SoftRoughNeutrosophicInfluence GraphswithApplication
HafsaMasoodMalik 1,MuhammadAkram 1, ∗ andFlorentinSmarandache 2
1 DepartmentofMathematics,UniversityofthePunjab,NewCampus,Lahore54590,Pakistan; hafsa.masood.malik@gmail.com
2 UniversityofNewMexicoMathematics&ScienceDepartment705GurleyAve.,Gallup,NM87301,USA; fsmarandache@gmail.com
* Correspondence:m.akram@pucit.edu.pk
Received:28June2018;Accepted:15July2018;Published:18July2018
Abstract: Inthispaper,weapplythenotionofsoftroughneutrosophicsetstographtheory. Wedevelopcertainnewconcepts,includingsoftroughneutrosophicgraphs,softroughneutrosophic influencegraphs,softroughneutrosophicinfluencecyclesandsoftroughneutrosophicinfluence trees.Weillustratetheseconceptswithexamples,andinvestigatesomeoftheirproperties.Wesolve thedecision-makingproblembyusingourproposedalgorithm.
Keywords: softroughneutrosophicgraphs;softroughneutrosophicinfluencegraphs;softrough neutrosophicinfluencecycles;softroughneutrosophicinfluencetrees
1.Introduction
Smarandache[1]introducedneutrosophicsetsasageneralizationoffuzzysetsandintuitionistic fuzzysets.Aneutrosophicsethasthreeconstituents:truth-membership,indeterminacy-membership andfalsity-membership,inwhicheachmembershipvalueisarealstandardornon-standardsubsetof theunitinterval
]0 ,1+[.Inreal-lifeproblems,neutrosophicsetscanbeappliedmoreappropriatelybyusing thesingle-valuedneutrosophicsetsdefinedbySmarandache[1]andWangetal.[2].Ye[3,4]and Pengetal.[5]furtherextendedthestudyofneutrosophicsets.Softsettheory[6]wasproposedby Molodtsovin1999todealwithuncertaintyinaparametricmanner.BabithaandSunildiscussedthe conceptofsoftsetrelation[7].Ontheotherhand,Pawlak[8]proposedthenotionofroughsets.Itisa rigidappearanceofmodelingandprocessingpartialinformation.Ithasbeeneffectivelyconnected todecisionanalysis,machinelearning,inductivereasoning,intelligentsystems,patternrecognition, signalanalysis,expertsystems,knowledgediscovery,imageprocessing,andmanyotherfields[9–12]. Inliterature,roughtheoryhasbeenappliedindifferentfieldofmathematics[13–16].Duboisand Prade[17]developedtwoconceptscalledroughfuzzysetsandfuzzyroughsetsandconcludedthat thesetwotheoriesaredifferentapproachestohandlevagueness.Fengetal.[18]combinedsoftsets withfuzzysetsandroughsets.Mengetal.[19]dealtwithsoftroughfuzzysetsandsoftfuzzyrough sets.Broumietal.[20]studiedroughneutrosophicsets.Yangetal.[21]proposedsingle-valued neutrosophicroughsets,andestablishedanalgorithmfordecision-makingproblembasedonsinglevaluedneutrosophicroughsetsontwouniverses.
Agraphisaconvenientwayofrepresentinginformationinvolvingrelationshipbetweenobjects. Theobjectsarerepresentedbyverticesandrelationsbyedges.Whenthereisvaguenessinthe descriptionoftheobjectsorinitsrelationshipsorinboth,itisnaturalthatweneedtodesignafuzzy graphmodel.Fuzzymodelshasvitalroleastheiraspirationindecreasingtheirregularitybetweenthe traditionalnumericalmodelsusedinengineeringandsciencesandthesymbolicmodelsusedinexpert
Mathematics 2018, 6,125;doi:10.3390/math6070125 www.mdpi.com/journal/mathematics
systems.ThefuzzygraphtheoryasageneralizationofEuler’sgraphtheorywasfirstintroducedby Kaufmann[22].Later,Rosenfeld[23]consideredfuzzygraphsandobtainedanalogsofseveralgraph theoreticalconcepts.MordesonandPeng[24]definedsomeoperationsonfuzzygraphs.Mathew andSunitha[25,26]presentedsomenewconceptsonfuzzygraphs.Ganietal.[27–30]discussed severalconcepts,includingsize,order,degree,regularityandedgeregularityinfuzzygraphsand intuitionisticfuzzygraphs.ParvathiandKarunambigai[31]describedsomeoperationonintuitionistic fuzzygraph.Recently,Akrametal.[32–36]hasintroducedseveralextensionsoffuzzygraphswith applications.Denish[37]consideredtheideaoffuzzyincidencegraph.Fuzzyincidencegraphswere furtherstudiedin[38,39].Duetothelimitationofhumansknowledgetounderstandthecomplex problems,itisverydifficulttoapplyonlyasingletypeofuncertaintymethodtodealwithsuch problems.Therefore,itisnecessarytodevelophybridmodelsbyincorporatingtheadvantagesof manyotherdifferentmathematicalmodelsdealinguncertainty.Recently,newhybridmodels,including roughfuzzygraphs[40,41],fuzzyroughgraphs[42],intuitionisticfuzzyroughgraphs[43,44],rough neutrosophicgraphs[45]andneutrosophicsoftroughgraphs[46]areconstructed.Forothernotations anddefinitions,thereadersarerefereedto[47–51].Inthispaper,weapplythenotionofsoftrough neutrosophicsetstographtheory.Wedevelopcertainnewconcepts,includingsoftroughneutrosophic graphs,softroughneutrosophicinfluencegraphs,softroughneutrosophicinfluencecyclesandsoft roughneutrosophicinfluencetrees.Weillustratetheseconceptswithexamples,andinvestigatesome oftheirproperties.Wesolvedecision-makingproblembyusingourproposedalgorithm.
Thispaperisorganizedasfollows.InSection 2,somedefinitionsandsomepropertiesofsoft roughneutrosophicgraphsaregiven.InSection 3,softroughneutrosophicinfluencegraphs,softrough neutrosophicinfluencecyclesandsoftroughneutrosophicinfluencetreesarediscussed.InSection 4, anapplicationispresented.Finally,weconcludeourcontributionwithasummaryinSection 5 andan outlookforthefurtherresearch.
2.SoftRoughNeutrosophicGraphs
Definition1. Let B beBooleansetand A asetofattributes.Foranarbitraryfullsoftset S over B such that Ss (a)⊂B,forsome a∈A,where Ss :A→P (B) isaset-valuedfunctiondefinedas Ss (a)={b∈B|(a,b)∈S}, forall a∈A.Let (B,S) beafullsoftapproximationspace.Foranyneutrosophicset N={(b,TN (b),IN (b),FN (b))| b∈B}∈N (B),where N (B) isneutrosophicpowersetofset B.Theupperandlowersoftroughneutrosophic approximationsofN w.r.t (B,S),denotedby S(N) andS(N),respectively,aredefinedasfollows: S(N)={(b, TS(N)(b), IS(N)(b), FS(N)(b)) | b ∈B} , S(N)={(b, TS(N)(b), IS(N)(b), FS(N)(b)) | b ∈B}, where TS(N)(b)= b∈Ss (a) t∈Ss (a) TN (t), TS(N)(b)= b∈Ss (a) t∈Ss (a) TN (t), IS(N)(b)= b∈Ss (a) t∈Ss (a) IN (t) , IS(N)(b)= b∈Ss (a) t∈Ss (a) IN (t),(1) FS(N)(b)= b∈Ss (a) t∈Ss (a) FN (t), FS(N)(b)= b∈Ss (a) t∈Ss (a) FN (t)
Inotherwords, TS(N)(b)= a∈A (1 S(a, b)) ∨ t∈B S(a, t) ∧ TN (t) , TS(N)(b)= a∈A S(a, b) ∧ t∈B (1 S(a, t)) ∨ TN (t) ,
IS(N) (b)= a∈A S(a, b) ∧ t∈B (1 S(a, t)) ∨ IN (t) ,
IS(N) (b)= a∈A (1 S(a, b)) ∨ t∈B S(a, t) ∧ IN (t) ,
FS(N) (b)= a∈A S(a, b) ∧ t∈B (1 S(a, t)) ∨ FN (t) ,
FS(N) (b)= a∈A (1 S(a, b)) ∨ t∈B S(a, t) ∧ FN (t)
Thepair (S(N),S(N)) iscalledsoftroughneutrosophicset(SRNS)ofN w.r.t (B,S)
Example1. Suppose N={(b1,0.8,0.3,0.16),(b2,0.85,0.24,0.2),(b3,0.79,0.2,0.2),(b4,0.85,0.36,0.25),(b5,0.82,0.25,0.25)} isaneutrosophicsetontheuniversalset B={b1,b2,b3,b4,b5} underconsideration.Let A={a1,a2,a3} beasetof parameteronB.AfullsoftsetoverB,denotedbyS,isdefinedinTable 1
Table1. Fullsoftset S. Sb1 b2 b3 b4 b5 a1 00101 a2 10100 a3 01111
Aset-valuedfunction Ss :A→P (B) isdefinedas Ss (a1)={b3,b5},Ss (a2)={b1,b3},Ss (a3)={b2,b3,b4,b5} FromEquation(1)ofDefinition 1,wehave
TS(A) (b1)= y∈Ss (a2) N(y)=∨{0.8,0.79}=0.80, IS(N) (b1)= y∈Ss (a2) N(y)=∧{0.3,0.2} =0.20, FS(N) (b1)= y∈Ss (a2) N(y)=∧{0.16,0.2}=0.16; TS(N)(b1)= y∈Ss (a2) N(y)=∧{0.8,0.79}=0.79, IS(N) (b1)= y∈Ss (a2) N(y)=∨{0.3,0.2} =0.30, FS(N) (b1)= y∈Ss (a2) N(y)=∨{0.16,0.2}=0.20.
Similarly,
TS(N)(b2)= 0.85, IS(N)(b2)= 0.20, FS(N)(b2)= 0.20, TS(N)(b3)= 0.80, IS(N)(b3)= 0.20, FS(N)(b3)= 0.20, TS(N)(b4)= 0.85, IS(N)(b4)= 0.20, FS(N)(b4)= 0.20, TS(N)(b5)= 0.82, IS(N)(b5)= 0.20, FS(N)(b5)= 0.20; TS(N)(b2)= 0.79, IS(N)(b2)= 0.36, FS(N)(b2)= 0.25, TS(N)(b3)= 0.79, IS(N)(b3)= 0.25, FS(N)(b3)= 0.20, TS(N)(b4)= 0.79, IS(N)(b4)= 0.36, FS(N)(b4)= 0.25,
TS(N)(b5)= 0.79, IS(N)(b5)= 0.25, FS(N)(b5)= 0.25. Thus,weobtain
S(N)={(b1,0.80,0.20,0.16), (b2,0.85,0.20,0.20), (b3,0.80,0.20,0.20), (b4,0.85,0.20,0.20), (b5,0.82,0.20,0.20)}, S(N)={(b1,0.79,0.30,0.20), (b2,0.79,0.36,0.25), (b3,0.79,0.25,0.20), (b4,0.79,0.36,0.25), (b5,0.79,0.25,0.25)}
Definition2. Asoftroughneutrosophicrelation(SRNR) (R(M),R(M)) on B=B×B isasoftrough neutrosophicset,R:A(A×A)→P (B) isafullsoftseton Banddefinedby R(akl , bij ) ≤ min{S(ak, bi ), S(al , bj )}, forall (akl ,bij )∈R,suchthat Rs (akl )⊂B forsome akl ∈A,where Rs :A→P (B) isaset-valuedfunction,forall akl ∈A,definedby Rs (akl )= {bij ∈ B | (akl , bij ) ∈ R}, bij ∈ B
Foranyneutrosophicset M∈N (B), theupperandlowersoftroughneutrosophicapproximationof M w.r.t (B,R) aredefinedasfollows:
R(M)={(bij, TR(M)(bij ), IR(M)(bij ), FR(M)(bij )) | bij ∈ ˜ B} , R(M)={(bij, TR(M)(bij ), IR(M)(bij ), FR(M)(bij )) | bij ∈B}, where
TR(M)(bij )= bij ∈Rs (akl ) tij ∈Rs (akl )
IR(M)(bij )= bij ∈Rs (akl ) tij ∈Rs (akl )
FR(M)(bij )= bij ∈Rs (akl ) tij ∈Rs (akl )
Inotherwords,
TM (tij ), TR(M)(bij )= bij ∈Rs (akl ) tij ∈Rs (akl )
IM (tij ) , IR(M)(bij )= bij ∈Rs (akl ) tij ∈Rs (akl )
TM (tij ),
IM (tij ),(2)
FM (tij ), FR(M)(bij )= bij ∈Rs (akl ) tij ∈Rs (akl ) FM (tij ).
TR(M)(bij )= akl ∈A (1 R(akl , bij )) ∨ tij ∈B R(akl , tij ) ∧ TM (tij ) ,
TR(M)(bij )= akl ∈A R(akl , bij ) ∧ tij ∈B (1 R(akl , tij )) ∨ TM (tij ) ,
IR(M) (bij )= akl ∈A R(akl , bij ) ∧ tij ∈B (1 R(akl , tij )) ∨ IM (tij ) ,
IR(M) (bij )= akl ∈A (1 R(akl , bij )) ∨ tij ∈B R(akl , tij ) ∧ IM (tij ) ,
FR(M) (bij )= akl ∈A R(akl , bij ) ∧ tij ∈B (1 R(akl , tij )) ∨ FM (tij ) ,
FR(M) (bij )= akl ∈A (1 R(akl , bij )) ∨ tij ∈B R(akl , tij ) ∧ FM (tij ) .
If R(M)=R(M),thenitiscalledinducedsoftroughneutrosophicrelationonsoftroughneutrosophicset, otherwise,softroughneutrosophicrelation.
Remark1. Foraneutrosophicset M on B andaneutrosophicset N on B,wehaveneutrosophicrelation asfollow
TM (bij ) ≤min i {TN (bi )}, IM (bij )≤ min i {IN (bi )}, FM (bij )≤ min i {FN (bi )}
FromDefinition 2,itfollowsthat:
TR(M)(bij ) ≤min{TS(N)(bi ), TS(N)(bj )}, TR(M)(bij ) ≤min{TS(N)(bi ), TS(N)(bj )}, IR(M) (bij ) ≤ max{IS(N)(bi ), IS(N)(bj )}, IR(M) (bij ) ≤ max{IS(N)(bi ), IS(N)(bj )}, FR(M) (bij ) ≤max{FS(N)(bi ), FS(N)(bj )}, FR(M) (bij ) ≤max{FS(N)(bi ), FS(N)(bj )}
Definition3. InDefinition 2 bij iscalledeffective,if
TR(M)(bij )=TS(N)(bi ) ∧ TSN (bj ), TR(M)(bij )=TS(N)(bi ) ∧ TSN (bj ),
IR(M)(bij )=IS(N)(bi ) ∨ ISN (bj ), IR(M)(bij )=IS(N)(bi ) ∨ ISN (bj ), FR(M)(bij )=FS(N)(bi ) ∨ FSN (bj ), FR(M)(bij )=FS(N)(bi ) ∨ FSN (bj ).
Definition4. Asoftroughneutrosophicinfluence(SRNI)isarelationfromsoftroughneutrosophicsettosoft roughneutrosophicrelation,denotedby (X(Q),X(Q)) on ˆ B=B×B,where X: ˆ A(A×A)→P ( ˆ B) isafullsoft seton ˆ Bdefinedby
X(al amn, bi bjk ) ≤ S(al , bi ) ∧ R(amn, bjk ), forall (al amn,bi bjk )∈X andforsome i=j=k and l=m=n.Let Xs : ˆ A→P ( ˆ B) beaset-valuedfunctiondefinedby
Xs (al amn )= {bi bjk ∈ ˆ B|(al amn, (bi,, bjk )) ∈ X}, ∀(a , lamn ) ∈ ˆ A,
Forany Q∈N ( ˆ B),theupperandlowersoftroughneutrosophicapproximationof Q w.r.t ( ˆ B,X),forall bi bjk ∈ ˆ B,aredefinedasfollows:
X(Q)={(bi bjk, TX(Q)(bi bjk ), IX(Q)(bi bjk ), FX(Q)(bi bjk ))},
X(Q)={(bi bjk, TX(Q)(bi bjk ), IX(Q)(bi bjk ), FX(Q)(bi bjk ))}, where
TX(Q)(bi bjk )= bi bjk ∈Xs (al amn ) ti tjk ∈Xs (al amn )
TX(Q)(bi bjk )= bi bjk ∈Xs (al amn ) ti tjk ∈Xs (al amn )
IX(Q)(bi bjk )= bi bjk ∈Xs (al amn ) ti tjk ∈Xs (al amn )
IX(Q)(bi bjk )= bi bjk ∈Xs (al amn ) ti tjk ∈Xs (al amn )
FX(Q)(bi bjk )= bi bjk ∈Xs (al amn ) ti tjk ∈Xs (al amn )
TQ (ti tjk ) ,
TQ (ti tjk ) ,
IQ (ti tjk ),
IQ (ti tjk ) ,(3)
FQ (ti tjk ),
FX(Q)(bi bjk )= bi bjk ∈Xs (al amn ) ti tjk ∈Xs (al amn ) FQ (ti tjk ) .
