Annalsof Fuzzy Mathematicsand Informatics Volume14,No.1,(July2017),pp.55–73
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Specialtypesofbipolarsinglevaluedneutrosophic graphs
AliHassan,MuhammadAslamMalik,SaidBroumi,AssiaBakali, MohamedTalea,FlorentinSmarandache
Received16February2017; Revised28February2017; Accepted20March2017
Abstract. Neutrosophictheoryhasmanyapplicationsingraphtheory, bipolarsinglevaluedneutrosophicgraphs(BSVNGs)isthegeneralization offuzzygraphsandintuitionisticfuzzygraphs,SVNGs.Inthispaperwe introducesometypesofBSVNGs,suchassubdivisionBSVNGs,middle BSVNGs,totalBSVNGsandbipolarsinglevaluedneutrosophiclinegraphs (BSVNLGs),alsoinvestigatetheisomorphism,coweakisomorphismand weakisomorphismpropertiesofsubdivisionBSVNGs,middleBSVNGs, totalBSVNGsandBSVNLGs.
2010AMSClassification: 05C99
Keywords: Bipolarsinglevaluedneutrosophiclinegraph,SubdivisionBSVNG, middleBSVNG,totalBSVNG.
CorrespondingAuthor: BroumiSaid(broumisaid78@gmail.com)
1. Introduction
eutrosophicsettheory(NS)isapartofneutrosophywhichwasintroduced bySmarandache[43]fromphilosophicalpointofviewbyincorporatingthedegree ofindeterminacyorneutralityasindependentcomponentfordealingproblemswith indeterminateandinconsistentinformation.Theconceptofneutrosophicsettheoryisageneralizationofthetheoryoffuzzyset[50],intuitionisticfuzzysets[5], interval-valuedfuzzysets[47]interval-valuedintuitionisticfuzzysets[6].Theconceptofneutrosophicsetischaracterizedbyatruth-membershipdegree(T),an indeterminacy-membershipdegree(I)andafalsity-membershipdegree(f)independently,whicharewithintherealstandardornonstandardunitinterval] 0, 1+[. Therefore,iftheirrangeisrestrainedwithintherealstandardunitinterval[0, 1]: Nevertheless,NSsarehardtobeapplyinpracticalproblemssincethevaluesofthe functionsoftruth,indeterminacyandfalsityliein] 0, 1+[ ThesinglevaluedneutrosophicsetwasintroducedforthefirsttimebySmarandache[43].Theconcept
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ofsinglevaluedneutrosophicsetsisasubclassofneutrosophicsetsinwhichthe valueoftruth-membership,indeterminacymembershipandfalsity-membershipdegreesareintervalsofnumbersinsteadoftherealnumbers.Lateron,Wangetal.[49] studiedsomepropertiesrelatedtosinglevaluedneutrosophicsets.Theconceptof neutrosophicsetsanditsextensionssuchassinglevaluedneutrosophicsets,interval neutrosophicsets,bipolarneutrosophicsetsandsoonhavebeenappliedinawide varietyoffieldsincludingcomputerscience,engineering,mathematics,medicineand economicandcanbefoundin[9, 15, 16, 30, 31, 32, 33, 34, 35, 36, 37, 51].Graphs arethemostpowerfultoolusedinrepresentinginformationinvolvingrelationship betweenobjectsandconcepts.Inacrispgraphstwoverticesareeitherrelatedor notrelatedtoeachother,mathematically,thedegreeofrelationshipiseither0or1. Whileinfuzzygraphs,thedegreeofrelationshiptakesvaluesfrom[0, 1]. Atanassov [42]definedtheconceptofintuitionisticfuzzygraphs(IFGs)usingfivetypesof Cartesianproducts.Theconceptfuzzygraphs,intuitionisticfuzzygraphsandtheir extensionssuchintervalvaluedfuzzygraphs,bipolarfuzzygraph,bipolarintuitionitsicfuzzygraphs,intervalvaluedintuitioniticfuzzygraphs,hesitancyfuzzygraphs, vaguegraphsandsoon,havebeenstudieddeeplybyseveralresearchersintheliterature.Whendescriptionoftheobjectortheirrelationsorbothisindeterminateand inconsistent,itcannotbehandledbyfuzzyintuitionisticfuzzy,bipolarfuzzy,vague andintervalvaluedfuzzygraphs.So,forthispurpose,Smaranadache[45]proposed theconceptofneutrosophicgraphsbasedonliteralindeterminacy(I)todealwith suchsituations.Lateron,Smarandache[44]gaveanotherdefinitionforneutrosphic graphtheoryusingtheneutrosophictruth-values(T,I,F)withoutandconstructed threestructuresofneutrosophicgraphs:neutrosophicedgegraphs,neutrosophic vertexgraphsandneutrosophicvertex-edgegraphs.Recently,Smarandache[46] proposednewversionofneutrosophicgraphssuchasneutrosophicoffgraph,neutrosophicbipolar/tripola/multipolargraph.Recentlyseveralresearchershavestudied deeplytheconceptofneutrosophicvertex-edgegraphsandpresentedseveralextensionsneutrosophicgraphs.In[1, 2, 3].Akrametal.introducedtheconceptof singlevaluedneutrosophichypergraphs,singlevaluedneutrosophicplanargraphs, neutrosophicsoftgraphsandintuitionsticneutrosophicsoftgraphs.Then,followed theworkofBroumietal.[7, 8, 9, 10, 11, 12, 13, 14, 15],MalikandHassan[38] definedtheconceptofsinglevaluedneutrosophictreesandstudiedsomeoftheir properties.Lateron,HassanetMalik[17]introducedsomeclassesofbipolarsingle valuedneutrosophicgraphsandstudiedsomeoftheirproperties,alsotheauthors generalizedtheconceptofsinglevaluedneutrosophichypergraphsandbipolarsinglevaluedneutrosophichypergraphsin[19, 20].In[23, 24]HassanetMalikgave theimportanttypesofsingle(interval)valuedneutrosophicgraphs,anotherimportantclassesofsinglevaluedneutrosophicgraphshavebeenpresentedin[22]andin [25]HassanetMalikintroducedtheconceptofm-Polarsinglevaluedneutrosophic graphsanditsclasses.Hassanetal.[18, 21]studiedtheconceptonregularityand totalregularityofsinglevaluedneutrosophichypergraphsandbipolarsinglevalued neutrosophichypergraphs.Hassanetal.[26, 27, 28]discussedtheisomorphism propertiesonSVNHGs,BSVNHGsandIVNHGs.Nasiretal.[40]introducedanew typeofgraphcalledneutrosophicsoftgraphsandestablishedalinkbetweengraphs
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andneutrosophicsoftsets.Theauthorsalsostudeiedsomebasicoperationsofneutrosophicsoftgraphssuchasunion,intersectionandcomplement.NasirandBroumi [41]studiedtheconceptofirregularneutrosophicgraphsandinvestigatedsomeof theirrelatedproperties.Ashrafetal.[4],proposedsomenovelsconceptsofedgeregular,partiallyedgeregularandfulledgeregularsinglevaluedneutrosophicgraphs andinvestigatedsomeoftheirproperties.Alsotheauthors,introducedthenotion ofsinglevaluedneutrosophicdigraphs(SVNDGs)andpresentedanapplicationof SVNDGinmulti-attributedecisionmaking.MehraandSingh[39]introduceda newconceptofneutrosophicgraphnamedsinglevaluedneutrosophicSignedgraphs (SVNSGs)andexaminedthepropertiesofthisconceptwithsuitableillustration. Ulucayetal.[48]proposedanewextensionofneutrosophicgraphscalledneutrosophicsoftexpertgraphs(NSEGs)andhaveestablishedalinkbetweengraphs andneutrosophicsoftexpertsetsandstudiessomebasicoperationsofneutrosophic softexpertsgraphssuchasunion,intersectionandcomplement.Theneutrosophic graphshavemanyapplicationsinpathproblems,networksandcomputerscience. StrongBSVNGandcompleteBSVNGarethetypesofBSVNG.Inthispaper,we introduceotherstypesofBSVNGssuchassubdivisionBSVNGs,middleBSVNGs, totalBSVNGsandBSVNLGsandtheseareallthestrongBSVNGs,alsowediscuss theirrelationsbasedonisomorphism,coweakisomorphismandweakisomorphism.
