TWMSJ.App.Eng.Math.V.8,N.1a,2018,pp.255-266
SINGLEVALUEDNEUTROSOPHICTREES
A.HASSAN1 ,M.A.MALIK2 , §
Abstract. Theedgeconnectivityplaysimportantroleincomputernetworkproblems andpathproblems.Inthispaper,weintroducespecialtypesofsinglevaluedneutrosophic (SVN)bridges,singlevaluedneutrosophiccut-vertices,singlevaluedneutrosophiccycles andsinglevaluedneutrosophictreesinsinglevaluedneutrosophicgraphs,andintroduced someoftheirproperties.
Keywords:SVN-cycles,SVN-trees,SVN-bridges,SVN-cut-verticesandSVN-levels.
AMSSubjectClassification:05C75.
1. Introduction
NeutrosopicsetsareintroducedbySmarandache[4]whicharethegeneralizationof fuzzysetsandintuitionisticfuzzysets.TheNeutrosophicsetshasmanyapplicationsin medical,managementsciences,lifesciencesandengineering,graphtheory,robotics,automatatheoryandcomputerscience.ThemeasureofSVNSsintroducedbySahinand kucuk[5].SinglevaluedneutrosophicgraphswasintroducedbyBrumi,Talea,Bakaliand Smarandache[1,2,3].Thedegree,orderandsizeoffuzzygraphdiscussedGani[10]. Cyclesandco-cycleswasintroducedbyMordesonandNair[9].SunithaandVijayakumar [6]givesthedefinitionofcomplementofafuzzygraphforunderstandandutilizeingeneralconceptoffuzzygraphswithrespecttocomplementproperties.SunithaandVijay kumar[7,8]alsointroducedpropertiesoffuzzycutvertex,fuzzytreeandfuzzybridges andobtainedmanyresultsonmetricspaces.
Singlevaluedneutrosophicgraphshavemanyapplicationsincomputersciencesuchas imagesegmentation,clusteringandnetworkproblems.Inthispaperwediscusstheconceptsofsinglevaluedneutrosophicbridges,singlevaluedneutrosophiccycles,singlevalued neutrosophictrees,singlevaluedneutrosophicfirmandsinglevaluedneutrosophicblocks onthebasisofweightofedgeconnectivity.
1 DepartmentofMathematics,UniversityofthePunjab,Quaid-e-AzamCampus,Lahore-54590, Pakistan.
e-mail:alihassan.iiui.math@gmail.com;ORCID:https://orcid.org/0000-0003-1648-337X.
2 DepartmentofMathematics,UniversityofthePunjab,Quaid-e-AzamCampus,Lahore-54590, Pakistan.
e-mail:malikpu@yahoo.com;ORCID:https://orcid.org/0000-0003-4434-7893.
§ Manuscriptreceived:November14,2016;accepted:January10,2017.
TWMSJournalofAppliedandEngineeringMathematics,Vol.8,No.1a c I¸sıkUniversity,Department ofMathematics,2018;allrightsreserved.
2. Preliminaries
Inthissection,werecalldefinitionstounderstandtheconceptsofSVNtrees.
Definition2.1. [4] Let Z beacrispset,Asinglevaluedneutrosophicset(SVNS) C is characterizedbytruthmembershipfunction TC (p), anindeterminacymembershipfunction IC (p) andafalsitymembershipfunction FC (p). Foreverypoint p ∈ Z; TC (p),IC (p),FC (p) ∈ [0,1].
Definition2.2. [1,2,3] Asinglevaluedneutrosophicgraph(SVNG)isapair G =(C,D) ofacrispgraph G∗ =(V,E), where C isSVNSon V and D isSVNSon E suchthat
TD(p,q) ≤ min(TC (p),TC (q))
ID(p,q) ≥ max(IC (p),IC (q))
FD(p,q) ≥ max(FC (p),FC (q)) where 0 ≤ TD(p,q)+ ID(p,q)+ FD(p,q) ≤ 3 ∀x,y ∈ V.
Definition2.3. [1,2,3] Let G =(C,D) beaSVNGofacrispgraph G∗ =(V,E),G is saidtobecompleteSVNG,if
TD(p,q)=min(TC (p),TC (q))
ID(p,q)=max(IC (p),IC (q))
FD(p,q)=max(FC (p),FC (q)) ∀p,q ∈ V.
Definition2.4. [1,2,3] Apath P inaSVNG G =(A,B) isasequenceofdistinct vertices p1,p2,p3,...,pm suchthat TB (pj pj+1) > 0,IB (pj pj+1) > 0,FB (pj pj+1) > 0 for 1 ≤ j ≤ m.
Definition2.5. [1,2,3] Ifthereisatleastonepathbetweeneverypairofverticesin SVNG G =(A,B) ThenGissaidtobeconnected,elseGisdisconnected.
Definition2.6. [1,2,3] ThepartialSVNsubgraphofSVNG G =(C,D) onacrispgraph G∗ =(V,E) isaSVNG H =(C ,D ), suchthat (1) C ⊆ C, thatis ∀x ∈ V
TC (p) ≤ TC (p),IC (p) ≥ IC (p),FC (p) ≥ FC (p).
