Bipolar single valued neutrosophic detour distance

2021·0.31 MB·English

by  G Upender Reddy, T Siva Nageswara Rao, N Srinivasa Rao, V Venkateswara Rao;

INDIAN JOURNAL OF SCIENCE AND TECHNOLOGY
RESEARCHARTICLE
Bipolar single valued neutrosophic detour
distance
(cid:3)
GUpenderReddy1 ,TSivaNageswaraRao2,NSrinivasaRao3,
VVenkateswaraRao4
OPENACCESS
1AssistantProfessor,DepartmentofMathematics,NizamCollege(A),OsmaniaUniversity,
Received:22.12.2020
Hyderabad,500001,TS,India
Accepted:02.02.2021 2AssociateProfessor,DivisionofMathematics,DepartmentofS&H,V.F.S.T.R(Deemedto
beUniversity),Vadlamudi,522207,Guntur(Dt.),A.P,India
Published:16.02.2021
3AssistantProfessor,DivisionofMathematics,DepartmentofS&H,V.F.S.T.R(Deemedtobe
University),Vadlamudi,522207,Guntur(Dt.),A.P,India
4ResearchScholar,DivisionofMathematics,DepartmentofS&H,V.F.S.T.R(Deemedtobe
University),Vadlamudi,522207,Guntur(Dt.),A.P,India
Citation:ReddyGU,RaoTSN,
RaoNS,RaoVV(2021)Bipolarsingle
valuedneutrosophicdetour
Abstract
distance. IndianJournalofScience
andTechnology14(5):427-431.http
s://doi.org/10.17485/IJST/v14i5.2102 Objectives: In the present article, we deduced a characterization of Bipolar
(cid:3) Single Valued Neutrosophic (BSVN) radius and eccentricity of the vertex
Correspondingauthor.
based on Bipolar Single Valued Neutrosophic set(BSVNS) detour. Method:
[email protected]
We obtained some definitions BSVN on a vertex like BSVN detour eccentric
Funding:None vertex, BSVN detour radius, BSVN detour diameter, BSVN detour centered
CompetingInterests:None and BSVN detour periphery. Findings: We derived some important results
basedontheseBSVNdetourradius,diameter,centerandperiphery.Novelty:
Copyright:©2021Reddyetal.This
isanopenaccessarticledistributed The detour distance of the BSVNS model is proposed and generalized by
underthetermsoftheCreative
this. An important and suitable condition for the graphs of the Single Valued
CommonsAttributionLicense,which
permitsunrestricteduse, NeutrosophicSet(SVNS)modeltoBSVNSdetourdistanceshasbeenidentified.
distribution,andreproductionin Keywords:Detourdistance;BSVNdetoureccentric;BSVNdetourdistance;
anymedium,providedtheoriginal
BSVNdetourperipheralnode;BSVNdetourpath
authorandsourcearecredited.
PublishedByIndianSocietyfor
EducationandEnvironment(iSee) 1 Introduction
ISSN
TheNeutrosophicsets(1)isagreatexactimplementforthesituationuncertaintyinthe
Print:0974-6846
Electronic:0974-5645 realworld.Theseuncertaintyideascomefromthetheoriesoffuzzysets(2),intuitionistic
fuzzysets(3) andinterval-valuedintuitionisticfuzzysets(4).Therepresentationofthe
neutrosophicsetsistruth,indeterminacyandfalsityvalue.TheseT,I,Fvaluesbelong
tostandardornonstandardunitintervaldenotedby]-0,1+[.
TheideaofasubclassoftheNSandSVNSbyintuitionisticfuzzysets(5),inthisthe
functionsTruth,Indeterminacy,Falsityarenotdependentandthesevaluesarepresent
within[0,1].NeutrosophictheoryiswidelyexpandedinallfieldsespeciallyinGraph
theoryandTopology.
Inagraphtheory,thenewgraphmodelwasinvitedbyusingBSVNsetisknownas
BSVNGraph(BSVNG)(6,7).TheideaofBSVNgraphsfromthefuzzy,bipolarfuzzyand
single-valuedneutrosophicgraphs.Theuncertaintyonthegraphofverticesandedges
orbothrepresentationstobeafuzzygraph.
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Reddyetal. /IndianJournalofScienceandTechnology2021;14(5):427–431
Inthismanuscript,wediscussaboutBSVNgraphsandneutrosophicdetourdistancebetweentwoverticesofthegraphbased