2. Preliminaries
InthissectionwerecallsomebasicconceptsonBSVNG.Let G denotesBSVNG and G∗ =(V,E)denotesitsunderlyingcrispgraph.
Definition2.1 ([10]). Let X beacrispset,thesinglevaluedneutrosophicset (SVNS) Z ischaracterizedbythreemembershipfunctions TZ (x),IZ (x)and FZ (x) whicharetruth,indeterminacyandfalsitymembershipfunctions, ∀x ∈ X TZ (x),IZ (x),FZ (x) ∈ [0, 1]
Definition2.2 ([10]) Let X beacrispset,thebipolarsinglevaluedneutrosophic set(BSVNS) Z ischaracterizedbymembershipfunctions T + Z (x),I + Z (x),F + Z (x), TZ (x),IZ (x), and FZ (x). Thatis ∀x ∈ X T + Z (x),I + Z (x),F + Z (x) ∈ [0, 1], TZ (x),IZ (x),FZ (x) ∈ [ 1, 0]
Definition2.3 ([10]). Abipolarsinglevaluedneutrosophicgraph(BSVNG)isa pair G =(Y,Z)of G∗ , where Y isBSVNSon V and Z isBSVNSon E suchthat T + Z (βγ) ≤ min(T + Y (β),T + Y (γ)),I + Z (βγ) ≥ max(I + Y (β),I + Y (γ)), IZ (βγ) ≤ min(IY (β),IY (γ)),FZ (βγ) ≤ min(FY (β),FY (γ)), F + Z (βγ) ≥ max(F + Y (β),F + Y (γ)),TZ (βγ) ≥ max(TY (β),TY (γ)), where 0 ≤ T + Z (βγ)+ I + Z (βγ)+ F + Z (βγ) ≤ 3 3 ≤ TZ (βγ)+ IZ (βγ)+ FZ (βγ) ≤ 0 ∀ β,γ ∈ V. 57
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Inthiscase, D isbipolarsinglevaluedneutrosophicrelation(BSVNR)on C. The BSVNG G =(Y,Z)iscomplete(strong)BSVNG,if
T + Z (βγ)=min(T + Y (β),T + Y (γ)),I + Z (βγ)=max(I + Y (β),I + Y (γ)),
IZ (βγ)=min(IY (β),IY (γ)),FZ (βγ)=min(FY (β),FY (γ)), F + Z (βγ)=max(F + Y (β),F + Y (γ)),TZ (βγ)=max(TY (β),TY (γ)), ∀ β,γ ∈ V (∀ βγ ∈ E). TheorderofBSVNG G =(A,B)of G∗ , denotedby O(G), is definedby
O(G)=(O+ T (G),O+ I (G),O+ F (G),OT (G),OI (G),OF (G)), where O+ T (G)= α∈V T + A (α),O+ I (G)= α∈V I + A (α),O+ F (G)= α∈V F + A (α), OT (G)= α∈V TA (α),OI (G)= α∈V IA (α),OF (G)= α∈V FA (α)
ThesizeofBSVNG G =(A,B)of G∗ , denotedby S(G), isdefinedby S(G)=(S+ T (G),S+ I (G),S+ F (G),ST (G),SI (G),SF (G)), where S+ T (G)= βγ∈E T + B (βγ),ST (G)= βγ∈E TB (βγ), S+ I (G)= βγ∈E I + B (βγ),SI (G)= βγ∈E IB (βγ), S+ F (G)= βγ∈E F + B (βγ),SF (G)= βγ∈E FB (βγ)
Thedegreeofavertex β inBSVNG G =(A,B)of G∗ ,,denotedby dG(β), is definedby dG(β)=(d+ T (β),d+ I (β),d+ F (β),dT (β),dI (β),dF (β)), where d+ T (β)= βγ∈E T + B (βγ),dT (β)= βγ∈E TB (βγ), d+ I (β)= βγ∈E I + B (βγ),dI (β)= βγ∈E IB (βγ), d+ F (β)= βγ∈E F + B (βγ),dF (β)= βγ∈E FB (βγ) 58
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3. TypesofBSVNGs
InthissectionweintroducethespecialtypesofBSVNGssuchassubdivision, middleandtotalandintersectionBSVNGs,forthisfirstwegivethebasicdefinitions ofhomomorphism,isomorphism,weakisomorphismandcoweakisomorphismof BSVNGswhichareveryusefultounderstandtherelationsamongthetypesof BSVNGs.