(2) D ⊆ D, thatis ∀pq ∈ E
TD (pq) ≤ TD(pq),ID (pq) ≥ ID(pq),FD (pq) ≥ FD(pq).
Definition2.7. [1,2,3] TheSVNsubgraphofSVNG G =(C,D) ofcrispgraph G∗ = (V,E) isaSVNG H =(C ,D ) ona H ∗ =(V ,E ), suchthat (1) C = C, thatis ∀p ∈ V
TC (p)= TC (p),IC (p)= IC (p),FC (p)= FC (p)
(2) D = D, thatis ∀pq ∈ E intheedgeset E
TD (pq)= TD(pq),ID (pq)= ID(pq),FD (pq)= FD(pq)
3. Singlevaluedneutrosophictrees
Theconceptofconnectivityplaysanimportantroleinpathandnetworkproblems,we introduceherethebasicconceptofbridge,cycle,tree,cutvertexandlevelsofSVNG.
Definition3.1. Let C =(TC ,IC ,FC ) beaSVNSon X, thesupportof C isdenotedand definedbysupp(C)= supp(TC ) ∪ supp(IC ) ∪ supp(FC ), where supp(TC )= {x : x ∈ X,TC (x) > 0}, supp(IC )= {x : x ∈ X,IC (x) > 0}, supp(FC )= {x : x ∈ X,FC (x) > 0}. Wecallsupp(TC ), supp(IC ) andsupp(FC ) truthsupport,indeterminacysupportandfalsity supportrespectively.
Definition3.2. Let C =(TC ,IC ,FC ) beaSVNSon X, the (ξ,η,ζ)-levelsubsetof C is denotedanddefinedby A(ξ,η,ζ) = C ξ ∪ C η ∪ C ζ , where
C ξ = {x : x ∈ X,TC (x) ≥ ξ}, C η = {x : x ∈ X,IC (x) ≤ η}, C ζ = {x : x ∈ X,FC (x) ≤ ζ}
Definition3.3. Let C =(TC ,IC ,FC ) beaSVNSon X, theheightof C isdenoteand definedby h(C)=(hT (C),hI (C),hF (C)), where
hT (C)=sup{TC (x): x ∈ X}, hT (C)=inf{IC (x): x ∈ X}, hT (C)=inf{FC (x): x ∈ X}.
TheSVNS C isnormalifthereis p ∈ X suchthat TC (p)=1,IC (p)=0 and FC (p)=0
Definition3.4. Let C =(TC ,IC ,FC ) beaSVNSon X, thedepthof A isdenoteand definedby d(C)=(dT (C),dI (C),dF (C)), where dT (C)=inf{TC (x): x ∈ X}, dT (C)=sup{IC (x): x ∈ X}, dT (C)=sup{FC (x): x ∈ X}.
Definition3.5. ThecrispgraphofaSVNG G =(A,B) is G∗ =(A∗,B∗), where A∗ = supp(A) and B∗ = supp(B). Let G(ξ,η,ζ) =(A(ξ,η,ζ),B(ξ,η,ζ)) where ξ,η,ζ ∈ [0, 1], A(ξ,η,ζ) = {x : x ∈ V,TA(x) ≥ ξ,IA(x) ≤ η,FA(x) ≤ ζ} isthe (ξ,η,ζ)-levelsubsetof A and B(ξ,η,ζ) = {xy : xy ∈ E,TB (xy) ≥ ξ,IB (xy) ≤ η,FB (xy) ≤ ζ} isthe (ξ,η,ζ)-levelsubsetof B. Notethat G(ξ,η,ζ) isacrispgraph.
Definition3.6. AbridgeinSVNG G =(A,B) issaidtobe T -bridge,ifremovingthe edge xy decreasesthe T -strengthofconnectivityofsometwovertices.Abridgein G is saidtobe I-bridge,ifremovingtheedge xy increasesthe I-strengthofconnectednessof twovertices.Abridgein G issaidtobe F -bridge,ifbyremovingtheedge xy increases the F -strengthofconnectednessofsometwovertices.AbridgeinSVNG G issaidtobe SVN-bridge xy ifitis T -bridge, I-bridgeand F -bridge.
Definition3.7. Let G =(A,B) beaSVNGonthecrispgraph G∗ =(V,E), the T -strength ofconnectednessbetween x and y in V is
T ∞ B (xy)=sup{T k B (xy): k =1, 2,...,n},
T ∞ B (xy)=sup{TB (xv1)∧TB (v1v2)∧ ∧TB (vk 1y): x,v1,v2,...,vk 1,y ∈ V,k =1, 2,...,n}, the I-strengthofconnectednessbetween x and y in V is
I ∞ B (xy)=inf{I k B (xy): k =1, 2,...,n},
I ∞ B (xy)=inf{IB (xv1)∨IB (v1v2)∨ ∨IB (vk 1y): x,v1,v2,...,vk 1,y ∈ V,k =1, 2,...,n} andthe F -strengthofconnectednessbetween x and y in V is
F ∞ B (xy)=inf{F k B (xy): k =1, 2,...,n},
F ∞ B (xy)=inf{FB (xv1)∨FB (v1v2)∨ ∨FB (vk 1y): x,v1,v2,...,vk 1,y ∈ V,k =1, 2,...,n}
The T -strength, I-strengthand F -strengthbetween x and y in G isdenotedby T ∞ G (xy), I ∞ G (xy) and F ∞ G (xy) respectively.Next T ∞ B (xy),I ∞ B (xy) and F ∞ B (xy) denote T ∞ G−{xy}(xy), I ∞ G−{xy}(xy) and F ∞ G−{xy}(xy) where G −{xy} isobtainedfrom G byremovingtheedge xy.
