ Q′ i (vw)and F ∞ Qi (vw) >F ∞ Q′ i (vw),forsome v,w ∈ S \{p} Notethat p isa Qi T SVNSfuzzycutvertexif T ∞ Qi (vw) >T ∞ Q′ i (vw), Qi I SVNSfuzzycutvertexif I ∞ Qi (vw) >I ∞ Q′ i (vw)and Qi F SVNSfuzzycutvertexif F ∞ Qi (vw) >F ∞ Q′ i (vw).
Example2.20. ConsidertheSVNSGS Gn =(Q,Q1,Q2)asshowninFig.2 5and G′ n =(Q′,Q′ 1,Q′ 2)beSVNS subgraphstructureofSVNSGS Gn foundbydeletingvertex n2.Deletedvertex n2 isaSVNSfuzzy Q1-Icut vertexsince I ∞ Q1 (n2 n5)=0 4 > 0 3= I ∞ Q′ 1 (n2n5), I ∞ Q1 (n3 n4)=0 7= I ∞ Q′ 1 (n3n4), I ∞ Q1 (n3 n5)=0.4 > 0.3= I ∞ Q′ 1 (n3n5).
Definition2.21. Suppose ˇ Gn =(Q,Q1,Q2,...,Qn)beaSVNSGSand mn be Qi edge Let(Q′,Q′ 1,Q′ 2,...,Q′ n)beaSVNSfuzzyspanningsubgraphstructureof ˇ Gn,suchthat ∀ edges mn = rs, TQ′ i (mn)=0= IQ′ i (mn)= FQ′ i (mn), TQ′ i (rs)= TQi (rs), IQ′ i (rs)= IQi (rs), FQ′ i (rs)= FQi (rs). Then mn isaSVNSfuzzy Qi-bridgeif T ∞ Qi (vw) >T ∞ Q′ i (vw), I ∞ Qi (vw) >I ∞ Q′ i (vw)and F ∞ Qi (vw) >F ∞ Q′ i (vw),forsome v,w ∈ S Notethat mn isa Qi T SVNSfuzzybridgeif T ∞ Qi (vw) >T ∞ Q′ i (vw), Qi I SVNSfuzzybridgeif I ∞ Qi (vw) > I ∞ Q′ i (vw)and Qi F SVNSfuzzybridgeif F ∞ Qi (vw) >F ∞ Q′ i (vw).
Q1(03,0 5,0.4) Q 1 (0 5 , 0 7 , 0 6) Q1(03, 0 3, 04)
Q2 (0 2, 0 4, 0 3)
Q2(0 3, 0 6, 04)
Q2 (0 1, 0 4, 0 2)
Example2.22. ConsidertheSVNSGS Gn =(Q,Q1,Q2)asshowninFig.2.5and G′ n =(Q′,Q′ 1,Q′ 2)beSVNS spanningsubgraphstructureofSVNSGS Gn foundbydeleting Q1-edge(n2 n5). n5 (0 4, 0 5, 0 6)
n3 (0 . 5, 0. 7, 0. 6) n1(0 .3 , 0 .6 , 0 .4) Q2(0 1,0.4,0 2) Q 1 (0 4 , 0 4 , 0 5) Q 1 (0 3 , 0 2 , 0 4)
n2 (0 4, 0 7, 0 5) n6(0. 3, 0. 4, 04) n4(0 .6 , 0 .9 , 0 .7)
Figure2.5:ASVNSGS ˇ Gn =(Q,Q1,Q2)
Edge(n2 n5)isaSVNSfuzzy Q1-bridge.Since T ∞ Q1 (n2 n5)=0 4 > 0 3= T ∞ Q′ 1 (n2n5), I ∞ Q1 (n2 n5)=0.4 > 0.3= I ∞ Q′ 1 (n2n5)and F ∞ Q1 (n2n5)=0.5 > 0= F ∞ Q′ 1 (n2 n5).