Definition3.1. Let G1 =(C1,D1)and G2 =(C2,D2)betwoBSVNGsof G∗ 1 = (V1,E1)and G∗ 2 =(V2,E2), respectively.Thenthehomomorphism χ : G1 → G2 is amapping χ : V1 → V2 whichsatisfiesthefollowingconditions:
T + C1 (p) ≤ T + C2 (χ(p)),I + C1 (p) ≥ I + C2 (χ(p)),F + C1 (p) ≥ F + C2 (χ(p)), TC1 (p) ≥ TC2 (χ(p)),IC1 (p) ≤ IC2 (χ(p)),FC1 (p) ≤ FC2 (χ(p)), ∀ p ∈ V1,
T + D1 (pq) ≤ T + D2 (χ(p)χ(q)),TD1 (pq) ≥ TD2 (χ(p)χ(q)), I + D1 (pq) ≥ I + D2 (χ(p)χ(q)),ID1 (pq) ≤ ID2 (χ(p)χ(q)), F + D1 (pq) ≥ F + D2 (χ(p)χ(q)),FD1 (pq) ≤ FD2 (χ(p)χ(q)), ∀ pq ∈ E1
Definition3.2. Let G1 =(C1,D1)and G2 =(C2,D2)betwoBSVNGsof G∗ 1 = (V1,E1)and G∗ 2 =(V2,E2), respectively.Thentheweakisomorphism υ : G1 → G2 isabijectivemapping υ : V1 → V2 whichsatisfiesfollowingconditions: υ isahomomorphismsuchthat
T + C1 (p)= T + C2 (υ(p)),I + C1 (p)= I + C2 (υ(p)),F + C1 (p)= F + C2 (υ(p)), TC1 (p)= TC2 (υ(p)),IC1 (p)= IC2 (υ(p)),FC1 (p)= FC2 (υ(p)), ∀ p ∈ V1 Remark3.3. TheweakisomorphismbetweentwoBSVNGspreservestheorders. Remark3.4. TheweakisomorphismbetweenBSVNGsisapartialorderrelation. Definition3.5. Let G1 =(C1,D1)and G2 =(C2,D2)betwoBSVNGsof G∗ 1 = (V1,E1)and G∗ 2 =(V2,E2), respectively.Thentheco-weakisomorphism κ : G1 → G2 isabijectivemapping κ : V1 → V2 whichsatisfiesfollowingconditions: κ isahomomorphismsuchthat
T + D1 (pq)= T + D2 (κ(p)κ(q)),TD1 (pq)= TD2 (κ(p)κ(q)), I + D1 (pq)= I + D2 (κ(p)κ(q)),ID1 (pq)= ID2 (κ(p)κ(q)), F + D1 (pq)= F + D2 (κ(p)κ(q)),FD1 (pq)= FD2 (κ(p)κ(q)), ∀ pq ∈ E1
Remark3.6. Theco-weakisomorphismbetweentwoBSVNGspreservesthesizes. Remark3.7. Theco-weakisomorphismbetweenBSVNGsisapartialorderrelation.
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Table1. BSVNSsofBSVNG.
AT + A I + A F + A TA IA FA a 0.20.10.4-0.3-0.1-0.4 b 0.30.20.5-0.5-0.4-0.6 c 0.40.70.6-0.2-0.6-0.2
BT + B I + B F + B TB IB FB p 0.20.40.5-0.2-0.5-0.6 q 0.30.80.6-0.1-0.7-0.8 r 0.10.70.9-0.1-0.8-0.5
Definition3.8. Let G1 =(C1,D1)and G2 =(C2,D2)betwoBSVNGsof G∗ 1 = (V1,E1)and G∗ 2 =(V2,E2), respectively.Thentheisomorphism ψ : G1 → G2 isa bijectivemapping ψ : V1 → V2 whichsatisfiesthefollowingconditions:
T + C1 (p)= T + C2 (ψ(p)),I + C1 (p)= I + C2 (ψ(p)),F + C1 (p)= F + C2 (ψ(p)),
TC1 (p)= TC2 (ψ(p)),IC1 (p)= IC2 (ψ(p)),FC1 (p)= FC2 (ψ(p)), ∀ p ∈ V1,
T + D1 (pq)= T + D2 (ψ(p)ψ(q)),TD1 (pq)= TD2 (ψ(p)ψ(q)), I + D1 (pq)= I + D2 (ψ(p)ψ(q)),ID1 (pq)= ID2 (ψ(p)ψ(q)), F + D1 (pq)= F + D2 (ψ(p)ψ(q)),FD1 (pq)= FD2 (ψ(p)ψ(q)), ∀ pq ∈ E1
Remark3.9. TheisomorphismbetweentwoBSVNGsisanequivalencerelation.
Remark3.10. TheisomorphismbetweentwoBSVNGspreservestheordersand sizes.
Remark3.11. TheisomorphismbetweentwoBSVNGspreservesthedegreesof theirvertices.
Definition3.12. ThesubdivisionSVNGbe sd(G)=(C,D)of G =(A,B), where C isaBSVNSon V ∪ E and D isaBSVNRon C suchthat (i) C = A on V and C = B on E, (ii)if v ∈ V lieonedge e ∈ E, then T + D (ve)=min(T + A (v),T + B (e)),I + D (ve)=max(I + A (v),I + B (e)) ID (ve)=min(IA (v),IB (e)),FD (ve)=min(FA (v),FB (e)) F + D (ve)=max(F + A (v),F + B (e)),TD (ve)=max(TA (v),TB (e)) else D(ve)= O =(0, 0, 0, 0, 0, 0).
Example3.13. ConsidertheBSVNG G =(A,B)ofa G∗ =(V,E), where V = {a,b,c} and E = {p = ab,q = bc,r = ac}, thecrispgraphof G isshowninFig. 1.TheBSVNSs A and B aredefinedon V and E respectivelywhicharedefined inTable 1.TheSDBSVNG sd(G)=(C,D)ofaBSVNG G, theunderlyingcrisp graphof sd(G)isgiveninFig. 2.TheBSVNSs C and D aredefinedinTable 2 60
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Figure1. CrispGraphofBSVNG. Figure2. CrispGraphofSDBSVNG.
Table2. BSVNSsofSDBSVNG.
CT + C I + C F + C TC IC FC a 0.20.10.4-0.3-0.1-0.4 p 0.20.40.5-0.2-0.5-0.6 b 0.30.20.5-0.5-0.4-0.6 q 0.30.80.6-0.1-0.7-0.8 c 0.40.70.6-0.2-0.6-0.2 r 0.10.70.9-0.1-0.8-0.5
DT + D I + D F + D TD ID FD ap 0.20.40.5-0.2-0.5-0.6 pb 0.20.40.5-0.2-0.5-0.6 bq 0.30.80.6-0.1-0.7-0.8 qc 0.30.80.6-0.1-0.7-0.8 cr 0.10.70.9-0.1-0.8-0.5 ra 0.10.70.9-0.1-0.8-0.5
Proposition3.14. Let G beaBSVNGand sd(G) betheSDBSVNGofaBSVNG G, then O(sd(G))= O(G)+ S(G) and S(sd(G))=2S(G).
Remark3.15. Let G beacompleteBSVNG,then sd(G)neednottobecomplete BSVNG.
AliHassanetal./Ann.FuzzyMath.Inform. 14 (2017),No.1,55–73 Figure3. CrispGraphofTSVNG.
Definition3.16. Thetotalbipolarsinglevaluedneutrosophicgraph(TBSVNG)is T (G)=(C,D)of G =(A,B), where C isaBSVNSon V ∪ E and D isaBSVNR on C suchthat
(i) C = A on V and C = B on E, (ii)if v ∈ V lieonedge e ∈ E, then
T + D (ve)=min(T + A (v),T + B (e)),I + D (ve)=max(I + A (v),I + B (e))
ID (ve)=min(IA (v),IB (e)),FD (ve)=min(FA (v),FB (e))
F + D (ve)=max(F + A (v),F + B (e)),TD (ve)=max(TA (v),TB (e)) else
D(ve)= O =(0, 0, 0, 0, 0, 0), (iii)if αβ ∈ E,then
T + D (αβ)= T + B (αβ),I + D (αβ)= I + B (αβ),F + D (αβ)= F + B (αβ)
TD (αβ)= TB (αβ),ID (αβ)= IB (αβ),FD (αβ)= FB (αβ), (iv)if e,f ∈ E haveacommonvertex,then
T + D (ef )=min(T + B (e),T + B (f )),I + D (ef )=max(I + B (e),I + B (f )) ID (ef )=min(IB (e),IB (f )),FD (ef )=min(FB (e),FB (f )) F + D (ef )=max(F + B (e),F + B (f )),TD (ef )=max(TB (e),TB (f )) else
D(ef )= O =(0, 0, 0, 0, 0, 0)
Example3.17. ConsidertheExample 3.13 theTBSVNG T (G)=(C,D)ofunderlyingcrispgraphasshowninFig. 3.TheBSVNS C isgiveninExample 3.13.The BSVNS D isgiveninTable 3
Proposition3.18. Let G bea BSVNG and T (G) betheTBSVNGofaBSVNG G, then O(T (G))= O(G)+ S(G)= O(sd(G)) and S(sd(G))=2S(G).