Inotherwords,
TX(Q)(bi bjk )= al amn ∈A
TX(Q)(bi bjk )= al amn ∈A
IX(Q) (bi bjk )= al amn ∈A
(1 X(al amn, bi bjk )) ∨ ti tjk ∈B X(al amn, ti tjk ) ∧ TQ (ti tjk ) ,
X(al amn, bi bjk ) ∧ ti tjk ∈B (1 X(al amn, ti tjk )) ∨ TQ (ti tjk ) ,
X(al amn, bi bjk ) ∧ ti tjk ∈B (1 X(al amn, ti tjk )) ∨ IQ (ti tjk ) ,
IX(Q) (bi bjk )= al amn ∈A (1 X(al amn, bi bjk )) ∨ ti tjk ∈B X(al amn, ti tjk ) ∧ IQ (ti tjk ) ,
FX(Q) (bi bjk )= al amn ∈A X(al amn, bi bjk ) ∧ ti tjk ∈B (1 X(al amn, ti tjk )) ∨ FQ (ti tjk ) ,
FX(Q) (bi bjk )= al amn ∈A (1 X(al amn, bi bjk )) ∨ ti tjk ∈B X(al amn, ti tjk ) ∧ FQ (ti tjk )
Remark2. Foraneutrosophicset Q on ˆ B andaneutrosophicset N and M on B and ˜ B,respectively,wehave neutrosophicrelationasfollow
TQ (bi bjk ) ≤min jk {TM (bjk )}, IQ (bi bjk ) ≤ min jk {IM (bjk )}, FQ (bi bjk ) ≤min jk {FM (bjk )}
FromDefinition 4,wehave
TX(Q)(bi bjk ) ≤min{TS(N)(bi ), TR(M)(bjk )}, TX(Q)(bi bjk )≤ min{TS(N)(bi ), TR(M)(bjk )}, IX(Q) (bi bjk ) ≤ max{IS(N)(bi ), IR(M)(bjk )}, IX(Q)(bi bjk )≤ max{IS(N)(bi ), IR(M)(bjk )}, FX(Q) (bi bjk ) ≤max{FS(N)(bi ), FR(M)(bjk )}, FX(Q)(bi bjk )≤ max{FS(N)(bi ), FR(M)(bjk )}. Definition5. InDefinition 4 bi bjk iscalledinfluenceeffective,if
TX(Q)(bi bjk )=TS(N)(bi ) ∧ TRM (bij ), TX(Q)(bi bjk )=TS(N)(bi ) ∧ TRM (bij ), IX(Q)(bi bjk )=IS(N)(bi ) ∨ IRM (bij ), IX(Q)(bi bjk )=IS(N)(bi ) ∨ IRM (bij ), FX(Q)(bi bjk )=FS(N)(bi ) ∨ FRM (bij ), FX(Q)(bi bjk )=FS(N)(bi ) ∨ FRM (bij )
Example2. Letafullsoftset S onanuniversalset B={b1,b2,b3,b4} with A={a1,a2,a3} asetofparameters canbedefinedintabularformasTable 2 asfollows:
Table2. Fullsoftset S Sb1 b2 b3 b4 a1 1101 a2 0011 a3 1111
Now,wecandefineset-valuedfunctionSs suchthat Ss (a1)= {b1, b2, b4}, Ss (a2)= {b3, b4}, Ss (a3)= {b1, b2, b3, b4}
Let N= {(b1,1.0,0.0,0.0),(b2,0.8,0.0,0.1),(b3,0.5,0.5,0.5),(b4,0.4,0.7,0.3)} beaneutrosophicseton B, thenbyusingEquation(1)ofDefinition 1,wehave
S(N)= {(b1,1.0,0.0,0.0), (b2,1.0,0.0,0.0), (b3,0.5,0.5,0.3), (b4,0.5,0.5,0.3)},
S(N)= {(b1,0.4,0.7,0.3), (b2,0.4,0.7,0.3), (b3,0.4,0.7,0.5), (b4,0.4,0.7,0.3)}.
Hence (S(N), S(N)) issoftroughneutrosophicset.Letafullsoftset R on C={b12,b22,b23, b32,b42}⊆ ˜ B withL={a13,a21,a32}⊆ ˜ AcanbedefinedinTable 3 (fromLtoC)asfollows: Table3. Fullsoftset R
Rb12 b22 b23 b32 b42 a13 11101 a21 00010 a32 00100
Now,wecandefineset-valuedfunctionRs suchthat Rs (a13)= {b12, b22, b23, b42}, Rs (a21)= {b32}, Rs (a32)= {b23} andM= {(b12,0.4,0.0,0.0),(b22,0.4,0.0,0.0),(b23,0.4,0.0,0.0),(b32,0.4,0.0,0.0),(b42,0.4,0.0,0.0)} aneutrosophic relationonB,thenbyusingEquation(2)ofDefinition 2,weget
R(M)={(b12,0.4,0.0,0.0),(b22,0.4,0.0,0.0),(b23,0.4,0.0,0.0),(b32,0.4,0.0,0.0),(b42,0.4,0.0,0.0)}, R(M)={(b12,0.4,0.0,0.0),(b22,0.4,0.0,0.0),(b23,0.4,0.0,0.0),(b32,0.4,0.0,0.0),(b42,0.4,0.0,0.0)}.
Hence (R(M), R(M)) isaninducedsoftroughneutrosophicrelation.Letafullsoftset X on D={b1b22,b1b23,b1b32,b1b42,b3b12,b3b22,b3b42,b4b12,b4b22,b4b23,b4b32}⊆ ˆ B with P={a13,a21,a32}⊆ ˆ A canbe definedinTable 4 (fromPtoD)asfollows: Table4. Fullsoftset X.
Example3. Let V={v1,v2,v3,v4,v5,v6} beavertexsetand A={a1,a2,a3} asetofparameters.Afullsoftset SfromAonVcanbedefinedintabularforminTable 5 asfollows:
Table5. Fullsoftset S
Sv1 v2 v3 v4 v5 v6 a1 111110 a2 001111 a3 110011
LetN={(v1,0.8,0.6,0.4),(v2,0.9,0.4,0.45),(v3,0.7,0.4,0.35),(v4,0.6,0.3,0.5),(v5,0.4,0.7,0.6),(v6,0.5,0.5,0.5)} beaneutrosophicsetonV.ThenfromEquation(1)ofDefinition 1,wehave
S(N)={(v1,0.9,0.4,0.4),(v2,0.9,0.4,0.4),(v3,0.7,0.3,0.5),(v4,0.7,0.3,0.5),(v5,0.7,0.4,0.5), (v6,0.7,0.4,0.5)},
S(N)={(v1,0.4,0.7,0.6),(v2,0.4,0.7,0.6),(v3,0.4,0.7,1.0),(v4,0.4,0.7,1.0),(v5,0.4,0.7,0.6), (v6,0.4,0.7,0.6)}
Hence, SN=(S(N),S(N)) isasoftroughneutrosophicseton V.Let E={v11,v15,v16,v23,v25,v34,v41,v43, v56,v62,v63}⊆ ˜ V and L={a12,a13,a21,a23,a31}⊆ ˜ A.Thenafullsoftset R on E (from L to E)canbedefinedin Table 6 asfollows:
Table6. Fullsoftset R Rv11 v15 v16 v23 v25 v34 v41 v43 v56 v62 v63 a12 01111101100 a13 11101010100 a21 00000111011 a23 00000010110 a31 11011000010
Let M={(v11,0.4,0.3,0.35),(v15,0.3,0.3,0.2),(v16,0.3,0.2,0.25),(v23,0.4,0.1,0.1),(v25,0.4,0.2,0.0),(v34,0.3, 0.1,0.3),(v41,0.2,0.1,0.2),(v43,0.4,0.28,0.2),(v56,0.4,0.3,0.3),(v62,0.35,0.25,0.32),(v63,0.4,0.12,0.34)} bea neutrosophicsetonE.ThenfromEquation(2)ofDefinition 2,wehave
R(M)={(v11,0.4,0.1,0.00), (v15,0.4,0.10,0.00), (v16,0.4,0.10,0.00), (v23,0.4,0.10,0.00), (v25,0.4,0.1,0.00), (v34,0.4,0.10,0.20), (v41,0.4,0.10,0.30), (v43,0.4,0.10,0.20), (v56,0.4,0.1,0.30), (v62,0.4,0.10,0.30), (v63,0.4,0.10,0.20)},
R(M)={(v11,0.3,0.3,0.35), (v15,0.3,0.30,0.35), (v16,0.3,0.30,1.00), (v23,0.3,0.30,0.35), (v25,0.3,0.3,0.35), (v34,0.3,0.28,0.34), (v41,0.2,0.28,0.32), (v43,0.3,0.28,0.34), (v56,0.3,0.3,0.32), (v62,0.3,0.28,0.32), (v63,0.2,0.28,0.34)}
Hence, RM=(R(M),R(M)) issoftroughneutrosophicseton E.Thus, G=(S(N),R(M)) and G=(S(N),R(M)) areLSRNAGandUSRNAG,respectively,asshowninFigure 1
v1 (0 4, 0 7, 0 6) (0 .3 ,0 .3 ,0 .35) (0 .3 , 0 .3 , 0 .35)
v2 (0 4, 0 7, 0 6)
(0.3, 0.3, 1.0) ‘
(0.3,0.3,0.35) (0 3, 0 3, 0.35) (0.3,0.28,0. 34) (0 . 2, 0 . 28, 0 . 32) (0.3,0.28,0.34) (0.3,0.3,0.32)
v4 (0.4, 0.7, 0.1) v5 (0 4, 0 7, 0 6)
v1 (0 9, 0 4, 0 4) (0 .4 ,0 .1 ,0 .0) (0 .4 , 0 .1 , 0 .0)
v3 (0 4, 0 7, 0 1)
(0 .3, 0.28, 0.32)
v6 (0 4, 0 7, 0 6)
(0 .2 ,0 .28 ,0 . 34) (S(N ),R(M ))
v2 (0 9, 0 4, 0 4)
(0.4, 0.1, 0.0)
(0.4,0.1,0.0) (0 .4 , 0 .1 , 0 . 0) (0. 4,0. 1,0. 2) (0 . 4, 0 . 1, 0 . 3) (0. 4,0. 1,0. 2) (0.4,0.1,0.3)
v4 (0 7, 0 3, 0 5) v5 (0.7, 0.4, 0.5)
v3 (0 7, 0 3, 0 5)
(0 .4, 0.1, 0.3)
v6 (0.7, 0.4, 0.5)
(0 .4 ,0 .1 ,0 . 2) (S(N ), R(M ))
Figure1. Softroughneutrosophicgraph G =(G, G)
Figure1:Softroughneutrosophicgraph G =(G, G)
Hence,G=(G,G) isSRNG.
Definition7. Anunderlyinggraph(supportinggraph) G∗=(G∗ ,G∗) ofasoftroughneutrosophicgraph G=(G,G) isoftheformG∗=(V∗ ,E∗) and G∗=(V∗ ,E∗),
V∗ =LowerVertexSet={v ∈ V|TS(N)(v) = 0, IS(N)(v) = 0, FS(N)(v) = 0}, V∗ =UpperVertexSet={v ∈ V|TS(N)(v) = 0, IS(N)(v) = 0, FS(N)(v) = 0}, E∗ =LowerEdgeSet ={vij ∈ E|TR(M)(vij ) = 0, IR(M)(vij ) = 0, FR(M)(vij ) = 0}, E∗ =UpperEdgeSet ={vij ∈ E|TR(M)(vij ) = 0, IR(M)(vij ) = 0, FR(M)(vij ) = 0}.
Definition2.9. A strength ofsoftroughneutrosophicgraph,denotedbystren,isdefinedas stren= ( vjk ∈E ∗ TR(M ) (vjk )) ∧ ( vjk ∈E∗ TR(M ) (vjk )), ( vjk ∈E∗ IR(M ) (vjk ))∨ ( vjk ∈E ∗ IR(M ) (vjk )), ( vjk ∈E ∗ FR(M ) (vjk )) ∨ ( vjk ∈E ∗ FR(M ) (vjk )) .
Definition8. Asoftroughneutrosophicgraphhasawalkifeachapproximationgraphhasanalternative sequenceoftheform v0, e0, v1, e1, v2, ··· , vn 1, en 1, vn suchthat vk ∈ V∗ , ek ∈ E∗ , vk ∈ V∗ , ek ∈ E∗ where ek = vk(k+1) ∈ E, ∀k=0,1,2, ,n 1.If v0 = vn,thenitiscalledclosedwalk.Iftheedgesaredistinct, thenitiscalledasoftroughneutrosophictrail(SRNtrail).Iftheverticesaredistinct,thenitiscalledasoft roughneutrosophicpath(SRNpath).IfapathinaSRNGisclosed,thenitiscalledacycle.
Definition2.10. Astrongestpathjoininganytwovertices vi and vk isthesoftroughneutrosophicpath whichhasmaximumstrengthfrom vi and vk ,denotedby CONNG(vi,vk )or E ∞(vi ,vk ),iscalled strength ofconnectedness from vi and vk . Definition2.11. Asoftroughneutrosophicgraphisa cycle ifandonlyiftheunderlyinggraphsofeach approximationisacycle.A softroughneutrosophiccycle isasoftroughneutrosophicgraphifandonlyif thesupportinggraphofeachapproximationgraphisacycleandthereexist vlm,vij ∈E ∗ ,vlm ,vij ∈E ∗ =vij suchthat R(M )(vij )= vlm ∈E∗ vij
R(M )(vlm ), R(M )(vij )= vlm ∈E ∗ vij
R(M )(vlm ).
Definition9. Astrengthofsoftroughneutrosophicgraph,denotedbystren,isdefinedas
stren= ( vjk ∈E∗ TR(M)(vjk )) ∧ ( vjk ∈E∗ TR(M)(vjk )), ( vjk ∈E∗ IR(M)(vjk ))∨ ( vjk ∈E∗ IR(M)(vjk )), ( vjk ∈E∗ FR(M)(vjk )) ∨ ( vjk ∈E∗ FR(M)(vjk ))
Definition10. Astrongestpathjoininganytwovertices vi and vk isthesoftroughneutrosophicpathwhichhas maximumstrengthfrom vi and vk,denotedby CONNG (vi, vk ) or E∞ (vi, vk ),iscalledstrengthofconnectedness fromvi andvk.
Definition11. Asoftroughneutrosophicgraphisacycleifandonlyiftheunderlyinggraphsofeach approximationisacycle.Asoftroughneutrosophiccycleisasoftroughneutrosophicgraphifandonly ifthesupportinggraphofeachapproximationgraphisacycleandthereexist vlm,vij ∈E∗,vlm,vij ∈E∗ and vlm =vij suchthat R(M)(vij )= vlm ∈E∗ vij
R(M)(vlm ), R(M)(vij )= vlm ∈E∗ vij
Equivalently,eachapproximationgraphisacyclesuchthat R(M)(vij )= vlm ∈E∗ vij
R(M)(vlm )
TR(M)(vlm ), vlm ∈E∗ vij
TR(M)(vlm ), vlm ∈E∗ vij
IR(M)(vlm ), vlm ∈E∗ vij
IR(M)(vlm ), vlm ∈E∗ vij
FR(M)(vlm ) .
FR(M)(vlm ) , R(M)(vij )= vlm ∈E∗ vij
Example4. Let V={v1,v2,v3,v4} beavertexsetand A={a1,a2,a3,a4} asetofparameters.Arelation S over A×VcanbedefinedintabularforminTable 7 asfollows: Table7. Fullsoftset S Sv1 v2 v3 v4 a1 1111 a2 0101 a3 1011 a4 1010
Let N={(v1,0.3,0.4,0.6),(v2,0.4,0.5,0.1),(v3,0.9,0.6,0.4),(v4,1.0,0.2,0.1)} beaneutrosophicseton V ThenfromEquation(1)ofDefinition 1,wehave S(N)= {(v1,0.9,0.4,0.4), (v2,1.0,0.2,0.1), (v3,0.9,0.4,0.4), (v4,1.0,0.2,0.1)}, S(N)= {(v1,0.3,0.6,0.6), (v2,0.4,0.5,0.1), (v3,0.3,0.6,0.6), (v4,0.4,0.5,0.1)}
Hence, SN =(S(N), S(N)) issoftroughneutrosophicseton V.Let E={v13,v32,v24,v41}⊆V and L={a13,a32,a43}⊆A.ThenafullsoftsetRonE(fromLtoE)canbedefinedinTable 8 asfollows:
Hence, SN =(S(N ), S(N ))issoftroughneutrosophicseton V .Let E={v13,v32 ,v24 ,v41 }⊆
13
,a32 ,a
Table8:Fullsoftset R R v13 v32 v24 v41 a13 1011 a32 0100 a43 1000
Table8. Fullsoftset R.