Definition3.8. Let G =(A,B) beaSVNGonthecrispgraph G∗ =(V,E), (i) xy ∈ E iscalledbridgeif xy isbridgeof G∗ =(A∗,B∗) (ii) xy ∈ E iscalledSVN-bridgeif
T ∞ B (uv) <T ∞ B (uv),I ∞ B (uv) >I ∞ B (uv),F ∞ B (uv) >F ∞ B (uv)
forsome uv ∈ E, where TB ,IB and FB are TB ,IB and FB restrictedto V × V −{xy,yx}. (iii) xy ∈ E iscalledaweakSVN-bridgeifthereexist (ξ,η,ζ) ∈ (0,h(B)] suchthat xy is bridgeof G(ξ,η,ζ) , where 0=(0, 0, 0) (iv) xy ∈ E iscalledpartialSVN-bridgeif xy isbridge ∀(ξ,η,ζ) ∈ (d(B),h(B)] ∪{h(B)}. (v) xy ∈ E iscalledfullSVN-bridgeif xy isbridgefor G(ξ,η,ζ) forall (ξ,η,ζ) ∈ (0,h(B)], where 0=(0, 0, 0)
Example3.1. ConsidertheconnectedSVNG G =(A,B) ofacrispgraph G∗ =(V,E), where A and B beSVNSsof V = {α,β,γ} and E = {αβ,βγ,γα} respectivelydefinedin Table.1.ThenitshowsconnectedSVNGhasnobridgesofanyoffivetypes.
A TA IA FA B TB IB FB
α 1.0 0.0 0.0 αβ 0.9 0.1 0.1
β 1.0 0.0 0.0 βγ 0.9 0.1 0.1
γ 1.0 0.0 0.0 γα 0.9 0.1 0.1
Table1. SVNSsofSVNGwithoutSVN-bridges.
Example3.2. ConsidertheconnectedSVNG G =(A,B) ofacrispgraph G∗ =(V,E), where A and B beSVNSsof V = {α,β,γ,δ} and E = {αβ,βγ,γδ,δα} respectivelydefinedinTable.2.Then, d(B)=(0 1, 0 5, 0 5) and h(B)=(0 9, 0 1, 0 1) Thus (ξ,η,ζ) ∈ (0,h(B)] whichmeansfor 0 <ξ ≤ 0 1, 0 <η ≤ 0 5 and 0 <ζ ≤ 0 5, weobtain G(ξ,η,ζ) =(V, {αβ,βγ,γδ,δα}), for 0.1 <ξ ≤ 0.9, 0 <η ≤ 0.1 and 0 <ζ ≤ 0.1, we obtain G(ξ,η,ζ) =(V, {αδ,γδ}). Thus γδ isfullSVN-bridgeand δα ispartialSVN-bridge butnotfullSVN-bridge.
A TA IA FA B TB IB FB
α 1.0 0.0 0.0 αβ 0.1 0.5 0.5
β 1.0 0.0 0.0 βγ 0.1 0.5 0.5
γ 1.0 0.0 0.0 γδ 0.9 0.1 0.1 δ 1.0 0.0 0.0 δα 0.9 0.1 0.1
Table2. SVNSsofSVNGwithoutfullSVN-bridge.
Remark3.1. Let xy beabridgein G∗ then xy isSVN-bridgeifandonlyif TB (xy) >T ∞ B (xy),IB (xy) <I ∞ B (xy),FB (xy) <F ∞ B (xy).
Remark3.2. The xy isSVNbridgeifandonlyif xy isnotweakestbridgeofanycycle.
Proposition3.1. Theedge xy isSVN-bridgeifandonlyif xy isbridgefor G∗ and TB (xy)= h(TB ),IB (xy)= h(IB ),FB (xy)= h(FB ).