Proposition3.19. Let G beaBSVNG,then sd(G) isweakisomorphicto T (G)
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Table3. BSVNSofTBSVNG.
DT + D I + D F + D TD ID FD ab 0.20.40.5-0.2-0.5-0.6 bc 0.30.80.6-0.1-0.7-0.8 ca 0.10.70.9-0.1-0.8-0.5 pq 0.20.80.6-0.1-0.7-0.8 qr 0.10.80.9-0.1-0.8-0.8 rp 0.10.70.9-0.1-0.8-0.6 ap 0.20.40.5-0.2-0.5-0.6 pb 0.20.40.5-0.2-0.5-0.6 bq 0.30.80.6-0.1-0.7-0.8 qc 0.30.80.6-0.1-0.7-0.8 cr 0.10.70.9-0.1-0.8-0.5 ra 0.10.70.9-0.1-0.8-0.5
Definition3.20. Themiddlebipolarsinglevaluedneutrosophicgraph(MBSVNG) M (G)=(C,D)of G, where C isaBSVNSon V ∪ E and D isaBSVNRon C such that
(i) C = A on V and C = B on E, else C = O =(0, 0, 0, 0, 0, 0), (ii)if v ∈ V lieonedge e ∈ E, then
T + D (ve)= T + B (e),I + D (ve)= I + B (e),F + D (ve)= F + B (e)
TD (ve)= TB (e),ID (ve)= IB (e),FD (ve)= FB (e) else
D(ve)= O =(0, 0, 0, 0, 0, 0), (iii)if u,v ∈ V, then
D(uv)= O =(0, 0, 0, 0, 0, 0),
(iv)if e,f ∈ E and e and f areadjacentin G, then
T + D (ef )= T + B (uv),I + D (ef )= I + B (uv),F + D (ef )= F + B (uv)
TD (ef )= TB (uv),ID (ef )= IB (uv),FD (ef )= FB (uv)
Example3.21. ConsidertheBSVNG G =(A,B)ofa G∗ , where V = {a,b,c} and E = {p = ab,q = bc} theunderlayingcrispgraphisshowninFig. 4.TheBSVNSs A and B aredefinedinTable 4.ThecrispgraphofMBSVNG M (G)=(C,D)is showninFig. 5.TheBSVNSs C and D aregiveninTable 5.
Remark3.22. Let G beaBSVNGand M (G)betheMBSVNGofaBSVNG G, then O(M (G))= O(G)+ S(G)
Remark3.23. Let G beaBSVNG,then M (G)isastrongBSVNG.
Remark3.24. Let G becompleteBSVNG,then M (G)neednottobecomplete BSVNG.
Proposition3.25. Let G beaBSVNG,then sd(G) isweakisomorphicwith M (G) 63
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Figure4. CrispGraphofBSVNG. Table4. BSVNSsofBSVNG.
AT + A I + A F + A TA IA FA a 0.30.40.5-0.2-0.1-0.3 b 0.70.60.3-0.3-0.3-0.2 c 0.90.70.2-0.5-0.4-0.6
BT + B I + B F + B TB IB FB p 0.20.60.6-0.1-0.4-0.3 q 0.40.80.7-0.3-0.5-0.6
Table5. BSVNSsofMBSVNG.
CT + C I + C F + C TC IC FC a 0.30.40.5-0.2-0.1-0.3 b 0.70.60.3-0.3-0.3-0.2 c 0.90.70.2-0.5-0.4-0.6 e1 0.20.60.6-0.1-0.4-0.3 e2 0.40.80.7-0.3-0.5-0.6
DT + D I + D F + D TD ID FD pq 0.20.80.7-0.1-0.5-0.6 ap 0.20.60.6-0.1-0.4-0.3 bp 0.20.60.6-0.1-0.4-0.3 bq 0.20.60.6-0.3-0.5-0.6 cq 0.40.80.7-0.3-0.5-0.6
Figure5. CrispGraphofMBSVNG.
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Proposition3.26. Let G beaBSVNG,then M (G) isweakisomorphicwith T (G). Proposition3.27. Let G beaBSVNG,then T (G) isisomorphicwith G ∪ M (G) Definition3.28. Let P (X)=(X,Y )betheintersectiongraphofa G∗ , let C1 and D1 beBSVNSson V and E,respectivelyand C2 and D2 beBSVNSson X and Y respectively.Thenbipolarsinglevaluedneutrosophicintersectiongraph(BSVNIG) ofaBSVNG G =(C1,D1)isaBSVNG P (G)=(C2,D2)suchthat,
T + C2 (Xi)= T + C1 (vi),I + C2 (Xi)= I + C1 (vi),F + C2 (Xi)= F + C1 (vi)
TC2 (Xi)= TC1 (vi),IC2 (Xi)= IC1 (vi),FC2 (Xi)= FC1 (vi)
T + D2 (XiXj )= T + D1 (vivj ),TD2 (XiXj )= TD1 (vivj ), I + D2 (XiXj )= I + D1 (vivj ),ID2 (XiXj )= ID1 (vivj ), F + D2 (XiXj )= F + D1 (vivj ),FD2 (XiXj )= FD1 (vivj ) ∀ Xi,Xj ∈ X and XiXj ∈ Y.
Proposition3.29. Let G =(A1,B1) beaBSVNGof G∗ =(V,E), andlet P (G)= (A2,B2) beaBSVNIGof P (S) ThenBSVNIGisaalsoBSVNGandBSVNGis alwaysisomorphictoBSVNIG.
Proof. BythedefinitionofBSVNIG,wehave
T + B2 (SiSj )= T + B1 (vivj ) ≤ min(T + A1 (vi),T + A1 (vj ))=min(T + A2 (Si),T + A2 (Sj )), I + B2 (SiSj )= I + B1 (vivj ) ≥ max(I + A1 (vi),I + A1 (vj ))=max(I + A2 (Si),I + A2 (Sj )), F + B2 (SiSj )= F + B1 (vivj ) ≥ max(F + A1 (vi),F + A1 (vj ))=max(F + A2 (Si),F + A2 (Sj )), TB2 (SiSj )= TB1 (vivj ) ≥ max(TA1 (vi),TA1 (vj ))=max(TA2 (Si),TA2 (Sj )), IB2 (SiSj )= IB1 (vivj ) ≤ min(IA1 (vi),IA1 (vj ))=min(IA2 (Si),IA2 (Sj )), FB2 (SiSj )= FB1 (vivj ) ≤ min(FA1 (vi),FA1 (vj ))=min(FA2 (Si),FA2 (Sj )).