Rv13 v32 v24 v41 a13 1011 a32 0100 a43 1000
Let M={(v13,0.3,0.2,0.1),(v32,0.2,0.1,0.1),(v24,0.3,0.2,0.1),(v41,0.3,0.1,0.1)} beaneutrosophicseton E
ThenfromEquation(2)ofDefinition 2,wehave
R(M)= {(v13,0.3,0.2,0.1), (v32,0.2,0.1,0.1), (v24,0.3,0.1,0.1), (v41,0.3,0.1,0.1)}, R(M)= {(v13,0.3,0.2,0.1), (v32,0.2,0.1,0.1), (v24,0.3,0.2,0.1), (v41,0.2,0.1,0.1)}.
M ={(v13,0.3,0.2,0.1),(v32,0.2,0.1,0.1),(v24,0.3,0.2,0.1),(v41,0.3,0.1,0.1)} beaneutrosophicseton E ThenfromtheEquations2.2ofDefinition2.2,wehave R(M )= {(v13 , 0 3, 0 2, 0 1), (v32, 0 2, 0 1, 0 1), (v24, 0 3, 0 1, 0 1), (v41, 0 3, 0 1, 0 1)}, R(M )= {(v13 , 0 3, 0 2, 0 1), (v32, 0 2, 0 1, 0 1), (v24, 0 3, 0 2, 0 1), (v41, 0 2, 0 1, 0 1)} Hence, RM =(R(M ), R(M ))issoftroughneutrosophicseton E.Thus, G =(S(N ),R(M ))and S(N ), R(M ))areLSRNAGandUSRNAG,respectively,asshowninFigure2.Hence, G =(G, G v1 (0 3, 0 6, 0 6) v2 (0 4, 0 5, 0 1)
Figure2. Softroughneutrosophiccycle G =(G, G)
.
,RM2) ofasoftroughneutrosophicgraph
1)(v), FS(N2)(v)≥FS(N1)(v), andvij ∈H,
(M2)
ij
(
(M1)(vij ), IR(M2)(vij )≥IR(M1)(vij ), FR(M2)(vij )≥FR(M1)(vij ),
R(M2)(vij )≤TR(M1)(vij ), IR(M2)(vij )≥IR(M1)(vij ), FR(M2)(vij )≥FR(M1)(vij )
Definition13. A H=(SN2,RM2) iscalledsoftroughneutrosophicspanningsubgraphofasoftrough neutrosophicgraphG=(SN1,RM1),ifHisasoftroughneutrosophicsubgraphsuchthat
TS(N2)(v)=TS(N1)(v), IS(N2)(v)=IS(N1)(v), FS(N2)(v)=FS(N1)(v), TS(N2)(v)=TS(N1)(v), IS(N2)(v)=IS(N1)(v), FS(N2)(v)=FS(N1)(v)
Definition14. Asoftroughneutrosophicgraphisaforestifandonlyifeachsupportingapproximationgraphis aforest.Asoftroughneutrosophicgraph G=(SN1,RM1) isasoftroughneutrosophicforestifandonlyifthere existasoftroughneutrosophicspanningsubgraphH=(SN1,RM2) isaforestsuchthatvij ∈G H
TR(M1)(vij )<TCONNH (vi, vj ), IR(M1)(vij )>ICONNH (vi, vj ), FR(M1)(vij ) >FCONNH (vi, vj ),
TR(M1)(vij )<TCONNH (vi, vj ), IR(M1)(vij )>ICONNH (vi, vj ), FR(M1)(vij ) >FCONNH (vi, vj )
Asoftroughneutrosophicgraphisatreeifandonlyifeachsupportingapproximationgraphisatree. Asoftroughneutrosophicgraph G=(SN1,RM1) isasoftroughneutrosophictreeifandonlyifthereexistasoft roughneutrosophicspanningsubgraphH=(SN1,RM2) isatreesuchthatvij ∈G H
TR(M1)(vij )<TCONNH (vi, vj ), IR(M1)(vij )>ICONNH (vi, vj ), FR(M1)(vij ) >FCONNH (vi, vj ), TR(M1)(vij )<TCONNH (vi, vj ), IR(M1)(vij )>ICONNH (vi, vj ), FR(M1)(vij ) >FCONNH (vi, vj ).
Definition15. Let G=(SN,RM) beasoftroughneutrosophicgraph,anedge vij isabridgeiftheedge vij isa bridgeinbothsupportinggraphof G and G,thatistheremovalof vij disconnectsboththe G and G.Anedge vij isasoftroughneutrosophicbridgeinasoftroughneutrosophicgraphG=(SN,RM),ifvlm ∈G
TCONNG vij (vl , vm )<TCONNG (vl , vm ), TCONNG vij (vl , vm )<TCONNG (vl , vm ), ICONNG vij (vl , vm ) >ICONNG (vl , vm ), ICONNG vij (vl , vm ) >ICONNG (vl , vm ), FCONNG vij (vl , vm )>FCONNG (vl , vm ), FCONNG vij (vl , vm )>FCONNG (vl , vm ).
Definition16. Let G=(SN1,RM1) beasoftroughneutrosophicgraphthenavertex vi in G isa cutnode(cutvertex)ifitisacutnodeineachsupportinggraphof G and G.Thatis,thedeletionof vi fromthe supportinggraphsof G and G increasethecomponentsinthesupportinggraphs.Avertex vi iscalledsoftrough neutrosophiccutnode(cutvertex)inasoftroughneutrosophicgraphiftheremovalof vi reducesthestrengthof theconnectednessfromnodesvj tovk ∈V∗ ,V∗
TCONNG vi (vj, vk )<TCONNG (vj, vk ), TCONNG vi (vj, vk )<TCONNG (vj, vk ), ICONNG vi (vj, vk ) >ICONNG (vj, vk ), ICONNG vi (vj, vk ) >ICONNG (vj, vk ),
Definition18. Anedge vij insoftroughneutrosophicgraph G iscalled α strongsoftroughneutrosophic edgeif
TR(M)(vij )>TCONNG vij (vi, vj ), TR(M)(vij ) >TCONNG vij (vi, vj ),
IR(M)(vij ) <ICONNG vij (vi, vj ) , IR(M)(vij ) < ICONNG vij (vi, vj ),
FR(M)(vij )<FCONNG vij (vi, vj ), FR(M)(vij ) < FCONNG vij (vi, vj )
Definition19. Anedge vij insoftroughneutrosophicgraph G iscalled β strongsoftroughneutrosophic edgeif
TR(M)(vij )=TCONNG vij (vi, vj ), TR(M)(vij )=TCONNG vij (vi, vj ),
IR(M)(vij )=ICONNG vij (vi, vj ) , IR(M)(vij )= ICONNG vij (vi, vj ),
FR(M)(vij )=FCONNG vij (vi, vj ), FR(M)(vij )= FCONNG vij (vi, vj )
Definition20. Anedge vij insoftroughneutrosophicgraph G iscalled δ strongsoftroughneutrosophic edgeif
TR(M)(vij )<TCONNG vij (vi, vj ), TR(M)(vij ) <TCONNG vij (vi, vj ),
IR(M)(vij ) >ICONNG vij (vi, vj ) , IR(M)(vij ) > ICONNG vij (vi, vj ),
FR(M)(vij )>FCONNG vij (vi, vj ), FR(M)(vij ) > FCONNG vij (vi, vj )
Example5. Let V={v1,v2,v3,v4} beavertexsetand A={a1,a2,a3,a4} asetofparameters.Arelation S over A × VcanbedefinedintabularforminTable 9 asfollows:
Table9. Fullsoftset S Sv1 v2 v3 v4 a1 1111 a2 0101 a3 1011 a4 1010
Let N={(v1,0.3,0.4,0.6),(v2,0.4,0.5,0.1),(v3,0.9,0.6,0.4),(v4,1.0,0.2,0.1)} beaneutrosophicseton V ThenfromEquation(1)ofDefinition 1,wehave
S(N)= {(v1,0.9,0.4,0.4), (v2,1.0,0.2,0.1), (v3,0.9,0.4,0.4), (v4,1.0,0.2,0.1)}, S(N)= {(v1,0.3,0.6,0.6), (v2,0.4,0.5,0.1), (v3,0.3,0.6,0.6), (v4,0.4,0.5,0.1)}
Hence, SN=(S(N),S(N)) issoftroughneutrosophicseton V.Let E={v13,v32,v43}⊆ ˜ V and L={a12,a24,a34}⊆ ˜ A.ThenafullsoftsetRonE(fromLtoE)canbedefinedinTable 10 asfollows:
Table10. Fullsoftset R
Rv13 v32 v43 a12 010 a24 101 a34 001
Hence, SN =(S(N ),S(N ))issoftroughneutrosophicseton V .Let E={v13,v32,v43 }⊆V and L
Thenafullsoftset R on E (from L to E)canbedefinedinTable10asfollows:
2018, 6,125
Table10:Fullsoftset R R v13 v32 v43 a12 010 a24 101 a34 001
Let M={(v13,0.3,0.2,0.0),(v32,0.3,0.0,0.1),(v43,0.3,0.2,0.1)} beaneutrosophicseton E.Thenfrom Equation(2)ofDefinition 2,wehave
R(M)= {(v13,0.3,0.2,0.0), (v32,0.3,0.0,0.1), (v43,0.3,0.2,0.1)},
Let M ={(v13,0.3,0.2,0.0),(v32,0.3,0.0,0.1),(v43,0.3,0.2,0.1)} beaneutrosophicseton E.Thenfromthe Equations2.2ofDefinition2.2,wehave
R(M)= {(v13,0.3,0.2,0.1), (v32,0.3,0.0,0.1), (v43,0.3,0.2,0.1)}.
R(M )= {(v13 , 0 3, 0 2, 0 0), (v32, 0 3, 0 0, 0 1), (v43, 0 3, 0 2, 0 1)}, R(M )= {(v13 , 0.3, 0.2, 0.1), (v32, 0.3, 0.0, 0.1), (v43, 0.3, 0.2, 0.1)}.
Hence, RM=(R(M),R(M)) issoftroughneutrosophicseton E.Thus, G=(S(N),R(M)) and G=(S(N),R(M)) areLSRNAGandUSRNAG,respectively,asshowninFigure 3.Hence, G =(G, G) is SRNGandatree.v13 isabridgeandv3 isacutenode
Hence, RM =(R(M ),R(M ))issoftroughneutrosophicseton E.Thus, G=(S(N ),R(M ))and G=(S(N ),R(M )) areLSRNAGandUSRNAG,respectively,asshowninFigure3.Hence, G =(G, G)isSRNGandatree.
v1(0 3, 0 6, 0 6)
(0 3, 0.2, 0 1) (0 3, 0 0, 0 1) (0.3,0.2,0.1) G =(S(N ),R(M ))
v2(0 4, 0 5, 0 1) v4(0.4,0.5,0. 1)
v3(0 3, 0 6, 0 6)
v1 (0 9, 0 4, 0 4) v2 (1 0, 0 2, 0 1) v4(1.0,0.2,0. 1)
v3 (0 9, 0 4, 0 4)
(0 3, 0.2, 0 0) (0 3, 0.0, 0 1) (0.3,0.2,0.1) G =(S(N ), R(M ))
Figure3. Softroughneutrosophicgraph G =(G, G)
Figure3:Softroughneutrosophicgraph G =(G, G)
v13 isabridgeand v3 isacutenode
WestatethefollowingTheoremswithouttheirproofs.
WestatethefollowingTheoremswithouttheirproofs.
Theorem1. Let G=(SN1,RM1) beasoftroughneutrosophicgraphtree.Anedge vij isthestrongestedgeif vij isanedgeofitssubgraphH=(SN1,RM2)
Theorem2.1. Let G=(SN1,RM1) beasoftroughneutrosophicgraphtree.Anedge vij isthestrongest edgeif vij isanedgeofitssubgraph H=(SN1,RM2)
Theorem2. If v isacommonnodeofatleasttwosoftroughneutrosophicbridges,then v isasoftrough neutrosophiccutnode.
Theorem2.2. If v isacommonnodeofatleasttwosoftroughneutrosophicbridges,then v isasoft roughneutrosophiccutnode.
Theorem3. Ifvij isasoftroughneutrosophicbridgeofG,then
Theorem2.3. If vij isasoftroughneutrosophicbridgeof G,then
TR(M )(vij )=TCONNG vij (vi,vj ),TR(M )(vij )=TCONNG vij (vi,vj ), IR(M )(vij )=ICONNG vij (vi,vj ) ,IR(M )(vij )= ICONNG vij (vi,vj ), FR(M )(vij )=FCONNG vij (vi,vj ),FR(M )(vij )=FCONNG vij (vi,vj )
TR(M)(vij )=TCONNG vij (vi, vj ), TR(M)(vij )=TCONNG vij (vi, vj ), IR(M)(vij )=ICONNG vij (vi, vj ) , IR(M)(vij )= ICONNG vij (vi, vj ), FR(M)(vij )=FCONNG vij (vi, vj ), FR(M)(vij )= FCONNG vij (vi, vj ).