Definition2.23. ASVNSGS ˇ Gn =(Q,Q1,Q2,...,Qn)isa Qi-treeif(supp(Q),supp(Q1),supp(Q2),...,supp(Qn)) isa Qi tree.Inotherwords, ˇ Gn isa Qi-treeifasubgraphof ˇ Gn inducedby supp(Qi)generatesatree. Definition2.24. ASVNSGS Gn =(Q,Q1,Q2,...,Qn)isaSVNSfuzzy Qi-treeif Gn hasaSVNSfuzzy spanningsubgraphstructure ˇ Hn =(Q′′,Q′′ 1 ,Q′′ 2 ,...,Q′′ n)suchthat ∀Qi-edges mn notin ˇ Hn, ˇ Hn isa Q′′ i -tree, TQi (mn) <T ∞ Q′′ i (mn), IQi (mn) <I ∞ Q′′ i (mn), FQi (mn) <F ∞ Q′′ i (mn). Inparticular, ˇ Gn isaSVNSfuzzy Qi-Ttreeif TQi (mn) <T ∞ Q′′ i (mn),aSVNSfuzzy Qi-Itreeif IQi (mn) <I ∞ Q′′ i (mn)andaSVNSfuzzy Qi-Ftreeif FQi (mn) >F ∞ Q′′ i (mn).
Example2.25. ConsidertheSVNSGS ˇ Gn =(Q,Q1,Q2)asshowninFig.2.6,whichisa Q2-tree.Itisnota Q1-treebutaSVNSfuzzy Q1-treesinceithasasingle-vlauedneutrosophicfuzzyspanningsubgraph(Q′,Q′ 1,Q′ 2) asa Q′ 1-tree,whichisobtainedbydeleting Q1-edge n2n5 from Gn.Moreover, TQ1 (n2 n5)=0 2 < 0 3= T ∞ Q′ 1 (n2n5), IQ1 (n2 n5)=0 1 < 0 3= I ∞ Q1 ′ (n2 n5 and FQ1 (n2 n5)=0 6 > 0 5= F ∞ Q1 ′ (n2n5).
n1( 3,.6,.5)
Q2(.2,.4,.4) Q2( 3,.2,.3)
Q2( .2,.6,.2)
n6( 3,.4,.3)
n2( 4,.7,.6)
n3( 5,.7,.6)
Q1( 5,.7,.8)
Q1( 3,.5,.5)
Q2(. 1,.4,.1)
n4( 6,.9,.8)
Q1( 3,.3,.7)
Q2(1,.4,.4)
Q1( 2,.1,.6)
n5( 4 5 4
Figure2.6:Asingle-valuedneutrosophicfuzzy Q1-tree
Definition2.26. ASVNSGS Gs1 =(Q1 ,Q11,Q12,...,Q1n)ofgraphstructure G1 =(S1 ,S11,S12,...,S1n)is isomorphictoSVNSGS ˇ Gs2 =(Q2,Q21,Q22,...,Q2n)ofgraphstructure ˇ G2 =(S2,S21,Q22,...,S2n)ifwehave (f,φ)where f : S1 → S2 isabijectionand φ isapermutationonset {1, 2,...,n} andfollowingrelationsare satisfied;
TQ1 (m)= TQ2 (f (m)), IQ1 (m)= IQ2 (f (m)), FQ1 (m)= FQ2 (f (m)), ∀m ∈ S1 and TQ1i (mn)= TQ2φ(i) (f (m)f (n)), IQ1i (mn)= IQ2φ(i) (f (m)f (n) FQ1i (mn)= FQ2φ(i) (f (m)f (n)), ∀mn ∈ S1i,i=1,2,...,n.