ThisshowsthatBSVNIGisaBSVNG.
Nextdefine f : V → S by f (vi)= Si for i =1, 2, 3,...,n clearly f isbijective. Now vivj ∈ E ifandonlyif SiSj ∈ T and T = {f (vi)f (vj ): vivj ∈ E}. Also
T + A2 (f (vi))= T + A2 (Si)= T + A1 (vi),I + A2 (f (vi))= I + A2 (Si)= I + A1 (vi), F + A2 (f (vi))= F + A2 (Si)= F + A1 (vi),TA2 (f (vi))= TA2 (Si)= TA1 (vi), IA2 (f (vi))= IA2 (Si)= IA1 (vi),FA2 (f (vi))= FA2 (Si)= FA1 (vi), ∀ vi ∈ V,
T + B2 (f (vi)f (vj ))= T + B2 (SiSj )= T + B1 (vivj ), I + B2 (f (vi)f (vj ))= I + B2 (SiSj )= I + B1 (vivj ), F + B2 (f (vi)f (vj ))= F + B2 (SiSj )= F + B1 (vivj ), TB2 (f (vi)f (vj ))= TB2 (SiSj )= TB1 (vivj ), IB2 (f (vi)f (vj ))= IB2 (SiSj )= IB1 (vivj ), FB2 (f (vi)f (vj ))= FB2 (SiSj )= FB1 (vivj ), ∀ vivj ∈ E. 65
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Table6. BSVNSsofBSVNG.
A1 T + A1 I + A1 F + A1 TA1 IA1 FA1
α1 0.20.50.5-0.1-0.4-0.5 α2 0.40.30.3-0.2-0.3-0.2 α3 0.40.50.5-0.3-0.2-0.6 α4 0.30.20.2-0.4-0.1-0.3 B1 T + B1 I + B1 F + B1 TB1 IB1 FB1 x1 0.10.60.7-0.1-0.4-0.5 x2 0.30.60.7-0.2-0.3-0.6 x3 0.20.70.8-0.3-0.2-0.6 x4 0.10.70.8-0.1-0.4-0.5
Definition3.30. Let G∗ =(V,E)and L(G∗)=(X,Y )beitslinegraph,where A1 and B1 beBSVNSson V and E, respectively.Let A2 and B2 beBSVNSson X and Y, respectively.Thebipolarsinglevaluedneutrosophiclinegraph(BSVNLG) ofBSVNG G =(A1,B1)isBSVNG L(G)=(A2,B2)suchthat, T + A2 (Sx)= T + B1 (x)= T + B1 (uxvx),I + A2 (Sx)= I + B1 (x)= I + B1 (uxvx), IA2 (Sx)= IB1 (x)= IB1 (uxvx),FA2 (Sx)= FB1 (x)= FB1 (uxvx), F + A2 (Sx)= F + B1 (x)= F + B1 (uxvx),TA2 (Sx)= TB1 (x)= TB1 (uxvx), ∀ Sx,Sy ∈ X and T + B2 (SxSy )=min(T + B1 (x),T + B1 (y)),I + B2 (SxSy )=max(I + B1 (x),I + B1 (y)), IB2 (SxSy )=min(IB1 (x),IB1 (y)),FB2 (SxSy )=min(FB1 (x),FB1 (y)), F + B2 (SxSy )=max(F + B1 (x),F + B1 (y)),TB2 (SxSy )=max(TB1 (x),TB1 (y)), ∀ SxSy ∈ Y.
Remark3.31. EveryBSVNLGisastrongBSVNG. Remark3.32. The L(G)=(A2,B2)isaBSVNLGcorrespondingtoBSVNG G = (A1,B1)
Example3.33. Considerthe G∗ =(V,E)where V = {α1,α2,α3,α4} and E = {x1 = α1α2,x2 = α2α3,x3 = α3α4,x4 = α4α1} and G =(A1,B1)isBSVNGof G∗ =(V,E)whichisdefinedinTable 6.Considerthe L(G∗)=(X,Y )suchthat X = {Γx1 , Γx2 , Γx3 , Γx4 } and Y = {Γx1 Γx2 , Γx2 Γx3 , Γx3 Γx4 , Γx4 Γx1 } Let A2 and B2 beBSVNSsof X and Y respectively,thenBSVNLG L(G)isgiveninTable 7
Proposition3.34. The L(G)=(A2,B2) isaBSVNLGofsomeBSVNG G = (A1,B1) ifandonlyif
T + B2 (SxSy )=min(T + A2 (Sx),T + A2 (Sy )),
TB2 (SxSy )=max(TA2 (Sx),TA2 (Sy )), I + B2 (SxSy )=max(I + A2 (Sx),I + A2 (Sy )), FB2 (SxSy )=min(FA2 (Sx),FA2 (Sy )), 66
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Table7. BSVNSsofBSVNLG. A1 T + A1 I + A1 F + A1 TA1 IA1 FA1 Γx1 0.10.60.7-0.1-0.4-0.5 Γx2 0.30.60.7-0.2-0.3-0.6 Γx3 0.20.70.8-0.3-0.2-0.6 Γx4 0.10.70.8-0.1-0.4-0.5 B1 T + B1 I + B1 F + B1 TB1 IB1 FB1 Γx1 Γx2 0.10.60.7-0.1-0.4-0.6 Γx2 Γx3 0.20.70.8-0.2-0.3-0.6 Γx3 Γx4 0.10.70.8-0.1-0.4-0.6 Γx4 Γx1 0.10.70.8-0.1-0.4-0.5
IB2 (SxSy )=min(IA2 (Sx),IA2 (Sy )), F + B2 (SxSy )=max(F + A2 (Sx),F + A2 (Sy )), ∀ SxSy ∈ Y. Proof. Assumethat,
T + B2 (SxSy )=min(T + A2 (Sx),T + A2 (Sy )),
TB2 (SxSy )=max(TA2 (Sx),TA2 (Sy )), I + B2 (SxSy )=max(I + A2 (Sx),I + A2 (Sy )),
FB2 (SxSy )=min(FA2 (Sx),FA2 (Sy )),
IB2 (SxSy )=min(IA2 (Sx),IA2 (Sy )), F + B2 (SxSy )=max(F + A2 (Sx),F + A2 (Sy )), ∀ SxSy ∈ Y. Define
T + A1 (x)= T + A2 (Sx),I + A1 (x)= I + A2 (Sx),F + A1 (x)= F + A2 (Sx),
TA1 (x)= TA2 (Sx),IA1 (x)= IA2 (Sx),FA1 (x)= FA2 (Sx) ∀ x ∈ E. Then
I + B2 (SxSy )=max(I + A2 (Sx),I + A2 (Sy ))=max(I + A2 (x),I + A2 (y)),
IB2 (SxSy )=min(IA2 (Sx),IA2 (Sy ))=min(IA2 (x),IA2 (y)), T + B2 (SxSy )=min(T + A2 (Sx),T + A2 (Sy ))=min(T + A2 (x),T + A2 (y)), TB2 (SxSy )=max(TA2 (Sx),TA2 (Sy ))=max(TA2 (x),TA2 (y)), FB2 (SxSy )=min(FA2 (Sx),FA2 (Sy ))=min(FA2 (x),FA2 (y)), F + B2 (SxSy )=max(F + A2 (Sx),F + A2 (Sy ))=max(F + A2 (x),F + A2 (y))
ABSVNS A1 thatyieldstheproperty
T + B1 (xy) ≤ min(T + A1 (x),T + A1 (y)),I + B1 (xy) ≥ max(I + A1 (x),I + A1 (y)),
IB1 (xy) ≤ min(IA1 (x),IA1 (y)),FB1 (xy) ≤ min(FA1 (x),FA1 (y)), F + B1 (xy) ≥ max(F + A1 (x),F + A1 (y)),TB1 (xy) ≥ max(TA1 (x),TA1 (y)) willsuffice.Converseisstraightforward. 67
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Proposition3.35. If L(G) beaBSVNLGofBSVNG G, then L(G∗)=(X,Y ) is thecrisplinegraphof G∗ .