3.SoftRoughNeutrosophicInfluenceGraphs
3SoftRoughNeutrosophicInfluenceGraphs
Definition21. Asoftroughneutrosophicinfluencegraph G onanonemptyset V isa7-orderedtuple (A,S,SN,R,RM,X,XQ) suchthat (i) Aisasetofparameters, (ii) SisanarbitraryfullsoftsetoverV, (iii) RisanarbitraryfullsoftsetoverE ⊆ V × V, (iv) XisanarbitraryfullsoftsetoverI ⊆ V × E, (v) SN=(S(N),S(N)) isasoftroughneutrosophicsetonV,
Definition3.1. A softroughneutrosophicinfluencegraph Gonanonemptyset V isa7-orderedtuple (A,S,SN,R,RM,X,XQ)suchthat
(vi) RM=(R(M),R(M)) isasoftroughneutrosophicsetonE, (vii) XQ=(X(Q),X(Q)) isasoftroughneutrosophicsetonI,
Thus, G=(G,G)=(SN,RM,XQ) isasoftroughneutrosophicinfluencegraph(SRNIG),where G=(S(N),R(M),X(Q)) and G=(S(N),R(M),X(Q)) areloweranduppersoftroughneutrosophicinfluence approximationgraphs(LSRNIAGs)and(USRNIAGs),respectively,of G
Example6. Let V={v1,v2,v3,v4,v5,v6} beavertexsetand A={a1,a2,a3,a4} asetofparameters.Afullsoft setSoverA × VcanbedefinedintabularforminTable 11 asfollows:
Table11. Fullsoftset S
Sv1 v2 v3 v4 v5 v6 a1 111001 a2 010011 a3 101111 a4 111111
LetN={(v1,1.0,0.4,0.7),(v2,0.9,0.6,0.55),(v3,0.7,0.9,0.5),(v4,0.6,0.5,0.6), (v5,0.65,0.8,0.65),(v6,0.8,0.7,0.8)} beaneutrosophicsetonV.ThenfromEquation(1)ofDefinition 1,wehave
S(N)={(v1,1.0,0.4,0.50), (v2,0.9,0.6,0.55), (v3,1.0,0.4,0.5), (v4,1.0,0.4,0.5), (v5,0.9,0.6,0.55), (v6,0.9,0.6,0.55)}, S(N)={ (v1,0.7,0.9,0.80), (v2,0.7,0.8,0.80), (v3,0.7,0.9,0.8), (v4,0.6,0.9,0.8), (v5,0.65,0.8,0.8) (v6,0.7,0.8,0.8)}
Hence, SN=(S(N),S(N)) issoftroughneutrosophicseton V.Let E={v12,v24,v32,v42,v52,v62,}⊆V and L={a13,a24,a34,a41}⊆A.ThenafullsoftsetRonE(fromLtoE)canbedefinedinTable 12 asfollows:
Table12. Fullsoftset R
Rv12 v24 v32 v42 v52 v62 a13 010001 a24 010011 a34 101111 a41 111111
LetM={(v12,0.6,0.3,0.4),(v24,0.58,0.38,0.46),(v32,0.56,0.37,0.47),(v42,0.54,0.34,0.38), (v52,0.52,0.32,0.5),(v62,0.5,0.4,0.45)} beaneutrosophicsetonE.ThenfromEquation(2)ofDefinition 2,wehave
R(M)={(v12,0.60,0.30,0.38), (v24,0.58,0.38,0.45), (v32,0.60,0.30,0.38), (v42,0.60,0.30,0.38), (v52,0.58,0.32,0.45), (v62,0.58,0.38,0.45)},
R(M)={(v12,0.50,0.40,0.50), (v24,0.50,0.40,0.46), (v32,0.50,0.40,0.50), (v42,0.50,0.40,0.50), (v52,0.50,0.40,0.50), (v62,0.50,0.40,0.46)}
Hence, RM=(R(M),R(M)) issoftroughneutrosophicseton E.Thus, G=(S(N),R(M)) and G=(S(N),R(M)) areLSRNAGandUSRNAG,respectively,asshownin Figure 4.Hence, G=(G,G) isSRNG.Let I={v1v24,v1v32,v1v42,v1v52,v1v62,v3v12,v3v24, v3v42,v3v52,v3v62,v4v12,v4v32,v4v52,v4v62,v5v12, v5v24,v5v32,v5v42,v5v62,v6v12,v6v24,v6v32,v6v42,v6v52} ⊆V × E and P={a1 a24,a1 a34,a2 a13,a2 a34,a2 a41,a3 a24,a3 a41,a4 a13}⊆ ˆ A.Thenand Q aneutrosophicseton I andafullsoftsetXonI(fromPtoI)canbedefinedinTable 13,respectivelyasfollows:
(v4, 0 6, 0 9, 0 8)
(v3 , 0 7, 0 9, 0 8)
(v1 , 0.7, 0.9, 0.8)
(v5, 0 65, 0 8, 0 8) (0 .50 , 0 .40 , 0 .50) (0. 50, 0. 40, 0. 46) (0 . 50, 0. 40, 0. 50) (0 .50 ,0 .40 ,0 . 50) (0 . 50 , 0 . 40 , 0 50) (0 . 50, 0 . 40, 0 46)
(v2 , 0 7, 0 8, 0 8)
G =(S(N ),R(M ))
(v4 , 1 0, 0 4, 0 5)
(v6 , 0 7, 0 8, 0 8)
(v3 , 1 0, 0 4, 0 5)
(v1 , 1 0, 0 4, 0 5)
(v5, 0 9, 0 6, 0 55) (0 .60 , 0 .30 , 0 .38) (0. 58, 0. 38, 0. 45) (0 . 58, 0. 32, 0. 45) (0 .60 ,0 .30 ,0 . 38) (0 . 60 , 0 . 30 , 0 . 38) (0 . 58, 0 . 38, 0 . 45)
(v6 , 0 9, 0 6, 0 55)
(v2, 0.9, 0.6, 0.55)
G =(S(N ), R(M ))
Figure4:Softroughneutrosophicgraph G =(G, G)
Figure4. Softroughneutrosophicgraph G =(G, G) Table13. Fullsoftset X
v
Hence, RM =(R(M ),R(M ))issoftroughneutrosophicseton E.Thus, G=(S(N ),R(M ))and G=(S(N ),R(M ))areLSRNAGandUSRNAG,respectively,asshowninFigure4.Hence, G=(G,G)is SRNG.Let I={v1v24 ,v1v32 ,v1 v42,v1 v52,v1 v62,v3 v12 ,v3v24 , v3 v42,v3 v52,v3 v62,v4 v12 ,v4v32 ,v4v52 ,v4v62 ,v5 v12, v5v24 ,v5 v32,v5 v42,v5 v62,v6 v12 ,v6v24 ,v6v32 ,v6v42 ,v6 v52}⊆V ×E and P ={a1a24,a1a34 ,a2a13,a2a34,a2 a41,a3a24, a3a41,a4 a13}⊆ ˆ A.Thenand Q aneutrosophicseton I andafullsoftset X on I (from P to I)canbe definedinTable13,respectivelyasfollows: Q={(v1 v24 , 0 42, 0 3, 0 38), (v1v32, 0 43, 0 28, 0 37), (v1v42, 0 49, 0 26, 0 33), (v1v52, 0 47, 0 29, 0 32), (v1 v62 , 0.46, 0.28, 0.36), (v3v12 , 0.4, 0.29, 0.37), (v3v24, 0.45, 0.24, 0.36), (v3v42, 0.48, 0.29, 0.35), (v3 v52, 0 41, 0 24, 0 36), (v3v62, 0 42, 0 26, 0 34), (v4v12, 0 5, 0 25, 0 3), (v4v32, 0 44, 0 27, 0 32), (v4 v52 , 0 45, 0 23, 0 31), (v4v62 , 0 48, 0 23, 0 38), (v5v12 , 0 46, 0 24, 0 3), (v5v24, 0 47, 0 26, 0 34), (v5 v32, 0.4, 0.3, 0.36), (v5v42, 0.48, 0.29, 0.38), (v5v62, 0.49, 0.3, 0.37), (v6v12 , 0.49, 0.3, 0.37), (v6 v24, 0.4, 0.28, 0.35), (v6v32 , 0.47, 0.27, 0.34), (v6v42 , 0.46, 0.29, 0.33), (v6v52 , 0.49, 0.3, 0.32)}
Thentheloweranduppersoftroughneutrosophicapproximationis directlycalculatedusingtheEquations2.3ofDefinition2.4,wehave
Q={(v1v24,0.42,0.3,0.38), (v1v32,0.43,0.28,0.37), (v1v42,0.49,0.26,0.33), (v1v52,0.47,0.29,0.32), (v1v62,0.46,0.28,0.36), (v3v12,0.4,0.29,0.37), (v3v24,0.45,0.24,0.36), (v3v42,0.48,0.29,0.35), (v3v52,0.41,0.24,0.36), (v3v62,0.42,0.26,0.34), (v4v12,0.5,0.25,0.3), (v4v32,0.44,0.27,0.32), (v4v52,0.45,0.23,0.31), (v4v62,0.48,0.23,0.38), (v5v12,0.46,0.24,0.3), (v5v24,0.47,0.26,0.34), (v5v32,0.4,0.3,0.36), (v5v42,0.48,0.29,0.38), (v5v62,0.49,0.3,0.37), (v6v12,0.49,0.3,0.37), (v6v24,0.4,0.28,0.35), (v6v32,0.47,0.27,0.34), (v6v42,0.46,0.29,0.33), (v6v52,0.49,0.3,0.32)}
ThentheloweranduppersoftroughneutrosophicapproximationisdirectlycalculatedusingEquation(3)of Definition 4,wehave
X(Q)={(v1v24,0.49,0.26,0.32), (v1v32,0.49,0.26,0.32), (v1v42,0.49,0.26,0.32), (v1v52,0.49,0.26, 0.32), (v1v62,0.49,0.26,0.34), (v3v12,0.5,0.23,0.3), (v3v24,0.49,0.23,0.34), (v3v42,0.5, 0.23,0.3), (v3v52,0.5,0.23,0.3), (v3v62,0.49,0.23,0.3), (v4v12,0.5,0.23,0.38), (v4v32, 0.5,0.23,0.3), (v4v52,0.49,0.23,0.31), (v4v62,0.49,0.23,0.34), (v5v12,0.49,0.24,0.3), (v5v24,0.49,0.26,0.34), (v5v32,0.49,0.24,0.3), (v5v42,0.49,0.24,0.3), (v5v62,0.49,0.26, 0.34), (v6v12,0.49,0.26,0.32), (v6v24,0.49,0.26,0.34), (v6v32,0.49,0.24,0.3), (v6v42,0.49, 0.26,0.33), (v6v52,0.49,0.26,0.32)};
X(Q)={(v1v24,0.4,0.3,0.38), (v1v32,0.4,0.3,0.38), (v1v42,0.46,0.3,0.37), (v1v52,0.46,0.3,0.37), (v1v62,0.46,0.3,0.37), (v3v12,0.4,0.3,0.38), (v3v24,0.4,0.3,0.38), (v3v42,0.4,0.3,0.38), (v3v52,0.4,0.3,0.38), (v3v62,0.4,0.3,0.38), (v4v12,0.4,0.3,0.38), (v4v32,0.4,0.3,0.38), (v4v52,0.4,0.3,0.38), (v4v62,0.4,0.3,0.38), (v5v12,0.4,0.3,0.38), (v5v24,0.4,0.3,0.37), (v5v32,0.4,0.3,0.38), (v5v42,0.4,0.3,0.38), (v5v62,0.4,0.3,0.37), (v6v12,0.46,0.3,0.37), (v6v24,0.4,0.3,0.37), (v6v32,0.4,0.3,0.38), (v6v42,0.46,0.3,0.37), (v6v52,0.46,0.3,0.37)}
Thus, G=(S(N),R(M),X(Q)) and G=(S(N),R(M),X(Q)) areLSRNIAGandUSRNIAG,respectively, asshowninFigures 5 and 6.Hence,G=(G,G) isSRNIG.
v4 (0 . 6, 0. 9, 08)
v5(0 65, 0 8, 0 8)
(0 46 ,0 3 ,0 37) (0 4, 0 3, 0 37) (0 4, 0. 3, 0 38) (0. 4, 0. 3, 038) (0 4, 0 3, 0 38)
(0 46, 0 3, 0 37) (0 46, 0. 3, 0. 37)
suchthat
(1 .2 ,0 .3 ,038)
v1(0 7, 0 9, 0 8) v2 (0 . 7, 0. 8, 0. 8)
(0 . 4, 0. 3, 0. 38) (046 ,03 ,0 .37)
(0 4, 0 3, 0 38) (0. 46, 0. 3, 037)
(0 .4 ,03 ,0 . 38) (04,03,038) (0 4 ,0 .3 ,0 38)
(04, 0. 3, 0. 38)
(04 ,0 .3 ,038)
(0 .5 ,04 ,046) (0 5, 0 4, 0 5) (05,0. 4,05) (0 5, 0 4, 0 46) (0 .5 ,0 .4 ,05)
(0 . 4, 0. 3, 0. 38) (0 4 , 0 03 , 0 37)
(04,0 3,038)
v6(0 7, 0 8, 0 8)
(0 4 , 0 3 , 0 38) (0 4 0 3 , 0 38) (0 46, 0 03, 0 38)
(0 .4 ,0 .3 ,038) (04 ,03 ,038)
v3(0 7, 0 9, 0 8)
Figure5. LowerSoftroughneutrosophicgraphG =(S(N), R(M), X(Q))
Figure5:LowerSoftroughneutrosophicgraphG =(S(N),R(M),X(Q))
vk ∈ V∗ ,ek ∈ E∗,ik,i′ k ∈ I∗ , vk ∈ V ∗ ,ek ∈ E ∗ ,ik,i′ k ∈ I ∗ where ik=(vkuv), ek=uv, i′ k=(vwvk+1)and ∀k=0,1,2, ,n 1.If v0 = vn,thenitiscalled closed.Ifthe pairsaredistinct,thenitiscalleda softroughneutrosophicinfluencetrail (SRNItrail).Iftheedges aredistinct,thenitiscalleda softroughneutrosophictrail (SRNtrail).Iftheverticesaredistinctin SRNtrail,thenitiscalleda softroughneutrosophicpath (SRNpath).Ifthevertices,edgeandpairsare distinctinawalkofSRNIG,thenitiscalleda softroughneutrosophicinfluencepath (SRNIpath).A pathisatrailandaninfluencetrail.Ifapathinasoftroughneutrosophicinfluencegraphisclosed,then
v1(1 0, 0 4, 0 7)
v4 (1 . 0, 0. 4, 0. 5)
v5(0 9, 0 6, 0 55)
(0 .49 ,0 .26 ,0 . 34) (0 . 49, 0. 26, 0. 34) (0 . 49, 0. 24, 0. 3) (0. 49, 0. 24, 0. 3) (0 49, 0 24, 0 3)
(0 . 49, 0 . 26, 0 32) (0 . 49, 0. 26, 0. 32)
(0 .6 ,0 .3 ,0 . 38)
(0 . 49, 0. 26, 0. 32) (0 .49 , 0 .26 , 0 .32)
(0 5, 0 23, 0 38) (0. 49, 0. 26, 0. 34)
(0 .49 ,0 .23 ,0 . 34) (0 .5, 0.23, 0.3) (0 .49 ,0 .23 ,0 . 34)
(0 . 49, 0. 26, 0. 32)
(0 . 49, 0. 24, 0. 3) (0 .49 , 0 .26 , 0 . 34)
(0 49,0 26,0.34)
(0 .58 ,0 .38 ,0 .45) (0 6, 0 3, 0 38) (0. 58,0. 32,0. 45) (0 . 58, 0 . 38, 0 45) (0 .6 ,0 .3 ,0 .38)
v6(0 9, 0 6, 0 55)
v2(0 9, 0 6, 0 55)
(0 .5 ,0 .23 ,0 .3) (0 .49 ,0 .23 ,0 .3)
(0 5, 0.23, 0.3)
(0 . 5 , 0 . 23 , 0 . 3) (0 . 49 , 0 . 23 , 0 . 31) (0 49, 0 26, 0 32)
v3(1 0, 0 4, 0 5)
Figure6. UpperSoftroughneutrosophicgraph G =(S(N), R(M), X(Q))
Definition22. Anunderlyinginfluencegraph(supportinginfluencegraph) G∗=(G∗,G∗) ofasoftrough neutrosophicinfluencegraph G=(G,G) isoftheform G∗=(V∗,E∗,I∗) and G∗=(V∗,E∗,I∗),where
V∗ =LowerVertexSet={v ∈ V|TS(N)(v) = 0, IS(N)(v) = 0, FS(N)(v) = 0},
V∗ =UpperVertexSet={v ∈ V|TS(N)(v) = 0, IS(N)(v) = 0, FS(N)(v) = 0}, E∗ =LowerEdgeSet ={vij ∈ E|TR(M)(vij ) = 0, IR(M)(vij ) = 0, FR(M)(vij ) = 0}, E∗ =UpperEdgeSet ={vij ∈ E|TR(M)(vij ) = 0, IR(M)(vij ) = 0, FR(M)(vij ) = 0},
I∗ =LowerInfluence ={vi vjk ∈ I|TX(Q)(vi vjk ) = 0, IX(Q)(vi vjk ) = 0, FX(Q)(vi vjk ) = 0}, I∗ =UpperInfluence ={vi vjk ∈ I|TX(Q)(vi vjk ) = 0, IX(Q)(vi vjk ) = 0, FX(Q)(vi vjk ) = 0}
Definition23. If vij ∈E∗ (E∗),then vij isaloweredge(upperedge)ofthesoftroughneutrosophicinfluence graph.If vi vjk ∈I∗ (I∗),then vi vjk islowerpairs(upperpair).If vjk ∈E∗(E∗)and vi vjk,isnotlowerpairs (upperpairs),thenitisalowernon-influenceedge(uppernon-influenceedge).
Figure6:UpperSoftroughneutrosophicgraph G=(S(N ), R(M ), X(Q)) vivjk ∈I ∗ IR(M )(vivjk )), ( vivjk ∈I ∗ FR(M )(vivjk ) ∨ vivjk ∈E∗ FR(M )(vivjk )) . Definition3.6. InasoftroughneutrosophicinfluencegraphG,ifineachapproximationgraph CONNG(vi,vk)=E∞(vi,vk)= ∨α{Eα(vi,vk)},CONNG(vi,vk)= E∞(vi,vk)= ∨α{Eα(vi,vk)} where Eα(vi,vk)=(Eα 1 ◦ E)(vi,vk), Eα(vi,vk)=(Eα 1 ◦ E)(vi,vk), and (E ◦ E)(vi,vk)= vj ∈V∗ (TR(M )(vij ) ∧ TR(M )(vjk ))), vj ∈V∗ (IR(M )(vij ) ∨ IR(M )(vjk )), vj ∈V∗ (FR(M )(vij ) ∨ FR(M )(vjk )) , (E ◦ E)(vi,vk)= vj ∈V∗ (TR(M )(vij ) ∧ TR(M )(vjk )), vj ∈V∗ (IR(M )(vij ) ∨ IR(M )(vjk )), vj ∈V∗ (FR(M )(vij ) ∨ FR(M )(vjk )) .
Definition24. Asoftroughneutrosophicinfluencegraphhasawalkifeachapproximationgraphhasan alternativesequenceoftheform v0, i0, e0, i0, v1, , vn 1, in 1, en 1, in 1, vn suchthat vk ∈ V∗ , ek ∈ E∗ , ik, ik ∈ I∗ , vk ∈ V∗ , ek ∈ E∗ , ik, ik ∈ I∗
where ik =(vk uv), ek =uv, ik =(vwvk+1) and ∀k=0,1,2, ,n 1.If v0 = vn,thenitiscalledclosed.Ifthe pairsaredistinct,thenitiscalledasoftroughneutrosophicinfluencetrail(SRNItrail).Iftheedgesaredistinct, thenitiscalledasoftroughneutrosophictrail(SRNtrail).IftheverticesaredistinctinSRNtrail,thenitis calledasoftroughneutrosophicpath(SRNpath).Ifthevertices,edgeandpairsaredistinctinawalkofSRNIG, thenitiscalledasoftroughneutrosophicinfluencepath(SRNIpath).Apathisatrailandaninfluencetrail. Ifapathinasoftroughneutrosophicinfluencegraphisclosed,thenitiscalledacycle.