Q1 ( . 1,. 3,. 5) Q 1( .2 ,.3 ,.6)
n3( 2,.7,.8) n4( 2,.3,.5)
Q 2(2 ,.2 ,.7) Q2 ( . 1,. 3,. 6)
m2( 5,.5,.6) m4( 2,.3,.5) m3( 2,.7,.8)
Q ′ 1 (1,. 3,. 6)
Q ′ 2 ( 1,. 3,. 5)
Q′2(2 ,.3 ,.6) Q′1( .2 ,.2 ,.7)
m1( 3,.3,.4)
Example2.27. Let ˇ Gn1 =(Q,Q1,Q2)and ˇ Gn2 =(Q′,Q′ 1,Q′ 2)betwoSVNSGSsasshowninFig.2.7. n1( 3,.3,.4) n2( 5,.5,.6)
Figure2.7:IsomorphicSVNSgraphstructures
Gn1 isisomorphic Gn2 under(f,φ)where f : S → S′ isabijectionand φ isapermutationonset {1, 2} definedas φ(1)=2, φ(2)=1andfollowingrelationsaresatisfied; TQ(ni)= TQ′ (f (ni)), IQ(ni)= IQ′ (f (ni)), FQ(ni)= FQ′ (f (ni)), ∀ni ∈ S and TQi (ninj )= TQ′ φ(i) (f (ni)f (nj )), IQi (ninj )= IQ′ φ(i) (f (ni)f (nj )), FQi (ninj )= FQ′ φ(i) (f (ni)f (nj )), ∀ninj ∈ Si,i=1,2.
Definition2.28. ASVNSGS ˇ Gs1 =(Q1 ,Q11,Q12,...,Q1n)ofgraphstructure ˇ G1 =(S1 ,S11,S12,...,S1n)is identicaltoSVNSGS ˇ Gs2 =(Q2,Q21,Q22,...,Q2n)ofgraphstructure ˇ G2 =(S2,S21,Q22,...,S2n)if f : S1 → S2 isabijectionandfollowingrelationsaresatisfied; TQ1 (m)= TQ2 (f (m)), IQ1 (m)= IQ2 (f (m)), FQ1 (m)= FQ2 (f (m)), ∀m ∈ S1 and TQ1i (mn)= TQ2i (f (m)f (n)), IQ1i (mn)= IQ2i (f (m)f (n)), FQ1i (mn)= FQ2(i) (f (m)f (n)), ∀mn ∈ S1i,i=1,2,...,n. Example2.29. LetLet ˇ Gn1 =(Q,Q1,Q2)and ˇ Gn2 =(Q′,Q′ 1,Q′ 2)betwoSVNSGSsofGSs ˇ G1 =(S,S1 ,S2), ˇ G2 =(S′,S′ 1,S′ 2)respectivelyasshowninFig.2.8andFig.2.9.
n5 (0 7, 0 6, 0 5)
Q2(0 5,0 4,0 5)
n6 (0 4, 0 5, 0 2)
n3 (0 5, 0 4, 0 3)
Q1(04, 0. 4, 03)
Q2(0 6, 0.5, 0 5)
n1 (0 2, 0 3, 0 4)
Q1(0 2, 0.2, 0 4) Q2(0 1,0 2,0 3)
n2(0 3, 0 4,0 5)
Q1(0 2,0 3,0 4)
Q2(0 1,0 3,0 5) Q1(0 3, 0.4, 0 2)
Q1(0 3, 0 2, 0 5) Q2(0 4,0 3,0 6)
n7 (0 5, 0 3, 0 6)
Q1(0. 4, 0. 5, 04)
n8 (0 4, 0 6, 0 3)
Figure2.8:ASVNSGS ˇ Gn1
Q′ 2(05,04,05)
m5 (0 7, 0 6, 0 5)
n4 (0 6, 0 5, 0 4)
Q′ 2(0 .6, 0.5, 05)
m3(06 , 0 .5 , 0 .4) m4(05, 0. 4, 03) m1 (0 3 0 4 0 5)
Q′1(0 .4 ,0 .4 ,0 . 3)
m8(04, 0 5, 02)