Proof. Since L(G)isaBSVNLG, T + A2 (Sx)= T + B1 (x),I + A2 (Sx)= I + B1 (x),F + A2 (Sx)= F + B1 (x), TA2 (Sx)= TB1 (x),IA2 (Sx)= IB1 (x),FA2 (Sx)= FB1 (x) ∀ x ∈ E,Sx ∈ X ifandonlyif x ∈ E, also T + B2 (SxSy )=min(T + B1 (x),T + B1 (y)),I + B2 (SxSy )=max(I + B1 (x),I + B1 (y)), IB2 (SxSy )=min(IB1 (x),IB1 (y)),FB2 (SxSy )=min(FB1 (x),FB1 (y)), F + B2 (SxSy )=max(F + B1 (x),F + B1 (y)),TB2 (SxSy )=max(TB1 (x),TB1 (y)), ∀ SxSy ∈ Y .Then Y = {SxSy : Sx ∩ Sy = φ,x,y ∈ E,x = y}
Proposition3.36. The L(G)=(A2,B2) beaBSVNLGofBSVNG G ifandonly if L(G∗)=(X,Y ) isthelinegraphand
T + B2 (xy)=min(T + A2 (x),T + A2 (y)),I + B2 (xy)=max(I + A2 (x),I + A2 (y)), IB2 (xy)=min(IA2 (x),IA2 (y)),FB2 (xy)=min(FA2 (x),FA2 (y)), F + B2 (xy)=max(F + A2 (x),F + A2 (y)),TB2 (xy)=max(TA2 (x),TA2 (y)), ∀ xy ∈ Y.
Proof. Itfollowsfrompropositions 3.34 and 3.35
Proposition3.37. Let G beaBSVNG,then M (G) isisomorphicwith sd(G)∪L(G). Theorem3.38. Let L(G)=(A2,B2) beBSVNLGcorrespondingtoBSVNG G = (A1,B1)
(1) If G isweakisomorphiconto L(G) ifandonlyif ∀ v ∈ V,x ∈ E and G∗ to beacycle,suchthat
T + A1 (v)= T + B1 (x),I + A1 (v)= T + B1 (x),F + A1 (v)= T + B1 (x), TA1 (v)= TB1 (x),IA1 (v)= TB1 (x),FA1 (v)= TB1 (x)
(2) If G isweakisomorphiconto L(G), then G and L(G) areisomorphic.
Proof. Byhypothesis, G∗ isacycle.Let V = {v1,v2,v3,...,vn} and E = {x1 = v1v2,x2 = v2v3,...,xn = vnv1},where P : v1v2v3 ...vn isacycle,characterizea BSVNS A1 by A1(vi)=(pi,qi,ri,pi,qi ,ri )and B1 by B1(xi)=(ai,bi,ci,ai,bi,ci) for i =1, 2, 3,...,n and vn+1 = v1 Thenfor pn+1 = p1,qn+1 = q1,rn+1 = r1, ai ≤ min(pi,pi+1),bi ≥ max(qi,qi+1),ci ≥ max(ri,ri+1), ai ≥ max(pi,pi+1),bi ≤ min(qi ,qi+1),ci ≤ min(ri ,ri+1), for i =1, 2, 3,...,n. Nowlet X = {Γx1 , Γx2 ,..., Γxn } and Y = {Γx1 Γx2 , Γx2 Γx3 ,..., Γxn Γx1 } Then for an+1 = a1, weobtain A2(Γxi )= B1(xi)=(ai,bi,ci,ai,bi,ci) 68
AliHassanetal./Ann.FuzzyMath.Inform. 14 (2017),No.1,55–73 and B2(Γxi Γxi+1 )=(min(ai,ai+1), max(bi,bi+1),max(ci,ci+1),max(ai,ai+1), min(bi,bi+1), min(ci,ci+1))for i =1, 2, 3,...,n and vn+1 = v1 Since f preservesadjacency,itinducepermutation π of {1, 2, 3,...,n}, f (vi)=Γvπ(i)vπ(i)+1 and vivi+1 → f (vi)f (vi+1)=Γvπ(i)vπ(i)+1 Γvπ(i+1)vπ(i+1)+1 , for i =1, 2, 3,...,n 1.Thus
pi = T + A1 (vi) ≤ T + A2 (f (vi))= T + A2 (Γvπ(i)vπ(i)+1 )= T + B1 (vπ(i)vπ(i)+1)= aπ(i).