Definition25. Astrengthofsoftroughneutrosophicinfluencegraph,denotedbystren,isdefinedas stren= ( vjk ∈E∗ TR(M)(vjk )) ∧ ( vjk ∈E∗ TR(M)(vjk )), ( vjk ∈E∗ IR(M)(vjk ))∨ ( vjk ∈E∗ IR(M)(vjk )), ( vjk ∈E∗ FR(M)(vjk )) ∨ ( vjk ∈E∗ FR(M)(vjk )) .
Aninfluencestrengthofsoftroughneutrosophicinfluencegraph,denotedbyInstren,isdefinedas Instren= ( vi vjk ∈I∗ TX(Q)(vi vjk ) ∧ vi vjk ∈I∗ TX(Q)(vi vjk )), ( (vi vjk )∈I∗ IR(M)(vi vjk )∨ vi vjk ∈I∗ IR(M)(vi vjk )), ( vi vjk ∈I∗ FR(M)(vi vjk ) ∨ vi vjk ∈E∗ FR(M)(vi vjk )) . Definition26. Inasoftroughneutrosophicinfluencegraph G,ifineachapproximationgraph
CONNG(vi, vk )= E∞ (vi, vk )= ∨α {Eα (vi, vk )}, CONNG(vi, vk )= E∞ (vi, vk )= ∨α {Eα (vi, vk )} where Eα (vi, vk )=(Eα 1 ◦ E)(vi, vk ), Eα (vi, vk )=(Eα 1 ◦ E)(vi, vk ), and (E ◦ E)(vi, vk )= vj ∈V∗ (TR(M)(vij ) ∧ TR(M)(vjk )), vj ∈V∗ (IR(M)(vij ) ∨ IR(M)(vjk )), vj ∈V∗ (FR(M)(vij ) ∨ FR(M)(vjk )) , (E ◦ E)(vi, vk )= vj ∈V∗ (TR(M)(vij ) ∧ TR(M)(vjk )), vj ∈V∗ (IR(M)(vij ) ∨ IR(M)(vjk )), vj ∈V∗ (FR(M)(vij ) ∨ FR(M)(vjk )) .
Thusitisthestrengthofstrongestpathfromvi tovk in G
Inasoftroughneutrosophicinfluencegraph G,ifineachapproximationgraph
ICONNG(vi, vk )= I∞ (vi, vk )= ∨α {Iα (vi, vk )}, ICONNG(vi, vk )= I∞ (vi, vk )= ∨α {Iα (vi, vk )}. where Iα (vi, vk )=(Iα 1 ◦ I)(vi, vk ), Iα (vi, vk )=(Iα 1 ◦ I)(vi, vk ),
and (I ◦ I)(vi, vk )= vm ∈V∗ (TX(Q)(vi vlm ) ∧ TX(Q)(vm vpk )), vm ∈V∗ (IX(Q)(vi vlm ) ∨ IX(Q)(vm vpk )), vm ∈V∗ (FX(Q)(vi vlm ) ∨ FX(Q)(vm vpk )) ,
(I ◦ I)(vi, vk )= vm ∈V∗ (TX(Q)(vi vlm ) ∧ TX(Q)(vm vpk )), vm ∈V∗ (IX(Q)(vi vlm ) ∨ IX(Q)(vm vpk )), vm ∈V∗ (FX(Q)(vi vlm ) ∨ FX(Q)(vm vpk )) .
Thusitisthestrengthofstrongestpathfromvi tovk in G.
Definition27. ASRNIGiscalledconnectedifeachtwovertex vj and vk arejoinedbyaSRN(SRNI)path. Maximalconnectedpartialsubgraphsineachapproximationsubgrapharecalledcomponent.
Definition28. Asoftroughneutrosophicinfluencegraphisacycleifandonlyiftheunderlyinggraphsofeach approximationisacycle.Asoftroughneutrosophicinfluencegraphisasoftroughneutrosophiccycleifandonly iftheunderlyinggraphsofeachapproximationsisacycleandthereexist vlm,vij ∈E∗,vlm,vij ∈E∗ and vlm =vij, suchthat R(M)(vij )= vlm ∈E∗ vij
TR(M)(vlm ), vlm ∈E∗ vij
TR(M)(vlm ), vlm ∈E∗ vij
IR(M)(vlm ), vlm ∈E∗ vij
IR(M)(vlm ), vlm ∈E∗ vij
FR(M)(vlm )
FR(M)(vlm ) , R(M)(vij )= vlm ∈E∗ vij
IX(Q)(vl vmn ), vl vmn ∈I∗ vi vjk
FX(Q)(vl vmn ) , X(Q)(vi vjk )= vl vmn ∈I∗ vi vjk
Example7.
IX(Q)(vl vmn ), vl vmn ∈I∗ vi vjk X Xv1v32 v1v24 v2v13 v3v24 v3v41 v4v13 v4v32 a1 a32 1000001 a2 a43 0010010 a4 a13 0101100
FX(Q)(vl vmn ) .
TX(Q)(vl vmn ), vl vmn ∈I∗ vi vjk ConsideringExample 4.Let I={v1v32,v1v24,v2v13,v3v24,v3v41,v4v13,v4v32}⊆ ˆ V and P={a1 a32,a2 a43,a4 a13}⊆ ˆ A.ThenafullsoftsetXonI(fromPtoI)canbedefinedinTable 14
asfollows: Table14. Fullsoftset
Let Q={(v1v32,0.2,0.1,0.0),(v1v24,0.1,0.0,0.1),(v2v13,0.2,0.1,0.0),(v3v24,0.2,0.1,0.0),(v4v13,0.1,0.1,0.0), (v4v32,0.0,0.1,0.0)} beaneutrosophicsetonI.ThenfromEquation(3)ofDefinition 4,wehave
Table14:Fullsoftset X X v1v32 v1v24 v2 v13 v3 v24 v3v41 v4 v13 v4 v32 a1a32 1000001 a2a43 0010010 a4a13 0101100
X(Q)={(v1v32,0.2,0.1,0.0), (v1v24,0.1,0.0,0.0), (v2v13,0.2,0.1,0.0), (v3v24,0.1,0.0,0.0), (v3v41,0.1,0.0,0.0), (v4v13,0.1,0.1,0.0), (v4v32,0.2,0.1,0.0)},
X(Q)={(v1v32,0.0,0.1,0.0), (v1v24,0.1,0.1,0.1), (v2v13,0.1,0.1,0.1), (v3v24,0.0,0.1,0.1), (v3v41,0.1,0.1,0.1), (v4v13,0.1,0.1,0.1), (v4v32,0.0,0.1,0.0)}
Thus, G =(S(N), R(M), X(Q)) and G =(S(N), R(M),X(Q)) areLSRNIAGandUSRNIAG, respectively,asshowninFigure 7.Hence, G =(G, G) isSRNIG.Theunderlyinggraph G∗=(G∗,G∗) suchthat G∗=(V∗,E∗,I∗), G∗=(V∗,E∗,I∗) where V∗ =V=V∗ , E∗ =E=E∗ and I∗ =I=I∗ v13,v32,v24,v41 arethelower edgeandupperedgeand v1v32,v4v32 isalowerpairandupperpair. v2v41 isbothloweranduppernon-influence edge. P(v1, v4) isapathofthesequenceoftheform v1,v1v24,v24,v2v13,v3,v3v41,v41,v1v32,v2,v2v13,v3v24,v4. Bydirectcalculations,thestrengthandinfluencestrengthofthispathare (0.2,0.2,0.1) and (0.0,0.1,0.1), respectively. G iscycle,softroughneutrosophiccycleandsoftroughneutrosophicinfluencecycle.
X (Q)={(v1 v32, 0.0, 0.1, 0.0), (v1v24 , 0.1, 0.1, 0.1), (v2v13, 0.1, 0.1, 0.1), (v3v24 , 0.0, 0.1, 0.1), (v3v41 , 0.1, 0.1, 0.1), (v4v13, 0.1, 0.1, 0.1), (v4v32 , 0.0, 0.1, 0.0)}.
v1(0.9, 0.4, 0.4)
∗=(V∗ ,E
(0 . 2, 0. 1, 0. 1) (0 . 1, 0. 1, 0. 1) (0 .0 , 0 .1 , 0 . 0) (0 . 1, 0. 1, 0. 1) (0. 0, 0. 1, 0. 1)
(0 .3 ,0 .2 ,0 . 1) (0 . 2, 0. 1, 0. 1) (0 .3 ,0 .2 ,0 . 1)
Thus,G =(S(N ),R(M ),X (Q))and G=(S(N ), R(M ), X(Q))areLSRNIAGandUSRNIAG,respectively,asshowninFigure7.Hence,G=(G, G)isSRNIG.TheunderlyinggraphG∗=(G∗ ,G∗)suchthat v1(0.3, 0.6, 0.6) v2 (0 4, 0 5, 0 1)
v 3 (0 . 3 , 0 . 6 , 0 . 6) v 4 (0 .4 ,0 .5 ,0 . 1)
(0 1, 0.1, 0.1) (0 .0 , 0 .1 , 0 .0) G =(S(N ),R(M ),X (Q))
(0.1,0.1,0.1)
SoftroughneutrosophicinfluencegraphG =(G, G)
(0.1,0.1,0.0) Figure7:SoftroughneutrosophicinfluencegraphG=(G, G) G
,I
), G
=(V
,E ∗ ,I
Definition29. Asoftroughneutrosophicinfluencesubgraph H
=(SN2,RM2,XQ2) ofasoftrough
neutrosophicinfluencegraph G=(SN1,RM1,XQ1),ifv∈Hsuchthat TS(N2)(v)≤TS(N1)(v), IS(N2)(v)≥IS(N1)(v), FS(N2)(v)≥FS(N1)(v), TS(N2)(v)≤TS(N1)(v), IS(N2)(v)≥IS(N1)(v), FS(N2)(v)≥FS(N1)(v), vij ∈H, TR(M2)(vij )≤TR(M1)(vij ), IR(M2)(vij )≥IR(M1)(vij ), FR(M2)(vij )≥FR(M1)(vij ), TR(M2)(vij )≤TR(M1)(vij ), IR(M2)(vij )≥IR(M1)(vij ), FR(M2)(vij )≥FR(M1)(vij ), andvi vjk ∈H, TX(Q2)(vi vjk )≤TX(Q1)(vi vjk ), IX(Q2)(vi vjk )≥IX(Q1)(vi vjk ), FX(Q2)(vij )≥FX(Q1)(vi vjk ),
Definition3.9. A softroughneutrosophicinfluencesubgraph H=(SN2,RM2,XQ2)ofasoftroughneutrosophicinfluencegraphG=(SN1,RM1,XQ1),if v∈H suchthat
TR(M2 ) (vij )≤TR(M1 ) (vij ),IR(M2 ) (vij )≥IR(M1 ) (vij ),FR(M2 ) (vij )≥FR(M1 ) (vij ), TR(M2 ) (vij )≤TR(M1 ) (vij ),IR(M2 ) (vij )≥IR(M1 ) (vij ),FR(M2 ) (vij )≥FR(M1 ) (vij ),
TX(Q2)(vi vjk )≤TX(Q1)(vi vjk ), IX(Q2)(vi vjk )≥IX(Q1)(vi vjk ), FX(Q2)(vij )≥FX(Q1)(vi vjk ).
Definition30. A H=(SN2,RM2,XQ2) iscalledsoftroughneutrosophicinfluencespanningsubgraphofasoft roughneutrosophicinfluencegraph G=(SN1,RM1,XQ1),if H isasoftroughneutrosophicinfluencesubgraph suchthat
TS(N2)(v)=TS(N1)(v), IS(N2)(v)=IS(N1)(v), FS(N2)(v)=FS(N1)(v),
TS(N2)(v)=TS(N1)(v), IS(N2)(v)=IS(N1)(v), FS(N2)(v)=FS(N1)(v).
Definition31. Asoftroughneutrosophicinfluencegraphisaforestifandonlyifeachsupportingapproximation graphisaforest.Asoftroughneutrosophicinfluencegraph G=(SN1,RM1,XQ1) isasoftroughneutrosophic forestifandonlyifthereexistasoftroughneutrosophicspanningsubgraph H=(SN1,RM2,XQ2) isaforest suchthatvij ∈G H
TR(M1)(vij )<TCONNH (vi, vj ), IR(M1)(vij )>ICONNH (vi, vj ), FR(M1)(vij ) >FCONNH (vi, vj ),
TR(M1)(vij )<TCONNH (vi, vj ), IR(M1)(vij )>ICONNH (vi, vj ), FR(M1)(vij ) >FCONNH (vi, vj )
Asoftroughneutrosophicinfluencegraph G=(SN1,RM1,XQ1) isasoftroughneutrosophicinfluence forestifandonlyifthereexistasoftroughneutrosophicspanningsubgraph H=(SN1,RM1,XQ2) isaforest suchthatvi vjk ∈G H
TX(Q1)(vi vjk )<TICONNH (vi, vk ), TX(Q1)(vi vjk )<TICONNH (vi, vk ), IX(Q1)(vi vjk ) >IICONNH (vi, vk ), IX(Q1)(vi vjk ) >IICONNH (vi, vk ), FX(Q1)(vi vjk )>FICONNH (vi, vk ), FX(Q1)(vi vjk )>FICONNH (vi, vk )
Definition32. Asoftroughneutrosophicinfluencegraphisatreeifandonlyifeachsupportingapproximation graphisatree.Asoftroughneutrosophicinfluencegraph G=(SN1,RM1,XQ1) isasoftroughneutrosophictree ifandonlyifthereexistasoftroughneutrosophicspanningsubgraph H=(SN1,RM2,XQ2) isatreesuchthat vij ∈G H
TR(M1)(vij )<TCONNH (vi, vj ), IR(M1)(vij )>ICONNH (vi, vj ), FR(M1)(vij ) >FCONNH (vi, vj ),
TR(M1)(vij )<TCONNH (vi, vj ), IR(M1)(vij )>ICONNH (vi, vj ), FR(M1)(vij ) >FCONNH (vi, vj ).