Proof. Supposethat xy isfullbridgethen xy isbridgeof G(ξ,η,ζ) forall(ξ,η,ζ) ∈ (0,h(B)]=(0,h(TB )] × (0,h(IB )] × (0,h(FB )] Hence xy ∈ Bh(B) andso TB (xy)= h(TB ),IB (xy)= h(IB ),FB (xy)= h(FB ) since xy isbridgefor G(ξ,η,ζ) forall(ξ,η,ζ) ∈ (0,h(B)] Itfollowsthat xy isbridgefor G∗ , since V = Ad(B) and E = Bd(B)
Conversely:Suppose xy isbridgefor G∗ and
TB (xy)= h(TB ),IB (xy)= h(IB ),FB (xy)= h(FB )
Then xy ∈ B(ξ,η,ζ) forall(ξ,η,ζ) ∈ (0,h(B)], thussince xy isbridgefor G∗,xy isbridge for G(ξ,η,ζ) forall(ξ,η,ζ) ∈ (0,h(B)], sinceeach G(ξ,η,ζ) issubgraphof G∗ Hence xy isa fullSVN-bridge.
Proposition3.2. Ifanarc xy isnotinthecycleofcrispgraph G∗ , thenthefollowing conditionsareequivalent.
(i) TB (xy)= h(TB ),IB (xy)= h(IB ),FB (xy)= h(FB ) (ii) xy ispartialSVN-bridge. (iii) xy isfullSVN-bridge.
Proof. Since xy isnotcontainedinacycleof G∗ and xy isbridgeof G∗.Henceby proposition3.1,(i) ⇔ (iii)obvious(iii) ⇔ (ii) Nextsupposethat(ii)holds,then xy isbridgefor G(ξ,η,ζ) forall(ξ,η,ζ) ∈ (d(B),h(B)]andso xy ∈ Bh(B) Thus TB (xy)= h(TB ),IB (xy)= h(IB ),FB (xy)= h(FB ), thus(i)holds.
Remark3.3. If xy isabridge,then xy isweakSVN-bridgeandSVN-bridge.
Proposition3.3. Anarc xy isSVN-bridgeifandonlyif xy isweakSVN-bridge.
Proof. Supposethat xy isaweakSVN-bridge,thenthereexists(ξ,η,ζ) ∈ (0,h(B)]such that xy isbridgefor G(ξ,η,ζ) Hencebyremoving xy itdisconnects G(ξ,η,ζ) , thusanypath from x to y in G hasanedge uv with TB (uv) <ξ,IB (uv) >η,FB (uv) >ζ. Henceby removalofarc xy impliesthat
T ∞ B (xy) <ξ ≤ T ∞ B (xy),I ∞ B (xy) >η ≥ I ∞ B (xy),F ∞ B (xy) >ζ ≥ F ∞ B (xy)
Hence xy isSVN-bridge. Conversely:Supposethat xy isSVN-bridge,thenthereisanarc uv suchthatbyremoving of xy impliesthat T ∞ B (uv) <T ∞ B (uv),I ∞ B (uv) >I ∞ B (uv),F ∞ B (uv) >F ∞ B (uv)
Hence xy isoneverystrongestpathjoining u and v andinfact TB (uv) ≥,IB (uv) ≤ and FB (uv) ≤ thisvalue.Thustheredoesnotexistapathotherthan xy connecting x and y in G(TB (xy),IB (xy),FB (xy)) , elsethisotherpathwithout xy wouldbeofstrength ≥ TB (xy), ≤ IB (xy)and ≤ FB (xy)andwouldbepartofapathconnecting u and v ofstrongest length,contrarytofactthat xy isoneverysuchpath.Hence xy isoneverysuchpath. Hence xy isabridgeof G(TB (xy),IB (xy),FB (xy)) and 0 <TB (xy) ≤ h(TB ), 0 <IB (xy) ≤ h(IB ), 0 <FB (xy) ≤ h(FB )
Thus(TB (xy),IB (xy),FB (xy))arethedesired(ξ,η,ζ).
Definition3.9. Avertex x ∈ V in G iscalled T -cutvertexifbyremovingitdecreasesthe T -strengthofconnectivitybetweensomepairofnodes.Avertex x ∈ V in G iscalled I-cut vertexifbyremovingitincreasesthe I-strengthofconnectivitybetweensomepairofnodes. Avertex x ∈ V in G iscalled F -cutvertexifbyremovingitincreasesthe F -strengthof connectivitybetweensomepairofnodes.Avertex x ∈ V isaSVN-cutvertexifitis T -cut vertex, I-cutvertexand F -cutvertex.