Q ′ 2(0.1,03,05) Q′ 1(03 , 04 , 02) Q′ 1(0 .2 , 02 , 04) Q ′ 2(01,02,03) Q ′ 1 (02, 0. 3, 0. 4)
m2 (0 2, 0 3, 0 4) Q′ 1(0 .3, 02, 0.5) Q ′ 2(04,0.3,06)
m7 (0 5, 0 3, 0 6)
Figure2.9:ASVNSGS Gn2
Q′1(0 .4 ,0 .5 ,04)
m6(04, 06, 0. 3)
SVNGS Gn1 isidenticalwith Gn2 under f : S → S′ definedas; f (n1)= m2, f (n2)= m1, f (n3)= m4, f (n4)= m3, f (n5)= m5, f (n6)= m8 , f (n7)= m7 , f (n8)= m6, TQ(ni) = TQ′ (f (ni)), IQ(ni)= IQ′ (f (ni)), FQ(ni)= FQ′ (f (ni)), ∀ni ∈ S and TQi (ninj )= TQ′ i (f (ni)f (nj )), IQi (ninj ) = IQ′ i (f (ni)f (nj )), FQi (ninj )= FQ′ i (f (ni)f (nj )), ∀ninj ∈ Si,i=1,2. Definition2.30. Suppose Gn =(Q,Q1,Q2,...,Qn)beaSVNSGSand φ beapermutationon {Q1,Q2,...,Qn} andon {1, 2,...,n} thatis φ(Qi)= Qj iff φ(i)= j ∀i.If mn ∈ Qi forany i and TQφ i (mn)= TQ(m) ∧ TQ(n) j=i Tφ(Qj )(mn), IQφ i (mn)= IQ(m) ∧ IQ(n) j=i Iφ(Qj )(mn), FQφ i (mn)= FQ(m) ∨ FQ(n) j=i Tφ(Qj )(mn), i =1, 2,...,n,then mn ∈ Qφ k ,where k isselectedsuchthat; TQφ k (mn) ≥ TQφ i (mn), IQφ k (mn) ≥ IQφ i (mn), FQφ k (mn) ≥ FQφ i (mn) ∀i. AndSVNSGS(Q,Qφ 1 ,Qφ 2 ,...,Qφ n)isnamedas φ-complementofSVNSGS ˇ Gn andsymbolizedas ˇ Gφc n . Example2.31. Let ˇ Gn =(Q,Q1,Q2,Q3)beaSVNSGSshowninFig.2.10, φ(1)=2,φ(2)=3,φ(3)=1.As aresultofsimplecalculations,
Proposition2.32. A φ-complementofaSVNSGS ˇ Gn = (Q,Q1,Q2,...,Qn) isalwaysastrongSVNSGS. Moreover,if φ(i)= k,where i,k ∈{1, 2,...,n},thenall Qk -edgesinSVNSGS (Q,Q1,Q2,...,Qn) become Qφ iedgesin (Q,Qφ 1 ,Qφ 2 ,...,Qφ n)
Proof. Accordingtothedefinitionof φ-complement, TQφ i (mn)= TQ(m) ∧ TQ(n) j=i Tφ(Qj )(mn),IQφ i (mn)= IQ(m) ∧ IQ(n) j=i Iφ(Qj )(mn), FQφ i (mn)= FQ(m) ∨ FQ(n) j=i Fφ(Qj )(mn), for i ∈{1, 2,...,n}