Similarly, pi ≥ a π(i),qi ≥ bπ(i),ri ≥ cπ(i),qi ≤ b π(i),ri ≤ c π(i) and ai = T + B1 (vivi+1) ≤ T + B2 (f (vi)f (vi+1)) = T + B2 (Γvπ(i)vπ(i)+1 Γvπ(i+1)vπ(i+1)+1 ) =min(T + B1 (vπ(i)vπ(i)+1),T + B1 (vπ(i+1)vπ(i+1)+1)) =min(aπ(i),aπ(i)+1)
Similarly, bi ≥ max(bπ(i),bπ(i)+1),ci ≥ max(cπ(i),cπ(i)+1),ai ≥ max(a π(i),aπ(i)+1), bi ≤ min(b π(i),bπ(i)+1)and ci ≤ min(c π(i),cπ(i)+1)for i =1, 2, 3,...,n. Therefore pi ≤ aπ(i),qi ≥ bπ(i),ri ≥ cπ(i),pi ≥ aπ(i),qi ≤ bπ(i),ri ≤ cπ(i) and
ai ≤ min(aπ(i),aπ(i)+1),ai ≥ max(aπ(i),aπ(i)+1), bi ≥ max(bπ(i),bπ(i)+1),bi ≤ min(bπ(i),bπ(i)+1), ci ≥ max(cπ(i),cπ(i)+1),ci ≤ min(cπ(i),cπ(i)+1) thus ai ≤ aπ(i),bi ≥ bπ(i),ci ≥ cπ(i),ai ≥ aπ(i),bi ≤ bπ(i),ci ≤ cπ(i) andso aπ(i) ≤ aπ(π(i)),bπ(i) ≥ bπ(π(i)),cπ(i) ≥ cπ(π(i)) aπ(i) ≥ aπ(π(i)),bπ(i) ≤ bπ(π(i)),cπ(i) ≤ cπ(π(i)) ∀ i =1, 2, 3,...,n. Nexttoextend, ai ≤ aπ(i) ≤ ... ≤ aπj (i) ≤ ai,ai ≥ aπ(i) ≥ ... ≥ aπj (i) ≥ ai bi ≥ bπ(i) ≥ ... ≥ bπj (i) ≥ bi,bi ≤ bπ(i) ≤ ... ≤ bπj (i) ≤ bi ci ≥ cπ(i) ≥ ≥ cπj (i) ≥ ci,ci ≤ cπ(i) ≤ ≤ cπj (i) ≤ ci where πj+1 identity.Hence
ai = aπ(i),bi = bπ(i),ci = cπ(i),ai = aπ(i),bi = bπ(i),ci = cπ(i) ∀ i =1, 2, 3,...,n. Thusweconcludethat ai ≤ aπ(i+1) = ai+1,bi ≥ bπ(i+1) = bi+1,ci ≥ cπ(i+1) = ci+1 ai ≥ aπ(i+1) = ai+1,bi ≤ bπ(i+1) = bi+1,ci ≤ cπ(i+1) = ci+1 69
AliHassanetal./Ann.FuzzyMath.Inform. 14 (2017),No.1,55–73 whichtogetherwith an+1 = a1,bn+1 = b1,cn+1 = c1,an+1 = a1,bn+1 = b1,cn+1 = c1 whichimpliesthat
ai = a1,bi = b1,ci = c1,ai = a1,bi = b1,ci = c1
∀ i =1, 2, 3,...,n. Thuswehave
a1 = a2 = = an = p1 = p2 = = pn a1 = a2 = ... = an = p1 = p2 = ... = pn b1 = b2 = ... = bn = q1 = q2 = ... = qn b1 = b2 = = bn = q1 = q2 = = qn c1 = c2 = = cn = r1 = r2 = = rn c1 = c2 = ... = cn = r1 = r2 = ... = rn
Therefore(a)and(b)holds,sinceconverseofresult(a)isstraightforward.
4. Conclusion
Theneutrosophicgraphshavemanyapplicationsinpathproblems,networksand computerscience.StrongBSVNGandcompleteBSVNGarethetypesofBSVNG.In thispaper,wediscussedthespecialtypesofBSVNGs,subdivisionBSVNGs,middle BSVNGs,totalBSVNGsandBSVNLGsofthegivenBSVNGs.Weinvestigated isomorphismpropertiesofsubdivisionBSVNGs,middleBSVNGs,totalBSVNGs andBSVNLGs.
References
[1] M.Akram,Single-ValuedNeutrosophicPlanarGraphs,InternationalJournalofAlgebraand Statistics2(2016)157–167.
[2] M.AkramandS.Shahzadi,Neutrosophicsoftgraphswithapplication,JournalofIntelligent andFuzzySystemsDOI:10.3233/JIFS-16090(2016)1–18.
[3] M.AkramandS.Shahzadi,Representationofgraphsusingintuitionisticneutrosophicsoft sets,JournalofMathematicalAnalysis7(2016)1–23.
[4] S.Ashraf,S.Naz,H.RashmanlouandM.A.Malik,Regularityofgraphsinsinglevalued neutrosophicenvironment(2016)(Inpress).
[5] K.Atanassov,Intuitionisticfuzzysets,FuzzySetsandSystems20(1986)87–96.
[6] K.AtanassovandG.Gargov,Intervalvaluedintuitionisticfuzzysets,FuzzySetsandSystems 31(1989)343–349.
[7] S.Broumi,M.Talea,A.BakaliandF.Smarandache,SingleValuedNeutrosophicGraphs, JournalofNewTheory10(2016)86–101.
[8] S.Broumi,M.Talea,F.SmarandacheandA.Bakali,SingleValuedNeutrosophicGraphs: Degree,OrderandSize.IEEEWorldCongressonComputationalIntelligence(2016)2444–2451.
[9] S.Broumi,M.Talea,A.BakaliandF.Smarandache,OnBipolarSingleValuedNeutrosophic Graphs,JournalofNewTheory11(2016)84–102.
[10] S.Broumi,F.Smarandache,M.TaleaandA.Bakali,AnIntroductiontoBipolarSingleValued NeutrosophicGraphTheory.AppliedMechanicsandMaterials841(2016)184–191.
[11] S.Broumi,A.Bakali,M.Talea,F.SmarandacheandL.Vladareanu,ApplyingDijkstraalgorithmforsolvingneutrosophicshortestpathproblem,Proceedingsofthe2016International ConferenceonAdvancedMechatronicSystems,Melbourne,Australia(2016)412–416.
AliHassanetal./Ann.FuzzyMath.Inform. 14 (2017),No.1,55–73
[12] S.Broumi,A.Bakali,M.TaleaandF.Smarandache,AnIsolatedIntervalValuedNeutrosophic Graphs,CriticalReviewXIII(2016)67–80.
[13] S.Broumi,F.Smarandache,M.TaleaandA.Bakali,Decision-MakingMethodBasedOnthe IntervalValuedNeutrosophicGraph,IEEEConfernceFutureTechnologie(2016)44–50.
[14]
S.Broumi,A.Bakali,M.Talea,F.SmarandacheandM.Ali,ShortestPathProblemunder BipolarNeutrosphicSetting,AppliedMechanicsandMaterials859(2016)59–66.
[15] S.Broumi,A.Bakali,M,TaleaandF,Smarandache,IsolatedSingleValuedNeutrosophic Graphs.NeutrosophicSetsandSystems11(2016)74–78.
[16] S.Fathi,H.ElchawalbyandA.A.Salama,Aneutrosophicgraphsimilaritymeasures,ChapterinBookentitledby:NewTrendsinNeutrosophicTheoryandApplications-Florentin SmarandacheandSurpatiPramanik(Editors)(2016)223–230.
[17] A.HassanandM.A.Malik,Theclassesofbipolarsinglevaluedneutrosophicgraphs,TWMS JournalofAppliedandEngineeringMathematics(2016)(Inpress).
[18] A.Hassan,M.A.Malik,S.BroumiandF.Smarandache,Regularsinglevaluedneutrosophic hypergraphs,NeutrosophicSetsandSystems13(2016)84–89.
[19] A.HassanandM.A.Malik,Generalizedbipolarsinglevaluedneutrosophichypergraphs, TWMSJournalofAppliedandEngineeringMathematics(2016)(Inpress).
[20] A.HassanandM.A.Malik,Generalizedneutrosophichypergraphs,TWMSJournalofApplied andEngineeringMathematics(2016)(Inpress).
[21] A.Hassan,M.A.Malik,S.BroumiandF.Smarandache,Regularbipolarsinglevaluedneutrosophichypergraphs,NeutrosophicSetsandSystems(2016)(Inpress).
[22] A.HassanandM.A.Malik,Studiesonneutrosophicgraphs,TWMSJournalofAppliedand EngineeringMathematics(2016)(Inpress).