Definition3.10. Let x ∈ V, (i) Thevertex x ∈ V iscalledacutvertex,if x isacutvertexof G∗ =(A∗,B∗) (ii) Thevertex x ∈ V iscalledSVN-cutvertexif T ∞ B (uv) <T ∞ B (uv),I ∞ B (uv) > I ∞ B (uv),F ∞ B (uv) >F ∞ B (uv) forsome u,v ∈ V, where TB ,IB and FB are TB ,IB and FB restrictedto V × V −{xz,zx : z ∈ V } (iii) Thevertex x ∈ V iscalledapartialsinglevaluedneutrosophiccutvertexif x isacut vertexfor G(ξ,η,ζ) ∀(ξ,η,ζ) ∈ (d(B),h(B)] ∪{h(B)}. (iv) Thevertex x ∈ V iscalledaweakSVN-cutvertexifthereexists (ξ,η,ζ) ∈ (0,h(B)] suchthat x isacutvertexof G(ξ,η,ζ) (v) Thevertex x ∈ V iscalledafullSVN-cutvertexif x isacutvertexfor G(ξ,η,ζ) ifthere exists (ξ,η,ζ) ∈ (0,h(B)]
Example3.3. ConsidertheconnectedSVNG G =(A,B) ofacrispgraph G∗ =(V,E), where A and B beSVNSsof V = {α,β,γ} and E = {αβ,βγ,γα} respectivelydefinedin Table.3.Then d(B)=(0 5, 0 4, 0 4) and h(B)=(0 9, 0 1, 0 1) Thus (ξ,η,ζ) ∈ (0,h(B)] whichmeansfor 0 <ξ ≤ 0.5, 0 <η ≤ 0.4 and 0 <ζ ≤ 0.4, weobtain G(ξ,η,ζ) = (V, {αβ,βγ,γα}), for 0 5 <ξ ≤ 0 9, 0 <η ≤ 0 1 and 0 <ζ ≤ 0 1, weobtain G(ξ,η,ζ) = (V, {αβ,γα}). Hence α isSVN-cutvertexandapartialSVN-cutvertexbutneitheracut vertexnorafullcutvertex.
A TA IA FA B TB IB FB
α 1.0 0.0 0.0 αβ 0.9 0.1 0.1 β 1.0 0.0 0.0 βγ 0.5 0.4 0.4 γ 1.0 0.0 0.0 γα 0.9 0.1 0.1
Table3. SVNSsofSVNGwithpartialSVN-cutvertex.
Remark3.4. Let G beaSVNGsuchthat G∗ isacycle,thenavertexisaSVN-cutvertex of G ifandonlyifitisasamevertexoftwoSVN-bridges.
Remark3.5. If z ∈ V isasamevertexofatleasttwoSVN-bridges,then z isaSVNcut vertex.
Remark3.6. If G isacompleteSVNG,then T ∞ B (uv)= TB (uv),I ∞ B (uv)= IB (uv) and F ∞ B (uv)= FB (uv)
Remark3.7. ThecompleteSVNGhasnoSVN-cutvertex.
Definition3.11. (i) TheSVNG G iscalledablockif G∗ isablock. (ii) TheSVNG G iscalledablockifithasnosinglevaluedneutrosophiccutvertices. (iii) TheSVNG G iscalledaweakblockifthereexists (ξ,η,ζ) ∈ (0,h(B)], suchthat G(ξ,η,ζ) isablock.
(iv) TheSVNG G iscalledapartialSVN-blockif G(ξ,η,ζ) isablock ∀(ξ,η,ζ) ∈ (d(B),h(B)]∪ {h(B)}
(v) TheSVNG G iscalledafullSVN-blockif G(ξ,η,ζ) isblock ∀(ξ,η,ζ) ∈ (0,h(B)]
Example3.4. ConsidertheconnectedSVNG G =(A,B) ofacrispgraph G∗ =(V,E), where A and B beSVNSsof V = {l,m,n} and E = {lm,mn,nl} respectivelydefinedin Table.4.Thenbyroutinecalculations d(B)=(0 5, 0 4, 0 4) and h(B)=(0 9, 0 1, 0 1) Thus (ξ,η,ζ) ∈ (0,h(B)] whichmeansfor 0 <ξ ≤ 0.5, 0 <η ≤ 0.4 and 0 <ζ ≤ 0.4, we obtain G(ξ,η,ζ) =(V, {lm,mn,nl}), for 0.5 <ξ ≤ 0.9, 0 <η ≤ 0.1 and 0 <ζ ≤ 0.1, we obtain G(ξ,η,ζ) =(V, {lm,ln}) Hence G isblockandaweakblockSVN-block,however G isnotSVNblocksince l isSVN-cutvertexof G, also G isnotapartialSVNblock,since l iscutvertexfor 0.5 <ξ ≤ 0.9, 0 <η ≤ 0.1 and 0 <ζ ≤ 0.1.
A TA IA FA B TB IB FB l 1.0 0.0 0.0 lm 0.9 0.1 0.1 m 1.0 0.0 0.0 mn 0.5 0.4 0.4 n 1.0 0.0 0.0 nl 0.9 0.1 0.1
Table4. SVNSsofSVN-Block.
Example3.5. ConsidertheconnectedSVNG G =(A,B) ofacrispgraph G∗ =(V,E), where A and B beSVNSsof V = {p,q,r} and E = {pq,qr,rp} respectivelydefinedin Table.5.Thenbyroutinecalculations d(B)=(0.9, 0.1, 0.1) and h(B)=(0.9, 0.1, 0.1). Thus (ξ,η,ζ) ∈ (0,h(B)] whichmeansfor 0 <ξ ≤ 0 9, 0 <η ≤ 0 1 and 0 <ζ ≤ 0 1, weobtain G(ξ,η,ζ) =(V, {pq,qr,rp}), for 0 5 <ξ ≤ 0 9, 0 <η ≤ 0 1 and 0 <ζ ≤ 0 1, we obtain G(ξ,η,ζ) =(V, {pq,rp}) Hence G isblock,aSVN-blockandafullSVN-block.