Forexpressionoftruthnessin φ-complementrequirementsareshownas: Since TQ(m) ∧ TQ(n) ≥ 0, j=i Tφ(Qj )(mn) ≥ 0and TQi (mn) ≤ TQ(m) ∧ TQ(n) ∀Qi. ⇒ j=i Tφ(Qj )(mn) ≤ TQ(m) ∧ TQ(n) ⇒ TQ(m) ∧ TQ(n) j=i Tφ(Qj )(mn) ≥ 0. Therefore, TQφ i (mn) ≥ 0 ∀i.Moreover, TQφ i (mn)achievesitsmaximumvaluewhen j=i Tφ(Qj )(mn)iszero.It isobviousthatwhen φ(Qi)= Qk and mn isa Qk -edgethen j=i Tφ(Qj )(mn)getszerovalue.So, TQφ i (mn)= TQ(m) ∧ TQ(n),for (mn) ∈ Qk,φ(Qi)= Qk . Similarly, IQφ i (mn)= IQ(m) ∧ IQ(n),for (mn) ∈ Qk,φ(Qi)= Qk Inthesimilarwayforexpressionoffalsityin φ-complementrequirementsareshownas: Since FQ(m) ∨ FQ(n) ≥ 0, j=i Fφ(Qj )(mn) ≥ 0and FQi (mn) ≤ FQ(m) ∨ FQ(n) ∀Qi. ⇒ j=i Fφ(Qj )(mn) ≤ FQ(m) ∨ FQ(n) ⇒ FQ(m) ∨ FQ(n) j=i Fφ(Qj )(mn) ≥ 0. Therefore, FQφ i (mn)isnon-negativeforall i.Moreover, FQφ i (mn)attainsitsmaximumvaluewhen j=i Fφ(Qj ) (mn) becomeszero.Itisclearthatwhen φ(Qi)= Qk and mn isa Qk-edgethen j=i Fφ(Qj ) (mn)getszerovalue.So, FQφ i (mn)= FQ(m) ∨ FQ(n),for (mn) ∈ Qk ,φ(Qi)= Qk Fromtheseexpressionsoftruthness,indeteminacy andfalsityrequiredresultsareachieved.
Definition2.33. Let Gn =(Q,Q1,Q2,...,Qn)beaSVNSGSand φ apermutationon {1, 2,...,n}.Then
(i)If ˇ Gn isisomorphicto ˇ Gφc n ,then ˇ Gn issaidtobeself-complementary. (ii)If ˇ Gn isidenticalto ˇ Gφc n ,then ˇ Gn issaidtobestrongself-complementary.
Definition2.34. Suppose ˇ Gn =(Q,Q1,Q2,...,Qn)beaSVNSGS.Then
(i)If Gn isisomorphicto Gφc n , ∀ permutations φ on {1,2,...,n},then Gn istotallyself-complementary. (ii)If Gn isidenticalto Gφc n , ∀ permutations φ on {1,2,...,n},then Gn istotallyself-complementary.
Example2.35. AllstrongSVNSGSsareself-complementaryortotallyself-complementarySVNSGSs.
Example2.36. SVNSGS ˇ Gn =(Q,Q1,Q2,Q3)inFig.2.11istotallystrongself-complementarySVNSGS.
n7(0 2, 0 3, 0 4)
Q 1 (0 4 , 0 4 , 0 6)
Q 2(04 ,0 .4 ,0 . 5)
n2(0 4, 0 5, 0 6) n1(0 7, 0 4, 0 5) Q3(02, 03, 05) Q3 (0 . 4, 0. 4, 0. 6) Q 2 (0 2 , 0 3 , 0 5)
n6(0 4, 0 5, 0 6) n5(0 2, 0 3, 0 3)
Q1(0 .2 ,0 .3 ,05)
n4(0 4, 0 5, 0 5) n3(0 2, 0 3, 0 4)
Figure2.11:Atotallystrongself-complementarySVNSGS
Theorem2.37. ASVNSGSistotallyself-complementaryifandonlyifitisstrongSVNSGS.