Asoftroughneutrosophicinfluencegraph G=(SN1,RM1,XQ1) isasoftroughneutrosophicinfluencetree ifandonlyifthereexistasoftroughneutrosophicspanningsubgraph H=(SN1,RM1,XQ2) isatreesuchthat vi vjk ∈G H
TX(Q1)(vi vjk )<TICONNH (vi, vk ), TX(Q1)(vi vjk )<TICONNH (vi, vk ), IX(Q1)(vi vjk ) >IICONNH (vi, vk ), IX(Q1)(vi vjk ) >IICONNH (vi, vk ), FX(Q1)(vi vjk )>FICONNH (vi, vk ), FX(Q1)(vi vjk )>FICONNH (vi, vk )
Definition33. Let G=(SN,RM,XQ) beasoftroughneutrosophicinfluencegraph,anedge vij isabridgeif edge vij isabridgeinbothunderlyinggraphsof G and G.Let G=(SN,RM,XQ) beasoftroughneutrosophic influencegraph,anedgevij isasoftroughneutrosophicbridgeifvlm ∈G
TCONNG vij (vl , vm )<TCONNG (vl , vm ), TCONNG vij (vl , vm )<TCONNG (vl , vm ),
ICONNG vij (vl , vm ) >ICONNG (vl , vm ), ICONNG vij (vl , vm ) >ICONNG (vl , vm ),
FCONNG vij (vl , vm )>FCONNG (vl , vm ), FCONNG vij (vl , vm )>FCONNG (vl , vm ),
Let G=(SN,RM,XQ) beasoftroughneutrosophicinfluencegraph,anedge vij isansoftrough neutrosophicinfluencebridgeifvlm ∈G
TICONNG vij (vl , vm )<TICONNG (vl , vm ), TICONNG vij (vl , vm )<TICONNG (vl , vm ),
IICONNG vij (vl , vm ) >IICONNG (vl , vm ),;IICONNG vij (vl , vm ) >IICONNG (vl , vm ), FICONNG vij (vl , vm )>FICONNG (vl , vm ), FICONNG vij (vl , vm )>FICONNG (vl , vm ),
Definition34. Let G=(SN,RM,XQ) beasoftroughneutrosophicinfluencegraph,avertexisacutnodeifa vertex vi isacutnodeinunderlyinggraphsof G and G.Let G=(SN,RM,XQ) beasoftroughneutrosophic influencegraphthenavertex vi in G isasoftroughneutrosophiccutnodeifthedeletionof vi from G reducesthe strengthoftheconnectednessfromnodesvj →vk ∈V∗,V∗
TCONNG vi (vj, vk )<TCONNG (vj, vk ), TCONNG vi (vj, vk )<TCONNG (vj, vk ),
ICONNG vi (vj, vk ) >ICONNG (vj, vk ), ICONNG vi (vj, vk ) >ICONNG (vj, vk ),
FCONNG vi (vj, vk )>FCONNG (vj, vk ), FCONNG vi (vj, vk )>FCONNG (vj, vk )
Let G=(SN,RM,XQ) beasoftroughneutrosophicinfluencegraphthenavertex vi in G isanneutrosophic influencecutnodeifthedeletionof vi from G reducestheinfluencestrengthoftheconnectednessfrom vj →vk ∈V∗,V∗
TICONNG vi (vj, vk )<TICONNG (vj, vk ), TICONNG vi (vj, vk )<TICONNG (vj, vk ),
IICONNG vi (vj, vk ) >IICONNG (vj, vk ), IICONNG vi (vj, vk ) >IICONNG (vj, vk ),
FICONNG vi (vj, vk )>FICONNG (vj, vk ), FICONNG vi (vj, vk )>FICONNG (vj, vk ),
Definition35. Let G=(SN,RM,XQ) beasoftroughneutrosophicinfluencegraph.Apair vi vjk iscalleda cutpairifandonlyif vi vjk isacutpairinbothsupportinginfluencegraphof G and G.Thatisafterremovingthe pair vi vjk thereisnopathfrom vi to vk inbothsupportinginfluencegraphof G and G.Let G=(SN,RM,XQ) beasoftroughneutrosophicinfluencegraph.Apair vi vjk iscalledasoftroughneutrosophiccutpairifand onlyififdeletingthepairvi vjk reducestheconnectednessfromvi tovk inbothgraph G and G.Thatis,
TCONNG vi vjk (vi, vk )<TCONNG (vi, vk ), TICONNG vi vjk (vi, vk )<TICONNG (vi, vk ),
ICONNG vi vjk (vi, vk ) >ICONNG (vi, vk ), IICONNG vi vjk (vi, vk ) >IICONNG (vi, vk ),
FCONNG vi vjk (vi, vk )>FCONNG (vi, vk ), FICONNG vi vjk (vi, vk )>FICONNG (vi, vk ),
Asoftroughneutrosophicinfluencecutpair vi vjk isapairinasoftroughneutrosophicinfluencegraph G=(SN,RM,XQ) ifthereisspanninginfluencesubgraph H = G vi vjk reducesthestrengthoftheinfluence connectednessfromvi tovk.Thatis,
TICONNG vi vjk (vi, vk )<TICONNG (vi, vk ), TICONNG vi vjk (vi, vk )<TICONNG (vi, vk ),
IICONNG vi vjk (vi, vk ) >IICONNG (vi, vk ), IICONNG vi vjk (vi, vk ) >IICONNG (vi, vk ), FICONNG vi vjk (vi, vk )>FICONNG (vi, vk ), FICONNG vi vjk (vi, vk )>FICONNG (vi, vk ),
Definition36. Anedge vij insoftroughneutrosophicinfluencegraph G iscalledstrongsoftroughneutrosophic edgeif
TR(M)(vij )≥TCONNG vij (vi, vj ), TR(M)(vij ) ≥TCONNG vij (vi, vj ),
IR(M)(vij ) ≤ICONNG vij (vi, vj ) , IR(M)(vij ) ≤ ICONNG vij (vi, vj ),
FR(M)(vij )≤FCONNG vij (vi, vj ), FR(M)(vij ) ≤ FCONNG vij (vi, vj ).
Apairvi vjk insoftroughneutrosophicinfluencegraph G iscalledstrongpairif
TX(Q)(vi vjk )≥TICONNG vi vjk (vi, vk ), TX(Q)(vi vjk ) ≥TICONNG vi vjk (vi, vk ),
IX(Q)(vi vjk ) ≤IICONNG vi vjk (vi, vk ) , IX(Q)(vi vjk ) ≤ IICONNG vi vjk (vi, vk ),
FX(Q)(vi vjk )≤FICONNG vi vjk (vi, vk ), FX(Q)(vi vjk ) ≤ FICONNG vi vjk (vi, vk )
Definition37. Anedge vij insoftroughneutrosophicinfluencegraph G iscalled α strongsoftrough neutrosophicedgeif
TR(M)(vij )>TCONNG vij (vi, vj ), TR(M)(vij ) >TCONNG vij (vi, vj ),
IR(M)(vij ) <ICONNG vij (vi, vj ) , IR(M)(vij ) < ICONNG vij (vi, vj ), FR(M)(vij )<FCONNG vij (vi, vj ), FR(M)(vij ) < FCONNG vij (vi, vj )
Apairvi vjk insoftroughneutrosophicinfluencegraph G iscalled α strongpairif
TX(Q)(vi vjk )>TICONNG vi vjk (vi, vk ), TX(Q)(vi vjk ) >TICONNG vi vjk (vi, vk ),
IX(Q)(vi vjk ) <IICONNG vi vjk (vi, vk ) , IX(Q)(vi vjk ) < IICONNG vi vjk (vi, vk ),
FX(Q)(vi vjk )<FICONNG vi vjk (vi, vk ), FX(Q)(vi vjk ) < FICONNG vi vjk (vi, vk )
Definition38. Anedge vij insoftroughneutrosophicinfluencegraph G iscalled β strongsoftrough neutrosophicedgeif
TR(M)(vij )=TCONNG vij (vi, vj ), TR(M)(vij )=TCONNG vij (vi, vj ),
IR(M)(vij )=ICONNG vij (vi, vj ) , IR(M)(vij )= ICONNG vij (vi, vj ),
FR(M)(vij )=FCONNG vij (vi, vj ), FR(M)(vij )= FCONNG vij (vi, vj )
Apairvi vjk insoftroughneutrosophicinfluencegraph G iscalled β strongpairif
TX(Q)(vi vjk )=TICONNG vi vjk (vi, vk ), TX(Q)(vi vjk )=TICONNG vi vjk (vi, vk ),
IX(Q)(vi vjk )=IICONNG vi vjk (vi, vk ) , IX(Q)(vi vjk )= IICONNG vi vjk (vi, vk ),
FX(Q)(vi vjk )=FICONNG vi vjk (vi, vk ), FX(Q)(vi vjk )= FICONNG vi vjk (vi, vk )
Definition39. Anedge vij insoftroughneutrosophicinfluencegraph G iscalled δ strongsoftrough neutrosophicedgeif
TR(M)(vij )<TCONNG vij (vi, vj ), TR(M)(vij ) <TCONNG vij (vi, vj ),
IR(M)(vij ) >ICONNG vij (vi, vj ) , IR(M)(vij ) > ICONNG vij (vi, vj ),
FR(M)(vij )>FCONNG vij (vi, vj ), FR(M)(vij ) > FCONNG vij (vi, vj )
Apairvi vjk insoftroughneutrosophicinfluencegraph G iscalled δ strongpairif
TX(Q)(vi vjk )<TICONNG vi vjk (vi, vk ), TX(Q)(vi vjk ) <TICONNG vi vjk (vi, vk ),
IX(Q)(vi vjk ) >IICONNG vi vjk (vi, vk ) , IX(Q)(vi vjk ) > IICONNG vi vjk (vi, vk ),
FX(Q)(vi vjk )>FICONNG vi vjk (vi, vk ), FX(Q)(vi vjk ) > FICONNG vi vjk (vi, vk ).
Definition40. A δ strongsoftroughneutrosophicedge vij iscalleda δ∗ strongsoftroughneutrosophic edgeif
TR(M)(vij )> vlm ∈E∗
TR(M)(vlm ), TR(M)(vij )> vlm ∈E∗ TR(M)(vlm ),
IR(M)(vij ) < vlm ∈E∗ IR(M)(vlm ), IR(M)(vij ) < vlm ∈E∗ IR(M)(vlm ),
FR(M)(vij )< vlm ∈E∗ FR(M)(vlm ), FR(M)(vij )< vlm ∈E∗ FR(M)(vlm )
A δ strongpairvi vjk iscalleda δ∗ strongpairif
TR(M)(vij )> vlm ∈E∗ TR(M)(vlm ), TR(M)(vij )> vlm ∈E∗ TR(M)(vlm ), IR(M)(vij ) < vlm ∈E∗ IR(M)(vlm ), IR(M)(vij ) < vlm ∈E∗ IR(M)(vlm ), FR(M)(vij )< vlm ∈E∗ FR(M)(vlm ), FR(M)(vij )< vlm ∈E∗ FR(M)(vlm ).
A δ strongpairvi vjk iscalleda δ∗ strongpairifvi vjk =vl vmn TX(Q)(vi vjk )> vl vmn ∈I∗ TX(Q)(vl vmn ), TX(Q)(vi vjk )> vl vmn ∈I∗ TX(Q)(vl vmn ), IX(Q)(vi vjk ) < vl vmn ∈I∗ IX(Q)(vl vmn ), IX(Q)(vi vjk ) < vl vmn ∈I∗ IX(Q)(vl vmn ), FX(Q)(vi vjk )< vl vmn ∈I∗ FX(Q)(vl vmn ), FX(Q)(vi vjk )< vl vmn ∈I∗ FX(Q)(vl vmn )
Definition41. Asoftroughneutrosophicinfluencegraphissaidtobeasoftroughneutrosophicinfluenceblock ifithasnosoftroughneutrosophicinfluencecutnodes.
Example8. ConsiderExample 5 LetI={v1v32,v1v43,v2v13,v3v32,v4v13}⊆ ˆ VandP={a1a34,a3a24,a4a12}⊆ ˆ A. ThenafullsoftsetXonI(fromPtoI)canbedefinedinTable 15 asfollows:
Table15. Fullsoftset X. Xv1v32 v1v43 v2v13 v3v32 v4v13 a1 a34 01101 a3 a24 01000 a4 a12 10010
Let Q={(vv32,0.3,0.0,0.0),(vv43,0.2,0.0,0.0),(v2v13,0.1,0.0,0.0),(v3v32,0.2,0.0,0.0),(v4v13,0.3,0.0,0.0} beaneutrosophicsetonI.ThenfromEquation(3)ofDefinition 4,wehave
X(Q)={(v1v32,0.3,0.0,0.0), (v1v43,0.2,0.0,0.0), (v2v13,0.3,0.0,0.0), (v3v32,0.3,0.0,0.0), (v4v13,0.3,0.0,0.0)},
X(Q)={(v1v32,0.2,0.0,0.0), (v1v43,0.2,0.0,0.0), (v2v13,0.1,0.0,0.0), (v3v32,0.2,0.0,0.0), (v4v13,0.1,0.0,0.0)}
X (Q)={(v1 v32, 0 2, 0 0, 0 0), (v1v43 , 0 2, 0 0, 0 0), (v2v13, 0 1, 0 0, 0 0), (v3v32 , 0 2, 0 0, 0 0), (v4v13 , 0.1, 0.0, 0.0)}.
Thus, G=(S(N),R(M),X(Q)) and G=(S(N),R(M),X(Q)) areLSRNIAGandUSRNIAG,respectively, asshowninFigure 8.Hence, G=(G,G) isSRNIG.Hence G isatree, v3 isacutvertex, v13 isabridge, v3v32 isacutpair.
Thus,G=(S(N ),R(M ),X(Q))and G=(S(N ),R(M ),X(Q))areLSRNIAGandUSRNIAG,respectively, asshowninFigure8.Hence,G=(G,G)isSRNIG.HenceGisatree, v3 isacutvertex, v13 isabridge, v3 (0 3, 0 6, 0 6)
v1 (0 3, 0 6, 0 6) v2 (0 4, 0 5, 0 1) v4(0.4,0.5,0. )
(0.3,0.2,0.1) (0.3,0.0,0.1) (0.3,0.2,0.1) G =(S(N ),R(M ),X(Q))
(0.3,0.0,0.0) (0.2,0.0,0.0) (0.3,0.0,0.0) (0.3,0.0,0.0)(0.3,0.0,0.0)
(0.2,0.0,0.0) (0.2,0.0,0.0) (0.1,0.0,0.0) (0.3,0.0,0.0)(0.1,0.0,0.0) v3(0 9, 0 4, 0 4)
(0.3,0.2,0.0) (0.3,0.0,0.0) (0.3,0.2,0.1) G=(S(N ), R(M ), X(Q))
v1(0 9, 0 4, 0 4) v2(0 9, 0 4, 0 4) v4(0.9,0.4,0. 4)
Figure8. SoftroughneutrosophicinfluencegraphG =(G, G)
Figure8:SoftroughneutrosophicinfluencegraphG=(G, G)
isacutpair Theorem3.1. G isasoftroughneutrosophicinfluenceforestifandonlyifin anycycleof G,thereisa pair vivjk suchthat
Theorem4. G isasoftroughneutrosophicinfluenceforestifandonlyifinanycycleof G,thereisapair vi vjk suchthat
TX(Q)(vi vjk )<TICONNG vi vjk (vi, vk ), TX(Q)(vi vjk ) <TICONNG vi vjk (vi, vk ),
IX(Q)(vi vjk ) >IICONNG vi vjk (vi, vk ) , IX(Q)(vi vjk ) > IICONNG vi vjk (vi, vk ),
FX(Q)(vi vjk )>FICONNG vi vjk (vi, vk ), FX(Q)(vi vjk ) > FICONNG vi vjk (vi, vk ).
Proof. Theproofisobvious.
Theorem5. Asoftroughneutrosophicgraph G isasoftroughneutrosophicinfluenceforestifthereisatmost onepathwiththemostinfluencestrength.
Proof. Let G benotasoftroughneutrosophicinfluenceforest.ThenbyTheorem 4,thereexistacycle C inGsuchthat
TX(Q)(vi vjk )≥TICONNG vi vjk (vi, vk ), TX(Q)(vi vjk ) ≥TICONNG vi vjk (vi, vk ),
IX(Q)(vi vjk ) ≤IICONNG vi vjk (vi, vk ) , IX(Q)(vi vjk ) ≤ IICONNG vi vjk (vi, vk ),
FX(Q)(vi vjk )≤FICONNG vi vjk (vi, vk ), FX(Q)(vi vjk ) ≤ FICONNG vi vjk (vi, vk ),
foreverypair vi vjk of C Therefore, vi vjk isthepathwithinthemostinfluencestrengthfrom vi to vk.Let vi vjk beapair suchthat
TX(Q)(vi vjk )> vl vmn ∈I∗
IX(Q)(vi vjk ) < vl vmn ∈I∗
TX(Q)(vl vmn ), TX(Q)(vi vjk )> vl vmn ∈I∗ TX(Q)(vl vmn ),
IX(Q)(vl vmn ), IX(Q)(vi vjk ) < vl vmn ∈I∗ IX(Q)(vl vmn ),
FX(Q)(vi vjk )< vl vmn ∈I∗ FX(Q)(vl vmn ), FX(Q)(vi vjk )< vl vmn ∈I∗ FX(Q)(vl vmn ),
in C.Thenremainingpartof C isapathwiththemostinfluencestrengthfrom vi to vjk.Thisisa contradictiontothethefactthereisatmostonepathwiththemostinfluencestrength.Hence, G isa softroughneutrosophicinfluenceforest.
Theorem6. Assumethat G isacycle.Then G isnotasoftroughneutrosophicinfluencetreeifandonlyif G isasoftroughneutrosophicinfluencecycle.
Proof. Let G=(SN,RM,XQ1) beasoftroughneutrosophicinfluencecycle.Thenthereexistatleast twodistinctedgeandtwodistinctpairsuchthat
R(M)(vij )= vlm ∈E∗ vij
R(M)(vij )= vlm ∈E∗ vij
TR(M)(vlm ), vlm ∈E∗ vij
TR(M)(vlm ), vlm ∈E∗ vij
X(Q)(vi vjk )= vl vmn ∈I∗ vi vjk
X(Q)(vi vjk )= vl vmn ∈I∗ vi vjk
IR(M)(vlm ), vlm ∈E∗ vij
IR(M)(vlm ), vlm ∈E∗ vij
TX(Q)(vl vmn ), vl vmn ∈I∗ vi vjk
TX(Q)(vl vmn ), vl vmn ∈I∗ vi vjk
FR(M)(vlm ) ,
FR(M)(vlm ) ,
IX(Q)(vl vmn ), vl vmn ∈I∗ vi vjk
IX(Q)(vl vmn ), vl vmn ∈I∗ vi vjk
FX(Q)(vl vmn ) ,
FX(Q)(vl vmn )
Let H=(SN,RM,XQ2) beaspanningsoftroughneutrosophicinfluencetreein G.Thenthere existsapathfrom vi to vk notinvolving vi vjk suchthat E∗ 1 E∗ 2 ={(vi vjk )}.Hencetheredoesnotexista pathinHfrom vi to vk suchthat
TX(Q2)(vi vjk )≤TICONNG (vi, vk ), TX(Q2)(vi vjk ) ≤TICONNG (vi, vk ), IX(Q2)(vi vjk ) ≥IICONNG (vi, vk ) , IX(Q2)(vi vjk ) ≥ IICONNG (vi, vk ),
FX(Q2)(vi vjk )≥FICONNG (vi, vk ), FX(Q2)(vi vjk ) ≥ FICONNG (vi, vk ).