A TA IA FA B TB IB FB p 1.0 0.0 0.0 pq 0.9 0.1 0.1 q 1.0 0.0 0.0 qr 0.5 0.4 0.4 r 1.0 0.0 0.0 rp 0.9 0.1 0.1
Table5. SVNSsoffullSVN-Block.
Definition3.12. TheconnectedSVNG G issaidtobeafirmif min{TA(x): x ∈ V }≥ max{TB (xy): xy ∈ E},
max{IA(x): x ∈ V }≤ min{IB (xy): xy ∈ E},
max{FA(x): x ∈ V }≤ min{FB (xy): xy ∈ E}
Definition3.13. Let G beaconnectedSVNG,then
(i) TheSVNG G issaidtobeacyclewhenever G∗ isacycle. (ii) TheSVNG G issaidtobeaSVN-cyclewhenever G∗ isacycleandthereisaunique pq ∈ E suchthat
TB (pq)=min{TB (uv): uv ∈ E},
IB (pq)=max{IB (uv): uv ∈ E},
FB (pq)=max{FB (uv): uv ∈ E}.
(iii) TheSVNG G issaidtobeaweakSVN-cycleifthereexists (ξ,η,ζ) ∈ (0,h(B)] such that G(ξ,η,ζ) isacycle.
(iv) TheSVNG G iscalledapartialSVN-cycleif G(ξ,η,ζ) isacycle ∀(ξ,η,ζ) ∈ (d(B),h(B)]∪ {h(B)}.
(v) TheSVNG G iscalledafullSVN-cycleif G(ξ,η,ζ) iscycle ∀(ξ,η,ζ) ∈ (0,h(B)].
Remark3.8. TheSVN-cycle G ispartialSVN-cycleifandonlyif G isafullSVN-cycle. Remark3.9. TheSVNG G isafullSVN-cycleifandonlyif B isconstanton E. and G isacycle.
Definition3.14. AconnectedSVNG G =(A,B) issaidtobeaSVN-treeifithasaSVN spanningsubgraph H =(A,C) whichisatree,whereforalledges xy notin H satisfying TB (xy) <T ∞ C (xy),IB (xy) >I ∞ C (xy),FB (xy) >F ∞ C (xy).
Definition3.15. (i) TheSVNG G iscalledaforestif G∗ isaforest. (ii) TheSVNG G =(A,B) issaidtobeaSVN-forestif G hasaSVNspanningsubgraph forest H =(A,C),whereallarcs uv ∈ E W, satisfying TB (uv) <T ∞ C (uv),IB (uv) > I ∞ C (uv),FB (uv) >F ∞ C (uv) (iii) TheSVNG G iscalledaweakSVN-forestif ∀(ξ,η,ζ) ∈ (0,h(B)] suchthat G(ξ,η,ζ) is aforest. (iv) TheSVNG G iscalledapartialSVN-forestif G(ξ,η,ζ) isaforest ∀(ξ,η,ζ) ∈ (d(B),h(B)]∪ {h(B)} (v) TheSVNG G iscalledafullSVN-forestif G(ξ,η,ζ) isforestforall (ξ,η,ζ) ∈ (0,h(B)].
Example3.6. ConsidertheconnectedSVNG G =(A,B) ofacrispgraph G∗ =(V,E), where A and B beSVNSsof V = {α,β,γ,δ} and E = {αβ,βγ,γδ,δα} respectivelydefined inTable.6.Then d(B)=(0 5, 0 4, 0 4) and h(B)=(0 9, 0 1, 0 1), for 0 <ξ ≤ 0 5, 0 <η ≤ 0 4 and 0 <ζ ≤ 0 4, weobtain G(ξ,η,ζ) =(V, {αβ,βγ,γδ,δα}), for 0 5 <ξ ≤ 0 9, 0 <η ≤ 0 1 and 0 <ζ ≤ 0 1, weobtain G(ξ,η,ζ) =(V, {αβ,γδ}) Hence G isapartial SVN-forestbutneitherSVN-forestnorfullSVNforest.
A TA IA FA B TB IB FB
α 1.0 0.0 0.0 αβ 0.9 0.1 0.1
β 1.0 0.0 0.0 βγ 0.5 0.4 0.4
γ 1.0 0.0 0.0 δα 0.5 0.4 0.4
δ 1.0 0.0 0.0 γδ 0.9 0.1 0.1
Table6. SVNSsofpartialSVN-forest.
Proposition3.4. TheSVNG G isfullSVN-forestifandonlyif G isforest.
Proof. Supposethat G isafullSVN-forest,then G∗ isaforest. Conversely:Supposethat G isforest,then G∗ isaforestandsomustbe G(ξ,η,ζ) forall (ξ,η,ζ) ∈ (0,h(B)], sinceeach G(ξ,η,ζ) isasubgraphof G∗ , thiscompletestheproof.
Proposition3.5. TheSVNG G isweakSVN-forestifandonlyif G doesnotcontaina cyclewhoseedgesareofstrength h(B).