Proof. ConsiderastrongSVNSGS ˇ Gn andapermutation φ on {1,2,...,n}.Byproposition2.32, φ-complementofaSVNSGS ˇ Gn =(Q,Q1,Q2,...,Qn)isalwaysastrongSVNSGS. Moreover,if φ(i)= k,where i,k ∈{1, 2,...,n},thenall Qk-edgesinSVNSGS(Q,Q1,Q2,...,Qn)become Qφ iedgesin(Q,Qφ 1 ,Qφ 2 ,...,Qφ n). Thisleads TQk (mn)= TQ(m) ∧ TQ(n)= TQφ i (mn), IQk (mn)= IQ(m) ∧ IQ(n)= IQφ i (mn), FQk (mn)= FQ(m)∨FQ (n)= FQφ i (mn).Henceunderthemapping(identitymapping) f : S → S, ˇ Gn and ˇ Gφ n areisomorphic suchthat, TQ(m)= TQ(f (m)), IQ(m)= IQ(f (m)), FQ(m)= FQ(f (m))and TQk (mn)= TQφ i (f (m)f (n))= TQφ i (mn), IQk (mn)= IQφ i (f (m)f (n))= IQφ i (mn), FQk (mn)= FQφ i (f (m)f (n))= FQφ i (mn), ∀mn ∈ Sk ,for φ 1(k)= i;i,k=1,2,...,n.Allthisissatisfiedforeverypermutation φ on {1,2,...,n}.Hence ˇ Gn istotallyself-complementarySVNSGS.Conversely,letforeverypermutation φ on {1,2,...,n}, ˇ Gn and ˇ Gφ n are isomorphic.ThenaccordingtothedefinitionofisomorphismofSVNSGSsand φ-complementofSVNSGS, TQk (mn)= TQφ i (f (m)f (n))= TQ(f (m)) ∧ TQ(f (n))= TQ(m) ∧ TQ(n), IQk (mn)= IQφ i (f (m)f (n))= IQ(f (m)) ∧ IQ(f (n))= TQ(m) ∧ IQ(n), FQk (mn)= FQφ i (f (m)f (n))= FQ(f (m)) ∨ TQ(f (n))= FQ(m) ∧ TQ(n), ∀mn ∈ Sk ,k=1,2,...,n.Hence Gn isstrongSVNSGS.
Remark2.38. EverySVNSGSwhichisself-complementaryisdefinitelytotallyself-complementary. Theorem2.39. If G = (S,S1,S2,...,Sn) isastrongandtotallyself-complementaryGSand Q =(TQ,IQ,FQ) isaSVNSsubsetof S where TQ,IQ,FQ areconstantvaluedfunctionsthenastrongSVNSGS of ˇ G withSVNSvertexset Q isalwaysastrongtotallyself-complementarySVNSGS.
Proof. Considerthreeconstants p, q, r ∈ [0, 1],suchthat TQ(m)= p,IQ(m)= q,FQ(m)= r ∀m ∈ S Since ˇ G is totallyself-complementarystrongGS,sothereisabijection f : S → S foranypermutation φ 1 on {1,2,...,n}, suchthatforany Sk -edge(mn),(f(m)f(n))[a Si-edgein ˇ G ]isa Sk-edgein ˇ Gφ 1 c.Henceforevery Qk -edge (mn),(f(m)f(n))[a Qi-edgein Gn ]isa Qφ k -edgein Gn φ 1 c . Moreover Gn isstrongSVNSGS,so
TQ(m)= p = TQ(f (m)), IQ(m)= q = IQ(f (m)), FQ(m)= r = FQ(f (m)) ∀m ∈ S and TQk (mn)= TQ(m) ∧ TQ(n)= TQ(f (m)) ∧ TQ(f (n))= TQφ i (f (m)f (n)), IQk (mn)= IQ(m) ∧ IQ(n)= IQ(f (m)) ∧ IQ(f (n))= IQφ i (f (m)f (n)), FQk (mn)= FQ(m) ∨ IQ(n)= FQ(f (m)) ∨ FQ(f (n))= FQφ i (f (m)f (n)), ∀mn ∈ Si,i =1, 2,...,n.Thisshows ˇ Gn isseif-complementarystrongSVNSGS.Everypermutation φ, φ 1 on {1,2,...,n} satisfiesaboveexpressions,thus ˇ Gn isstrongtotallyself-complementarySVNSGS.Hencerequired resultisobtained.
Remark2.40. Converseoftheorem2.39maynotbetrue,asaSVNSGSshowninFig. 2.39isstrongtotally self-complementary,itisstronganditsunderlyingGSisastrongtotallyself-complementarybut TQ, IQ, FQ arenotconstantfunctions.
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