ThusGisnotasoftroughneutrosophicinfluencetree.
Conversely,supposethat G isnotasoftroughneutrosophicinfluencetree.Since, G isasoft roughneutrosophicinfluencecycle.Soforall vi vjk ∈I∗ and vi vjk ∈I∗,wehaveasoftroughneutrosophic spanninginfluencesubrgraph H=(SN,RM,XQ2) whichistreeand X(Q2)(vi vjk )=0, X(Q2)(vi vjk )=0 suchthat ∀vi vlm = vl vmn
TX(Q2)(vi vjk )≤TICONNH (vi, vk ), TX(Q2)(vi vjk ) ≤TICONNG (vi, vk ), IX(Q2)(vi vjk ) ≥IICONNG (vi, vk ) , IX(Q2)(vi vjk ) ≥ IICONNG (vi, vk ), FX(Q2)(vi vjk )≥FICONNG (vi, vk ) , FX(Q2)(vi vjk ) ≥ FICONNG (vi, vk ), ∀vl vmn ∈ I∗ vi vjk and vl vmn ∈ I∗ vi vjk
TX(Q2)(vi vjk )= vl vmn ∈I∗
TX(Q1)(vl vmn ), TX(Q2)(vi vjk )= vl vmn ∈I∗
TX(Q1)(vl vmn ), IX(Q2)(vi vjk )= vl vmn ∈I∗
IX(Q1)(vl vmn ), IX(Q2)(vi vjk )= vl vmn ∈I∗ IX(Q1)(vl vmn ), FX(Q2)(vi vjk )= vl vmn ∈I∗
Therefore, X(Q)(vi vjk )= vl vmn ∈I∗ vi vjk
FX(Q1)(vl vmn ), FX(Q2)(vi vjk )= vl vmn ∈I∗
FX(Q1)(vl vmn )
TX(Q)(vl vmn ), vl vmn ∈I∗ vi vjk
IX(Q)(vl vmn ), vl vmn ∈I∗ vi vjk
FX(Q)(vl vmn ) , X(Q)(vi vjk )= vl vmn ∈I∗ vi vjk
TX(Q)(vl vmn ), vl vmn ∈I∗ vi vjk
IX(Q)(vl vmn ), vl vmn ∈I∗ vi vjk
FX(Q)(vl vmn ) where vi vjk = vl vmn notuniquely.ThereforeGisasoftroughneutrosophicinfluencecycle.
Theorem7. If
TX(Q)(vi vjk )>TICONNG vi vjk (vi, vk ), TX(Q)(vi vjk ) >TICONNG vi vjk (vi, vk ),
IX(Q)(vi vjk ) <IICONNG vi vjk (vi, vk ) , IX(Q)(vi vjk ) < IICONNG vi vjk (vi, vk ),
FX(Q)(vi vjk )<FICONNG vi vjk (vi, vk ), FX(Q)(vi vjk ) < FICONNG vi vjk (vi, vk ), inasoftroughneutrosophicgraph.Thenvi vjk isacutpairinsoftroughneutrosophicinfluencegraph G Proof. Suppose vi vjk isnotacutapirinsoftroughneutrosophicinfluencegraph,then TICONNG vi ,vk (vi, vk )=TICONNG (vi, vk ), TICONNG vi ,vk (vi, vk )=TICONNG (vl vmn ), IICONNG vi ,vk (vi, vk )=IICONNG (vi, vk ), IICONNG vi ,vk (vi, vk )=IICONNG (vl vmn ), FICONNG vi ,vk (vi, vk )=FICONNG (vi, vk ), FICONNG vi ,vk (vi, vk )=FICONNG (vl vmn )
Since, TX(Q)(vi, vk )≤TICONNG (vi, vk ), TX(Q)(vi, vk )≤TICONNG (vl vmn ), IX(Q)(vi, vk ) ≥IICONNG (vi, vk ), IX(Q)(vi, vk ) ≥IICONNG (vl vmn ), FX(Q)(vi, vk )≥FICONNG (vi, vk ), FX(Q)(vi, vk )≥FICONNG (vl vmn )
Therefore, TICONNG vi ,vk (vi, vk )≥TX(Q)(vi, vk ), TICONNG vi ,vk (vi, vk )≥TX(Q)((vi, vk )), IICONNG vi vk (vi, vk ) ≤IX(Q)(vi, vk ), IICONNG vi ,vk (vi, vk ) ≤IX(Q)((vi, vk )), FICONNG vi ,vk (vi, vk )≤FX(Q)(vi, vk ), FICONNG vi ,vk (vi, vk )≤FX(Q)((vi, vk )), whichisacontradiction.Hence,itisproved. Theorem8. If
TX(Q)(vi vjk )>TX(Q)(vl vmn ), TX(Q)(vi vjk ) >TX(Q)(vl vmn ), IX(Q)(vi vjk ) <IX(Q)(vl vmn ) , IX(Q)(vi vjk ) < IX(Q)(vl vmn ), FX(Q)(vi vjk )<FX(Q)(vl vmn ), FX(Q)(vi vjk ) < FX(Q)(vl vmn ), forsomevi vjk amongallcyclesinsoftroughneutrosophicinfluencegraph G.Then
TX(Q)(vi vjk )>TICONNG vi vjk (vi, vk ), TX(Q)(vi vjk ) >TICONNG vi vjk (vi, vk ), IX(Q)(vi vjk ) <IICONNG vi vjk (vi, vk ) , IX(Q)(vi vjk ) < IICONNG vi vjk (vi, vk ), FX(Q)(vi vjk )<FICONNG vi vjk (vi, vk ), FX(Q)(vi vjk ) < FICONNG vi vjk (vi, vk )
Proof. Since
TICONNG vi vjk (vi vjk )≥TICONNG (vi vjk ), TICONNG vi vjk (vi vjk )≥TICONNG ((vi vjk )),
IICONNG vi vjk (vi vjk ) ≤IICONNG (vi vjk ), IICONNG vi vjk (vi vjk ) ≤IICONNG ((vi vjk )),
FICONNG vi vjk (vi vjk )≤FICONNG (vi vjk ), FICONNG vi vjk (vi vjk )≤FICONNG ((vi vjk )).
Therefore,thereexistsapathfrom vi to vk notinvolving (vi vjk ) suchthat
TICONNG vi vjk (vi vjk )≥TX(Q)(vi vjk ), TICONNG vi vjk (vi vjk )≥TX(Q)((vi vjk )),
IICONNG vi vjk (vi vjk ) ≤IX(Q)(vi vjk ), IICONNG vi vjk (vi vjk ) ≤IX(Q)((vi vjk )),
FICONNG vi vjk (vi vjk )≤FX(Q)(vi vjk ), FICONNG vi vjk (vi vjk )≤FX(Q)((vi vjk )),
Thisalongwith vi vjk isacycleand vi vjk isleastvalue.
Theorem9. If vi vjk isasoftroughneutrosophicinfluencecutpairinsoftroughneutrosophicinfluence graph G.Then
TX(Q)(vi vjk )>TX(Q)(vl vmn ), TX(Q)(vi vjk ) >TX(Q)(vl vmn ),
IX(Q)(vi vjk ) <IX(Q)(vl vmn ) , IX(Q)(vi vjk ) < IX(Q)(vl vmn ), FX(Q)(vi vjk )<FX(Q)(vl vmn ), FX(Q)(vi vjk ) < FX(Q)(vl vmn ), forsomevi vjk amongallcyclesof G. Proof. Supposeoncontraryinacycle,we
TX(Q)(vi vjk )>TX(Q)(vl vmn ), TX(Q)(vi vjk ) >TX(Q)(vl vmn ), IX(Q)(vi vjk ) <IX(Q)(vl vmn ) , IX(Q)(vi vjk ) < IX(Q)(vl vmn ), FX(Q)(vi vjk )<FX(Q)(vl vmn ), FX(Q)(vi vjk ) < FX(Q)(vl vmn ).
Thenanypathinvolvingitcanbeconvertedintoapathnotinvolvingitwithinfluencestrength greaterthanandequaltothevalueof XQ forpreviouslydeletedpairs.So vi vjk isnotacutpair.Thisisa contradictiontoourassumption.Hence vi vjk isnotapairwiththeleastvalueamongallcycle.
Theorem10. If G=(SN1,RM1,XQ1) isasoftroughneutrosophicforest,thenthepairsofneutrosophic spanningsubgraphH=(SN1,RM1,XQ2) suchthat TX(Q1)(vi vjk )<TICONNH (vi, vk ), TX(Q1)(vi vjk )<TICONNH (vi, vk ), IX(Q1)(vi vjk ) >IICONNH (vi, vk ), IX(Q1)(vi vjk ) >IICONNH (vi, vk ), FX(Q1)(vi vjk )>FICONNH (vi, vk ), FX(Q1)(vi vjk )>FICONNH (vi, vk ),
areexactlythecutpairsof G
Theorem11. Asoftroughneutrosophicinfluencegraph G isacycle.Thenanedge vjk isasoftrough neutrosophicinfluencebridgeifandonlyifitisanedgecommontoatmosttwocutpair.
Theorem12. Let G beasoftroughneutrosophicinfluencegraph.Thenthefollowingconditionsareequivalent.
1. Forapairvi vjk ∈I∗ ∩ I∗ TX(Q)(vi vjk )>TICONNG vi vjk (vi, vk ), TX(Q)(vi vjk ) >TICONNG vi vjk (vi, vk ), IX(Q)(vi vjk ) <IICONNG vi vjk (vi, vk ) , IX(Q)(vi vjk ) < IICONNG vi vjk (vi, vk ), FX(Q)(vi vjk )<FICONNG vi vjk (vi, vk ), FX(Q)(vi vjk ) < FICONNG vi vjk (vi, vk ) 2.
4.ApplicationtoDecision-Making
Decisionmakingisaprocessthatplaysanimportantroleinourdailylives.Decisionmaking processcanhelpusmakemorepurposeful,thoughtfuldecisionsbysystemizingrelevantinformation stepbystep.Theprocessofdecisionmakinginvolvesmakingachoiceamongdifferentalternatives, itstartswhenwedonotknowwhattodo.
Theselectionoftherightpathfortransferringgoodsfromonestatetoanotherstatesillegally. Everystatehasdifferentpoliceswithinoroutsidethestate,thereareseveralfactorstotakeinto considerationforselectingtherightpath.Whethertheeconomyofacountryisgood,havingjob opportunityorasafety.
Supposeatraderwantstoextendhisbusinesstothecountries C1,C2,C3,C4,C5 and C6.Initially, hetakes C1 andextendshisbusinessonebyone.Assume A issetoftheparameters,consistingof element a1 = job, a2 = economyaboveaverage, a3 = safety, a4 = other. Let S beafullsoftsetfrom A toparameterset V,asshowninTable 16.
Table16. SoftNeutrosophicSet S SC1 C2 C3 C4 C5 C6 a1 111011 a2 001111 a3 111001 a3 111111
Suppose N={(C1,0.8,0.6,0.7),(C2,0.9,0.5,0.65),(C3,0.75,0.6,0.65),(C4,1.0,0.55,0.85),(C5,0.95,0.63,0.8), (C6,0.85,0.65,0.95)} ismostfavorableobjectdescribesmembershipofsuitablecountriesforeignpolices correspondingtothebooleanset V,whichisaneutrosophicsetontheset V underconsideration. SN =(S(N), S(N)) isafullsoftroughsetinfullsoftapproximationspace (V, S) where S(N)={(C1,0.90,0.50,0.65), (C2,0.90,0.50,0.65), (C3,0.90,0.55,0.65), (C4,1.00,0.55,0.65), (C5,0.95,0.55,0.65), (C6,0.9,0.55,0.65)}, S(N)={(C1,0.75,0.65,0.95), (C2,0.75,0.65,0.95), (C3,0.75,0.65,0.95), (C4,0.75,0.65,0.95), (C5,0.75,0.65,0.95), (C6,0.75,0.65,0.95)} .
Let M={(C12,0.74,0.5,0.62),(C14,0.75,0.45,0.63),(C15,0.74,0.54,0.61),(C23,0.72,0.48,0.65),(C26,0.71, 0.49,0.64),(C34,0.72,0.53,0.64),(C35,0.73,0.52,0.63),(C45,0.7,0.51,0.61),(C46,0.74,0.55,0.6),(C56,0.73,0.47, 0.64)} bemostfavorableobjectdescribesmembershipofcountriesforeignpolicestoward otherscountriescorrespondingtothebooleanset E,whichisaneutrosophicsetontheset V underconsideration.
RM =(RM, RM) isasoftneutrosophicroughrelation,where
RM={(C12,0.75,0.45,0.61), (C14,0.75,0.45,0.61), (C15,0.75,0.45,0.61), (C23,0.75,0.45,0.61), (C26,0.75,0.45,0.61), (C34,0.75,0.45,0.61), (C35,0.74,0.47,0.61), (C45,0.74,0.47,0.6), (C46,0.74,0.47,0.6), (C56,0.74,0.47,0.61)},
RM={(C12,0.71,0.54,0.65), (C14,0.71,0.54,0.65), (C15,0.71,0.54,0.65), (C23,0.71,0.54,0.65), (C26,0.71,0.54,0.65), (C34,0.71,0.54,0.65), (C35,0.71,0.54,0.64), (C45,0.70,0.55,0.64), (C46,0.70,0.55,0.64), (C56,0.71,0.54,0.64)} Let
15
e1e42 11111110 0011101 e2e14 00001100 0011011 e2e34 00000001 0000000 e3e34 11111100 0000011 e4e21 00100010 1100100 e4e42 11111110 1111101
Let Q = {(C1C15,0.7,0.43,0.58), (C1C23,0.65,0.39,0.54), (C1C35,0.66,0.37,0.56), (C2C34,0.68,0.38, 0.59), (C3C14,0.6,0.4,0.6), (C3C26,0.62,0.42,0.58), (C3C45,0.64,0.45,0.54), (C4C23,0.7,0.45,0.60), (C4C45, 0.7,0.36,0.48), (C4C46,0.68,0.35,0.5), (C5C23,0.69,0.45,0.54), (C5C34,0.65,0.42,0.58), (C5C46,0.64,0.41, 0.59), (C6C12,0.63,0.4,0.6), (C6C15,0.62,0.39,0.5)} bemostfavorableobjectdescribesmembershipof countriesimpacttowardotherscountriesregardingtradecorrespondingtothebooleanset I,whichis aneutrosophicsetontheset I underconsideration.