Proof. Supposethat G containsacyclewhoseedgesareofstrength h(B), then G(ξ,η,ζ) for (ξ,η,ζ) ∈ (0,h(B)]thatcontainsthiscycleandsoisnotaforest,thus G isnotaweak SVN-forest.
Conversely:Suppose G doesnotcontainacyclewhoseedgesareofstrength h(B), then Gh(B) doesnotcontainacycleandsoitisforest.
Remark3.10. If G isaSVN-forest,then G isaweakSVN-forest.
Theorem3.1. Let G beaforestand B isaconstanton E ifandonlyif G isafull SVN-forest, G∗ and Gh(B) havethesamenumberofconnectedcomponents,and G isa firm.
Proof. Supposethat G isaforestand B isconstanton E, thenforall(ξ,η,ζ) ∈ (0,h(B)], then G(ξ,η,ζ) = G∗ andso G isfullSVN-forestalso G∗ and Gh(B) havethesamenumber ofconnectedcomponents,clearly G isafirm,since B isconstanton E. Conversepartisobvious.
Corollary3.1. TheSVNG G isatreeand B isconstanton E ifandonlyif G isafull SVN-treeand G isafirm.
Definition3.16. (i) TheSVNG G iscalledatreeif G∗ isatree. (ii) TheSVNG G =(A,B) issaidtobeaSVN-treeifithasaSVNspanningsubgraph H =(A,C) whichisatree,whereforalledges uv ∈ E W, satisfying TB (uv) < T ∞ C (uv),IB (uv) >I ∞ C (uv),FB (uv) >F ∞ C (uv)
(iii) TheSVNG G iscalledaweakSVN-treeif ∀(ξ,η,ζ) ∈ (0,h(B)] suchthat G(ξ,η,ζ) isa tree.
(iv) TheSVNG G iscalledapartialSVN-treeif G(ξ,η,ζ) isatree ∀(ξ,η,ζ) ∈ (d(B),h(B)]∪ {h(B)}
(v) TheSVNG G iscalledafullSVN-treeif G(ξ,η,ζ) istreeforall (ξ,η,ζ) ∈ (0,h(B)]
Example3.7. ConsidertheconnectedSVNG G =(A,B) ofacrispgraph G∗ =(V,E), where A and B beSVNSsof V = {α,β,γ} and E = {αβ,βγ,γα} respectivelydefined inTable.7.Then d(B)=(0 5, 0 4, 0 4) and h(B)=(0 9, 0 1, 0 1), for 0 <ξ ≤ 0 5, 0 <η ≤ 0 4 and 0 <ζ ≤ 0 4, weobtain G(ξ,η,ζ) =(V, {αβ,βγ,γα}), for 0 5 <ξ ≤ 0 9, 0 <η ≤ 0 1 and 0 <ζ ≤ 0 1, weobtain G(ξ,η,ζ) =({α,β}, {αβ}) Hence G isapartial SVN-treebutneitherSVN-treenorfullSVN-tree.
A TA IA FA B TB IB FB
α 1.0 0.0 0.0 αβ 0.9 0.1 0.1 β 1.0 0.0 0.0 βγ 0.5 0.4 0.4 γ 0.5 0.2 0.2 γα 0.5 0.4 0.4
Table7. SVNSspartialSVN-tree.
Remark3.11. If G isaSVN-tree,then G isnotcompleteSVNG.
Remark3.12. If G isaSVN-tree,thenarcsofspanningsubgraph H aretheSVN-bridges of G.
Remark3.13. If G isaSVN-tree,theninternalverticesofspanningsubgraph H arethe SVN-cutverticesof G.
Remark3.14. If G isaSVN-tree,then xy isSVN-bridgeifandonlyif T ∞ B (xy)= TB (xy),I ∞ B (xy)= IB (xy),F ∞ B (xy)= FB (xy)
Remark3.15. TheSVNG G isaSVN-treeifandonlyifthereisauniquemaximum spanningtreeof G
Remark3.16. Let G isafirm,if G isaweakSVN-tree,then G isaSVN-tree.
Definition3.17. (i) TheSVNG G iscalledaconnectedif G∗ isaconnected. (ii) TheSVNG G =(A,B) issaidtobeaSVNconnectedif G isSVN-block.
(iii) TheSVNG G iscalledaweakSVNconnectedifthereexists (ξ,η,ζ) ∈ (0,h(B)] such that G(ξ,η,ζ) isaconnected.
(iv) TheSVNG G iscalledapartialSVNconnectedif G(ξ,η,ζ) isaconnected ∀(ξ,η,ζ) ∈ (d(B),h(B)] ∪{h(B)}.
(v) TheSVNG G iscalledafullSVNconnectedif G(ξ,η,ζ) istree ∀(ξ,η,ζ) ∈ (0,h(B)]
Proposition3.6. If G isconnectedthen G isweaklyconnected.
Proof. Since G isconnectedimpliesthat G∗ isconnected.Now G∗ = Gh(B) andso G is weakconnected.
Proposition3.7. If G isfirmandweakconnectedthen G isconnected.