[23] A.HassanandM.A.Malik,Specialtypesofsinglevaluedneutrosophicgraphs,TWMSJournal ofAppliedandEngineeringMathematics(2016)(Inpress).
[24] A.HassanandM.A.Malik,Specialtypesofintervalvaluedneutrosophicgraphs,Punjab UniversityJournalofMathematics(2017)(submitted).
[25] A.HassanandM.A.Malik,Them-Polarsinglevaluedneutrosophicgraphs,TWMSJournal ofAppliedandEngineeringMathematics(2016)(Inpress).
[26] A.Hassan,M.A.Malik,S.Broumi,M.Talea,A.BakaliandF.Smarandache,Isomorphism onsinglevaluedneutrosophicHypergraphs,CriticalReviewXIII(2016)19–40.
[27] A.Hassan,M.A.Malik,S.Broumi,M.Talea,A.BakaliandF.Smarandache,Isomorphism onintervalvaluedneutrosophicHypergraphs,CriticalReviewXIII(2016)41–65.
[28] A.Hassan,M.A.Malik,S.Broumi,M.Talea,A.BakaliandF.Smarandache,Isomorphism onbipolarsinglevaluedneutrosophicHypergraphs,CriticalReviewXIII(2016)79–102.
[29] K.Hur,P.K.Lim,J.G.LeeandJ.Kim,Thecategoryofneutrosophiccrispsets,Ann.Fuzzy Math.Inform.(2017)12pages.
[30] P.D.LiuandY.M.Wang,MultipleAttributeDecision-MakingMethodBasedonSingleValuedNeutrosophicNormalizedWeightedBonferroniMean,NeuralComputingandApplications (2014)2001–2010.
[31] P.D.LiuandY.C.Chu,Y.W.LiandY.B.Chen,Somegeneralizedneutrosophicnumber HamacheraggregationoperatorsandtheirapplicationtoGroupDecisionMaking,InternationalJournalofFuzzySystems16(2014)242–255.
[32] P.D.LiuandY.M.Wang,IntervalneutrosophicprioritizedOWAoperatoranditsapplication tomultipleattributedecisionmaking,JournalofSystemsScienceandComplexity29(2016) 681–697.
[33] P.D.LiuandL.L.Shi,TheGeneralizedHybridWeightedAverageOperatorBasedonInterval NeutrosophicHesitantSetandItsApplicationtoMultipleAttributeDecisionMaking,Neural ComputingandApplications26(2015)457–471.
[34] P.D.LiuandG.L.Tang,Somepowergeneralizedaggregationoperatorsbasedontheinterval neutrosophicnumbersandtheirapplicationtodecisionmaking,JournalofIntelligentand FuzzySystems30(2016)2517–2528.
[35] P.D.LiuandH.G.Li,Multipleattributedecisionmakingmethodbasedonsomenormal neutrosophicBonferronimeanoperators,NeuralComputingandApplications28(2017)179–194.
AliHassanetal./Ann.FuzzyMath.Inform. 14 (2017),No.1,55–73
[36] P.D.LiuandF.Teng,Multipleattributedecisionmakingmethodbasedonnormalneutrosophicgeneralizedweightedpoweraveragingoperator,internationaljournalofmachine learningandcybernetics10.1007/s13042-015-0385-y.
[37] P.D.LiuandL.L.Shi,SomeNeutrosophicUncertainLinguisticNumberHeronianMeanOperatorsandTheirApplicationtoMulti-attributeGroupDecisionmaking,NeuralComputing andApplicationsdoi:10.1007/s00521-015-2122-6.
[38] M.A.MalikandA.Hassan,Singlevaluedneutrosophictrees,TWMSJournalofAppliedand EngineeringMathematics(2016)(Inpress).
[39] S.MehraandM.Singh,Singlevaluedneutrosophicsignedgarphs,InternationalJournalof computerApplications9(2017)31–37.
[40] N.ShahandA.Hussain,NeutrosophicSoftGraphs,NeutrosophicSetsandSystems11(2016) 31–44.
[41] N.ShahandS.Broumi,IrregularNeutrosophicGraphs,NeutrosophicSetsandSystems13 (2016)47–55.
[42] A.ShannonandK.Atanassov,AFirstSteptoaTheoryoftheIntuitionisticFuzzyGraphs, Proc.oftheFirstWorkshoponFuzzyBasedExpertSystems(D.akov,Ed.),Sofia(1994) 59–61.
[43] F.Smarandache,Neutrosophy,NeutrosophicProbability,SetandLogic,ProQuestInformation andLearning,AnnArbor,Michigan,USA,http://fs.gallup.unm.edu/eBookneutrosophics6.pdf (lasteditiononline).
[44] F.Smarandache,Refinedliteralindeterminacyandthemultiplicationlawofsubindeterminacies,NeutrosophicSetsandSystems9(2015)58–63.
[45] F.Smarandache,SymbolicNeutrosophicTheory(Europanovaasbl,Brussels,Belgium(2015) 195–200.
[46] F.Smarandache,Neutrosophicoverset,neutrosophicunderset,Neutrosophicoffset,Similarly forNeutrosophicOver-/Under-/OffLogic,Probability,andStatistic,PonsEditions,Brussels (2016)170–170.
[47] I.Turksen,Intervalvaluedfuzzysetsbasedonnormalforms,FuzzySetsandSystems20(1986) 191–210.
[48] V.Ulucay,M.Sahin,SBroumi,A.Bakali,M.TaleaandF.Smarandache,Decision-Making MethodbasedonNeutrosophicSoftExpertGraphs,(submited).
[49] H.Wang,F.Smarandache,Y.ZhangandR.Sunderraman,SinglevaluedNeutrosophicSets, MultisspaceandMultistructure,4(2010)410–413.
[50] L.Zadeh,Fuzzysets,InformandControl8(1965)338–353.
[51] Moreinformationonhttp://fs.gallup.unm.edu/NSS/.
AliHassan (alihassan.iiui.math@gmail.com)
DepartmentofMathematics,UniversityofPunjab(Newcampus),Lahore(Pakistan)
MuhammadAslamMalik (aslam@math.pu.edu.pk)
DepartmentofMathematics,UniversityofPunjab(Newcampus),Lahore(Pakistan)
SaidBroumi (broumisaid78@gmail.com)
LaboratoryofInformationprocessing,FacultyofScienceBenMÆSik,University HassanII,B.P7955,SidiOthman,Casablanca,Morocco
AssiaBakali (assiabakali@yahoo.fr)
EcoleRoyaleNavale,BoulevardSourJdid,B.P16303Casablanca,Morocco
MohamedTalea (taleamohamed@yahoo.fr)
LaboratoryofInformationprocessing,FacultyofScienceBenMÆSik,University HassanII,B.P7955,SidiOthman,Casablanca,Morocco
FlorentinSmarandache (fsmarandache@gmail.com) 72
AliHassanetal./Ann.FuzzyMath.Inform. 14 (2017),No.1,55–73
DepartmentofMathematics,UniversityofNewMexico,705GurleyAvenue,Gallup, NM87301,USA