XQ =(XQ, XQ) isasoftneutrosophicroughinfluence,where
XQ={(C1C15,0.70,0.37,0.50), (C1C23,0.70,0.37,0.50), (C1C35,0.70,0.37,0.50), (C2C34,0.70,0.37,0.50), (C3C14,0.69,0.39,0.50), (C3C26,0.69,0.39,0.50), (C3C45,0.70,0.37,0.50), (C4C23,0.7,0.45,0.60), (C4C45,0.70,0.35,0.48), (C4C46,0.70,0.35,0.48), (C5C23,0.69,0.39,0.50), (C5C34,0.69,0.39,0.50), (C5C46,0.70,0.37,0.50), (C6C12,0.69,0.39,0.50), (C6C15,0.69,0.39,0.50)},
XQ={(C1C15,0.60,0.43,0.60), (C1C23,0.60,0.43,0.60), (C1C35,0.64,0.43,0.59), (C2C34,0.60,0.43,0.60), (C3C14,0.60,0.43,0.60), (C3C26,0.60,0.43,0.60), (C3C45,0.64,0.45,0.59), (C4C23,0.7,0.45,0.60), (C4C45,0.64,0.45,0.59), (C4C46,0.64,0.45,0.59), (C5C23,0.60,0.45,0.60), (C5C34,0.60,0.45,0.60), (C5C46,0.64,0.45,0.59), (C6C12,0.60,0.43,0.60), (C6C15,0.60,0.43,0.60)}
Thus, G =(G, G) isasoftneutrosophicroughinfluencegraphasshowninFigure 9.Hefindsthe strengthofeachpathfrom C1 to C6.Thepathsare
P1 : C1, C5, C2, C3, C6, P2 : C1, C4, C5, C6,
C3(0 75, 0 65, 0 95) (0. 71, 0. 54, 065) (0 .71 ,0 .55 ,0 . 64)
(06 ,0 .43 ,0 .6) (06,0 43,0 6) (064, 0.43, 059)
(0 .71 ,0 .54 ,0 . 65) (0 . 71, 0. 54, 0. 65)
(071, 055, 064) (071 ,054 ,0 .65) (071,0. 54,0. 65)
(064 ,0 .45 ,0 .59) (0 64 , 0 45 , 0 59)
(0 .71 , 0 .54 , 0 65)
(0.6,0.43,06) (0 .6 ,0 .43 ,0 .6) (0 . 64, 0. 45, 0. 59)
(0 71, 0 54, 0 65) (0 . 71, 0. 54, 0. 65)
(0 . 6, 0. 43, 0. 6) (0 .6 ,0 .45 ,0 . 6)
(0 .64 , 0 45 , 0 . 59) (0 .6 ,0 .45 ,0 .6)
P3 : C1, C3, C5, C2, C6 withtheirinfluencestrengthas (0.6,0.45,0.5),respectively. C 1 (0 75 , 0 65 , 0 95) C 6 (0 75 , 0 65 , 0 95) C4(075, 0. 65, 0. 95) C2(0 75, 0 65, 0 95) C5(0 75, 0 65, 0 95)
(06,0 43,0 6) (0 .7 ,0 .45 ,0 . 6)
(0 .6 ,0 .45 ,0 . 6) (0. 6, 0. 43, 0. 6) G =(S(N),R(M),X(Q)) C 1 (0 . 9, 0 5, 0 65)
(0. 75, 0. 45, 0. 61)
(0 . 7, 0 . 37, 0 . 5) (0 .7 ,0.37 ,0.5)
(0. 7, 037, 05) (0.7,0.37,05) (0 . 7, 0. 37, 0. 5) (0 . 7, 0. 45, 0. 6) (0 7, 0 35, 0 48)
(0. 69,0. 39,0. 5) (0 .69 ,0 .39 ,0 . 5)
C2(0 9, 0 5, 0 65) C3(0 9, 0 55, 0 65) C4(1 0, 0 55, 0 65)
(069,0 39,05) (0. 69, 0. 39, 0. 5)
(0 .69 ,0 .39 ,0 . 5)
(0.7,035,048) (0 . 7, 0. 37, 0. 5)
(0 75, 0 45, 0 61) (075, 045, 061) (0 .75 ,0 .45 ,0 . 61) (0 .75 ,045 ,0 .61) (0 . 75, 0. 45, 0. 61) (0 75, 0 45, 0 61)
(0 . 75 , 0 45 , 0 61) (0. 75, 0. 45, 0. 61)
(0.69, 0 39, 05)
(0.75,045,0.61)
C5(0 95,0 55,0 65)
C 6 (0 9 , 0 55 , 0 65) G=(S(N), R(M), X(Q))
Figure9. SoftroughneutrosophicinfluencegraphG =(G, G)
Figure9:SoftroughneutrosophicinfluencegraphG=(G, G) withtheirinfluencestrengthas(0 6,0 45,0 5),respectively. Since,thereismorethanonepath,therefore,thetradercalculatesthescorefunctionwhichisformulated inequation4.1:
ScoreFunction(Ci)= TS(N )(Ci)+ TS(N )(Ci)+ TR(M)(Cij )+ TR(M)(Cij )+ TX(Q)(CiCjk)+
Since,thereismorethanonepath,therefore,thetradercalculatesthescorefunctionwhichis formulatedinEquation(4):
ScoreFunction(Ci )= TS(N)(Ci )+ TS(N)(Ci )+ TR(M)(Cij )+ TR(M)(Cij )+ TX(Q)(Ci Cjk )+ TX(Q)(Ci Cjk ), IS(N)(Ci )IS(N)(Ci )+ IR(M)(Cij )IR(M)(Cij )+ (4) IX(Q)(Ci Cjk )IX(Q)(Ci Cjk ), FS(N)(Ci )FS(N)(Ci )+ FR(M)(Cij )FR(M)(Cij )+ FX(Q)(Ci Cjk )FX(Q)(Ci Cjk ) .
Foreach Ci,thescorevaluesof Ci iscalculateddirectlyandasshowninTable 19. Table19. ScoreFunction. V ScoreValues
C1 (9.97,1.054,2.702) C2 (5.87,1.2979,1.7105) C3 (8.48,1.3562,2.2994) C4 (6.73,1.392,2.3119) C5 (7.07,1.3673,1.9029) C6 (4.23,0.6929,1.2175)
So,hechoosesthepath P3:C1,C3,C5,C2,C6.TheAlgorithm 1 oftheapplicationisalsobegivenin Algorithm1.TheflowchartisgiveninFigure 10.
Algorithm1: Influencestrengthofeachpathinroughneutrosophicinfluencegraph
1.Inputtheuniversalsets C and P
2.Inputthefullsoftset S andneutrosophicset N on V
3.CalculatetheSoftroughneutrosophicsetson V
4.Inputtheuniversalsets E and L
5.Inputthefullsoftset R andneutrosophicset M on E.
6.CalculatetheSoftroughneutrosophicsetson E
7.Inputtheuniversalsets I and F
8.Inputthefullsoftset X andneutrosophicset Q on I
9.CalculatetheSoftroughneutrosophicsetson I
10.Findthenumberofpathandcalculatetheirinfluencestrengthof eachpathfrom C1 to Cn
11.Choosethatpathwhichhasmaximummembership,minimum indeterminacyandfalsityvalue.If i > 1,thancalculatethe scorevaluesofeach Ci,choosethat Ci whichhasmaximum membershipandcomeimmediatelyafter C1 inoneofthepaths.
5.Conclusions
Graphtheoryhasbeenappliedwidelyinvariousareasofengineering,computerscience,database theory,expertsystems,neuralnetworks,artificialintelligence,signalprocessing,patternrecognition, robotics,computernetworks,andmedicaldiagnosis.Presentresearchhasshownthattwoormore theoriescanbecombinedintoamoreflexibleandexpressiveframeworkformodelingandprocessing incompleteinformationininformationsystems.Variousmathematicalmodelsthatcombineroughsets, softsetsandneutrosophicsetshavebeenintroduced.Asoftroughneutrosophicsetisahybridtoolfor handlingindeterminate,inconsistentanduncertaininformationthatexistinreallife.Wehaveapplied thisconcepttographtheory.Wehavepresentedcertainconcepts,includingsoftroughneutrosophic graphs,softroughneutrosophicinfluencegraphs,softroughneutrosophicinfluencecycles,softrough
neutrosophicinfluencetrees.Wealsohaveconsideredanapplicationofsoftroughneutrosophic influencegraphindecision-makingtoillustratethebestpathinthebusiness.Inthefuture,we willstudy,(1)Neutrosophicroughhypergraphs,(2)Bipolarneutrosophicroughhypergraphs,(3) Neutrosophicsoftroughhypergraphs,(4)Decisionsupportsystemsbasedonsoftroughneutrosophic information.
Start
Input C,P ,S,N
TS(N )(b)= b∈Ss(a) t∈Ss(a) TN (t),
TS(N )(b)= b∈Ss(a) t∈Ss(a) TN (t),
IS(N )(b)= b∈Ss(a) t∈Ss(a) IN (t),
IS(N )(b)= b∈Ss(a) t∈Ss(a) IN (t),
FS(N )(b)= b∈Ss(a) t∈Ss(a) FN (t), FS(N )(b)= b∈Ss(a) t∈Ss(a) FN (t)
If S(N ) = S(N )
Yes
Input E,L,R,M
TR(M )(bij )= bij ∈R (alm) tij ∈R (alm) TM (tij ), TR(M )(bij )= bij ∈R (alm) tij ∈R (alm) TM (tij ), IR(M )(bij )= bij ∈Rs(alm) tij ∈Rs(alm) IM (tij ), IR(M )(bij )= bij ∈R (alm) tij ∈R (alm) IM (tij ), FR(M )(bij )= bij ∈Rs(alm) tij ∈Rs(alm) FM (tij ), FR(M )(bij )= bij ∈Rs(alm) tij ∈Rs(alm) FM (tij )
If R(M ) = R(M ) Input I,F ,X,Q
If X(Q) X( Yes
Q) CalculatetheinfluencestrengthInstren ofeachpathfrom C1 to Cn
AuthorContributions: H.M.M.,M.A.andF.S.conceivedanddesignedtheexperiments;M.A.andF.S.analyzed thedata;H.M.M.wrotethepaper.
ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.
References
1. Smarandache,F. NeutrosophyNeutrosophicProbability,Set,andLogic;AmerResPress:Rehoboth,MA, USA,1998.
2. Wang,H.;Smarandache,F.;Zhang,Y.;Sunderraman,R.Single-valuedneutrosophicsets. Multispace Multistruct. 2010, 4,410–413.
3. Ye,J.Multicriteriadecision-makingmethodusingthecorrelationcoefficientundersingle-valued neutrosophicenvironment. Int.J.Gen.Syst. 2013, 42,386–394.
4. Ye,J.ImprovedcorrelationcoefficientsofsinglevaluedNSsandintervalNSsformultipleattributedecision making. J.Intell.FuzzySyst. 2014, 27,2453–2462.
5. Qeng,J.J.;Wang,J.Q.;Zhang,H.Y.;Chen,X.H.Anoutrankingapproachformulti-criteriadecision-making problemswithsimplifiedneutrosophicsets. Appl.SoftComput. 2014, 25,336–346.
6. Molodtsov,D.A.Softsettheory-firstresults. Comput.Math.Appl. 1999, 37,19–31.
7. Babitha,K.V.;Sunil,J.J.Softsetrelationsandfunctions. Comput.Math.Appl. 2010, 60,1840–1849.
8. Pawlak,Z.Roughsets. Int.J.Comput.Inf.Sci. 1982, 11,341–356.
9. Pawlak,Z. RoughSets-TheoreticalAspectstoReasoningaboutData;KluwerAcademicPublisher:Boston,MA, USA,1991.
10. Pawlak,Z.;Skowron,A.Rudimentsofroughsets. Inf.Sci. 2007, 177,3–27.
11. Pawlak,Z.;Skowron,A.Roughsets:Someextensions. Inf.Sci. 2007, 177,28–40.
12. Pawlak,Z.;Skowron,A.RoughsetsandBooleanreasoning. Inf.Sci. 2007, 177,41–73.
13. Biswas,R.;Nanda,S.Roughgroupsandroughsubgroups. Bull.PolishAcad.Sci. 2000, 42,247–251.
14. Davvaz,B.Roughnessinrings. Inf.Sci. 2004, 164,147–163.
15. Kuroki,N.Roughidealinsemigroup. Inf.Sci. 1997, 100,139–163.
16. Shahzamanian,M.H.;Shirmohammadi,M.;Davvaz,B.RoughnessinCayleygraphs. Inf.Sci. 2010, 180, 3362–3372.
17. Dubios,D.;Prade,H.Roughfuzzy,fuzzyroughsets. Int.J.Gen.Syst. 1990, 17,191–209.
18. Feng,F.;Li,C.;Davvaz,B.;Ali,M.I.Softsetscombinedwithfuzzysetsandroughsets:Atentativeapproach. SoftComput. 2010, 14,899–911.
19. Meng,D.;Zhang,X.;Qin,K.Softroughfuzzysetsandsoftfuzzyroughsets. Comput.Math.Appl. 2011, 62, 4635–4645.
20. Broumi,S.;Smarandache,F.;Dhar,M.Roughneutrosophicsets. NeutrosophicSetsSyst. 2014, 3,62–67.
21. Yang,H.L.;Zhang,C.L.;Guo,Z.L.;Liu,Y.L.;Liao,X.Ahybridmodelofsinglevaluedneutrosophic setsandroughsets:Singlevaluedneutrosophicroughsetmodel. SoftComput. 2016, 21,6253–6267, doi:10.1007/s00500-016-2356-y.
22. Kauffmann,A. IntroductionalaTheorieDesSous-EmsemblesFlous;MassonetCie:Paris,France,1973; Volume1 23. Rosenfeld,A.Fuzzygraphs.In FuzzySetsandTheirApplications;Zadeh,L.A.,Fu,K.S.,Shimura,M.,Eds.; Elsevier:NewYork,NY,USA,1975;pp.77–95. 24. Mordeson,J.N.;Peng,C.S.Operationsonfuzzygraphs. Inf.Sci. 1994, 79,159–170. 25. Mathew,S.;Sunitha,M.Typesofarcsinafuzzygraph. Inf.Sci. 2009, 179,1760–1768.
26. Mathew,S.;Sunitha,M.Strongeststrongcycles,fuzzygraphs. IEEETrans.FuzzySyst. 2013, 21,1096–1104.
27.
Gani,A.N.;Latha,S.R.Onirregularfuzzygraphs. Appl.Math.Sci. 2012, 6,517–523.
28. Gani,A.N.;Radha,K.Onregularfuzzygraphs. J.Phys.Sci. 2008, 12,33–40.
29. Gani,A.N.;Radha,K.Thedegreeofavertexinsomefuzzygraphs. Int.J.AlgorithmsComput.Math. 2009, 2, 107–116.
30. Gani,A.N.;Begum,S.S.Degree,order,sizeinintuitionisticfuzzygraphs. Int.J.AlgorithmsComput.Math. 2010, 3,11–16.
31. Parvathi,R.;Karunambigai,M.G.;Atanassov,K.T.Operationsonintuitionisticfuzzygraphs. InProceedingsoftheFUZZ-IEEE2009IEEEInternationalConferenceonFuzzySystems,JejuIsland,Korea, 20–24August2009.
32.
Akram,M.;Ashraf,A.;Sarwar,M.Novelapplicationsofintuitionisticfuzzydigraphsindecisionsupport systems. Sci.WorldJ. 2014,doi:10.1155/2014/904606. 33. Akram,M.;Nawaz,S.Fuzzysoftgraphswithapplications. J.Intell.FuzzySyst. 2016, 30,3619–3632. 34.
Akram,M.;Shahzadi,S.Novelintuitionisticfuzzysoftmultiple-attributedecision-makingmethods. NeuralComput.Appl. 2016,doi:10.1007/s00521-016-2543-x. 35.
Akram,M.;Alshehri,N.;Davvaz,B.;Ashraf,A.Bipolarfuzzydigraphsindecisionsupportsystems. J.Multiple-ValuedLogicSoftComput. 2016, 27,531–551. 36.
Akram,M.;Dudek,W.A.Regularbipolarfuzzygraphs. NeuralComput.Appl. 2012, 21,197–205. 37.
Dinesh,T.Fuzzyincidencegraphandintroduction. Adv.FuzzySetsSyst. 2016, 21,33–48. 38. Mordeson,J.N.Fuzzyincidencegraphs. Adv.FuzzySetsSyst. 2016, 21,1–13. 39. Mathew,S.;Mordeson,J.N.Fuzzyinfluencegraphs. NewMath.Nat.Comput. 2017, 13,311–325 40. Zafar,F.;Akram,M.Anoveldecision-makingmethodbasedonroughfuzzyinformation. Int.J.FuzzySyst. 2018, 20,1000–1014.
41. Akram,M.;Zafar,F.Roughfuzzydigraphswithapplication. J.Appl.Math.Comput. 2018, doi:10.1007s12190-018-1171-2.
42. Akram,M.;Shumaiza;Arshad,M.Anewapproachbasedonfuzzyroughdigraphsfordecision-making. J.Intell.FuzzySyst. 2018,doi:10.3233/JIFS-172069.
43. Malik,H.M.;Akram,M.Anewapproachbasedonintuitionisticfuzzyroughgraphsfordecision-making. J.Intell.FuzzySyst. 2018, 34,2325–2342.
44. Zhan,J.;Malik,H.M.;Akram,M.Noveldecision-makingalgorithmsbasedonintuitionisticfuzzyrough environment. Int.J.Mach.Learn.Cybern. 2018,doi:10.1007/s13042-018-0827-4.
45. Ishfaq,N.;Sayed,S.;Akram,M.;Smarandache,F.Notionsofroughneutrosophicdigraphs. Mathematics 2018, 6,doi:10.3390/math6020018.
46. Akram,M.;Malik,H.M.;Shahzadi,S.;Smarandache,F.Neutrosophicsoftroughgraphswithapplication. Axioms 2018, 7,14,doi:10.3390/axioms7010014.
47. Zhang,X.;Dai,J.;Yu,Y.Ontheunionandintersectionoperationsofroughsetsbasedonvarious approximationspaces. Inf.Sci. 2015, 292,214–229.
48. Bao,Y.L.;Yang,H.L.Onsinglevaluedneutrosophicrefinedroughsetmodelanditsapplication. J.Intell. FuzzySyst. 2017, 33,1235–1248,doi:10.3233/JIFS-17094.
49. Maji,Q.K.Neutrosophicsoftset. Ann.FuzzyMath.Inform. 2013, 5,157–168.
50. Broumi,S.;Smarandache,F.Interval-valuedneutrosophicsoftroughsets. Int.J.Comput.Math. 2015, doi:10.1155/2015/232919.
51. Akram,M.;Shahzadi,S.;Smarandache,F.Multi-attributedecision-makingmethodbasedonneutrosophic softroughinformation. Axioms 2018, 7,19.
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