Proof. If G(ξ,η,ζ) isconnectedforsome(ξ,η,ζ) ∈ (0,h(B)], then G∗ isconnected,since G isfirm.
Proposition3.8. (i) If G isaweakSVN-tree,then G isweakconnectedand G isaweak SVN-forest,converselyifthereare (ξ1,η1,ζ1), (ξ2,η2,ζ2) ∈ (0,h(B)], with ξ1 <ξ2,η1 <η2 and ζ1 <ζ2 suchthat G(ξ1 ,η1 ,ζ1 ) isaforestand G(ξ2 ,η2 ,ζ2 ) isconnected,then G isweak SVN-tree.
(ii) TheSVNG G isatreeifandonlyif G isaforestand G isconnected.
(iii) TheSVNG G ispartialSVN-treeifandonlyif G isapartialSVN-forestand G is partiallyconnectedSVNG.
(iv) TheSVNG G isfullSVN-treeifandonlyif G isafullSVN-forestand G isfully connectedSVNG.
Proof. (i) If G(ξ,η,ζ) isatreeforsome(ξ,η,ζ) ∈ (0,h(B)], then G(ξ,η,ζ) isconnectedand isaforest.Forconverse,notethat G(ξ2 ,η2 ,ζ2 ) mustalsobeaforest,sincealso G(ξ2 ,η2 ,ζ2 ) is connected, G(ξ2 ,η2 ,ζ2 ) isatree. (ii), (iii) and (iv) areobvious.
Proposition3.9. TheSVNG G isfirmifandonlyif G(ξ,η,ζ) isfirmforall (ξ,η,ζ) ∈ (0,h(B)].
Proof. Suppose G isfirm,let(ξ,η,ζ) ∈ (0,h(B)], for xy ∈ T (ξ,η,ζ) then ξ ≤ TB (xy) ≤ min{TA(x): x ∈ V }≤ min{TA(x): x ∈ T ξ A} η ≥ IB (xy) ≥ max{IA(x): x ∈ V }≥ max{IA(x): x ∈ I η A} ζ ≥ FB (xy) ≥ max{FA(x): x ∈ V }≥ max{FA(x): x ∈ F ζ A}
therefore
max{TB (xy): xy ∈ T ξ B }≤ min{TA(x): x ∈ T ξ A} min{IB (xy): xy ∈ I η B }≤ max{IA(x): x ∈ I η A} min{FB (xy): xy ∈ F ζ B }≤ max{FA(x): x ∈ F ζ A} thusweconcludethat B(ξ,η,ζ)∗ = B(ξ,η,ζ),A(ξ,η,ζ)∗ = A(ξ,η,ζ) and G(ξ,η,ζ) isafirm. Conversely:Supposethat G(ξ,η,ζ) isafirmforall(ξ,η,ζ) ∈ (0,h(B)] Let
min{TA(x): x ∈ V } = ξ0 > 0 max{IA(x): x ∈ V } = η0 > 0 max{FA(x): x ∈ V } = ζ0 > 0 next
max{TB (xy): xy ∈ T ξ0 B }≤ ξ0 min{IB (xy): xy ∈ I η0 B }≥ η0 min{FB (xy): xy ∈ F ζ0 B }≥ ζ0 since G(ξ0 ,η0 ,ζ0 ) isfirmand V = A(ξ0 ,η0 ,ζ0 ) = A(ξ0 ,η0 ,ζ0 )∗ Let xy ∈ E B(ξ,η,ζ)∗ , then TB (xy) <ξ0,IB (xy) >η0 and FB (xy) >ζ0. Thus max{TB (xy): xy ∈ E}≤ ξ0 =min{TA(x): x ∈ V }, min{IB (xy): xy ∈ E}≥ η0 =max{IA(x): x ∈ V }, min{FB (xy): xy ∈ E}≥ ζ0 =max{FA(x): x ∈ V }. Hence G isfirm.
4. Conclusion
Theneutrosophicgraphshavemanyapplicationsinpathproblems,networksandcomputerscience.TheedgeconnectivityinSVNGisbasicconcepttounderstandtheconnectionsofconnectednessbetweentwosystemsofcomputers.TheSVN-bridges,cycles,trees, cut-VerticesandLevelsareintroducedhere,alsotheSVN-Blocksandfirmsareintroduced withitspropertiesandcriteriatoprovetheSVNGtobefirmorBlock.
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AliHassan iscurrentlyworkingasaresearchstudentintheDepartmentofMathematicsinKarachiUniversity.HereceivedhismasterdegreefromUniversityofthe Punjab,Lahore,Pakistan.HisareaofinterestincludesFuzzygrouptheoryandfuzzy graphtheory.
MuhammadAslamMalik iscurrentlyworkingasanassociateprofessorinthe DepartmentofMathematicsUniversityofthePunjab,Lahore.HereceivedhisPh.D degreefromUniversityofthePunjab,Lahore.Hecompletedhispostdoctoratein GraphtheoryatBirminghamUniversity,UK.HisareaofinterestincludesGraph theory,FuzzygrouptheoryandFuzzygraphtheory.