onBSVNeccentricity,radius,diameter,andtheperipherywithrespecttodetourdistance.
2 Preliminaries
Explanation2.1: BSVNsets
{ A BSVN set is explained as}the membership functions represented as an object in W is denoted by
<w;TP;IP;FP;TN;IN;FN >:weW the functions TP; IP; FPare mapping from W to [0; 1] and TN; IN; FNare
mappingfromW to[(cid:0)1; 0].
Example.LetW=fw1; w2; w3gandA=f<w1;0:4;0:2;0:6;(cid:0)0:03;(cid:0)0:2;(cid:0)0:01>;<w2; 0:6;0:4;0:2;(cid:0)0:5;(cid:0)0:3;(cid:0)0:03>;
<w3;0:7;0:04;0:3;(cid:0)0:8;(cid:0)0:4;(cid:0)0:05>gisaBSVNsetinW.
Explanation2.2: SVNrelationonW:-
LetW beanon-emptyset.ThenwecallmappingZ=(W; Tp; Ip; FP; TN; IN; FN);FN(w):W XW ![(cid:0)1; 0]X [0; 1]
isaBSVNrelationonW suchthat
TZP(w1;w2)e[0;1];IZP(w1;w2)e[0;1];FZP(w1;w2)e[0;1]andTZN(w1;w2)e[(cid:0)1;0];IZN(w1;w2)e[(cid:0)1;0];FZN(w1;w2)e[(cid:0)1;0]
Explanation2.3:LetZ1=(TzP1; IzP1; FzP1; TzN1; IzN1; FzN1)andZ2=(TzP2; IzP2; FzP2; TzN2; IzN2; FzN2)beaBSVNgraphsona
setW.IfZ2isaBSVNrelationonZ1,then
TZP2(w1; w2)(cid:20)min(TZP1(w1); TZP1(w2)),IZP2(w1; w2)(cid:21)max(IZP1(w1); IZP1(w2));FZP2(w1; w2)(cid:21)max(FZP1(w1); FZP1(w2))
aanlldw1T;NZ2w(w221;Ww2)(cid:21)max(TZN1(w1); TZN1(w2));IZP2(w1;w2)(cid:20)min(IZN1(w1); IZN1(w2));FZN2(w1;w2)(cid:20)min(FZN1(w1); FZN1(w2))for
Explanation2.4: ThesymmetricpropertydefinedonBSVNrelationZonW isexplainedby
TZP(w1; w2)=TZP(w2; w1), IZP(w1; w2)=IZP(w2; w1), FZP(w1; w2)=FZP(w2; w1) and TZN(w1; w2)=TZN(w2; w1),
IZN(w1; w2)=IZN(w2; w1),FZN(w1; w2)=FZN(w2; w1)forallw1; w22W
Explanation2.5:BSVNgraph
The new graph in SVN is denoted by for all w1; w2 2 W , G(cid:3) = (V; E) is a pair G = (Z1; Z2), where Z1 =
(TzP1; IzP1; FzP1; TzN1; IzN1; FzN1)isaBSVNSinV andZ2=(TzP2; IzP2; FzP2; TzN2; IzN2; FzN2)isBSVNSinV2definedas
TZP2(w1; w2)(cid:20)min(TZP1(w1); TZP1(w2)),IZP2(w1; w2)(cid:21)max(IZP1(w1); IZP1(w2)); FZP2(w1; w2)(cid:21)max(FZP1(w1); FZP1(w2))
aanlldw1T;NZ2w(w221;Ww2)(cid:21)max(TZN1(w1); TZN1(w2));IZP2(w1;w2)(cid:20)min(IZN1(w1); IZN1(w2));FZN2(w1;w2)(cid:20)min(FZN1(w1); FZN1(w2))for
TheBSVNSGofanedgedenotedbyw1w22V2
Explanation 2.6 Let G = (Z1; Z2) be a BSVNSG anda; b 2V. A path P : a = w0; w1; w2; :::;wk(cid:0)1; wk = b in G
is a sequence of distinct vertices such that TZP2(wi(cid:0)1; wi)>0, IZP2(wi(cid:0)1; wi)>0, FZP2(wi(cid:0)1; wi)>0,TZN2(wi(cid:0)1; wi)>0,
IZN2(wi(cid:0)1; wi)>0,FZN2(wi(cid:0)1; wi)>0wherei=1; 2; 3; :::; kandlengthofthepathisk2N(a positiveinteger);whereais
calledinitialvertexandbiscalledterminalvertexinthepath.
Explanation 2.7. A BSVN graph G = (Z1; Z2) of G(cid:3) = (V; E) is called a strong BSVN graph if TZP2(w1; w2) =
min(TZP1(w1); TZP1(w2)),IZP2(w1; w2)=max(IZP1(w1); IZP1(w2));
FZP2(w1; w2) = max(FZP1(w1); FZP1(w2)) and TNZ2(w1; w2) = max(TZN1(w1); TZN1(w2));IZP2(w1; w2) =
min(IZN1(w1); IZN1(w2));FZN2(w1; w2)=min(FZN1(w1); FZN1(w2))forallw1; w22W
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IfP:a=w0; w1; w2; :::;wk(cid:0)1; wk=bbeapathoflengthkbetweenaandbthen(TZP2(a; b);IZP2(a; b);FZP2(a; b))kand
(TN(a; b);IN (a; b);FN(a; b))kisdefinedas
Z2 Z2 Z2
8 {
( ) >>>< sup{TZP2(a;w1)^TZP2(w1;;w2)^:::^TZP2(wk(cid:0)1; bg
TZP2(a;b);IZP2(a;b);FZP2(a;b) k=>>>: iinnff{IFZPZP22((aa;;ww11))__IFZPZ2P2(w(w1;1;;w;w22))__::::::__IZPF2ZP(2w(wk(cid:0)k1(cid:0);1; bbgg
8 {
( ) >>>< sup{TZN2(a;w1)_TZN2(w1;;w2)_:::_TZN2(wk(cid:0)1; bg
TZN2(a;b);IZN2(a;b);FZN2(a;b) k=>>>: iinnff{IFZNZN22((aa;;ww11))^^IFZNZ2N2(w(w1;1;;w;w22))^^::::::^^IZNF2ZN(2w(wk(cid:0)k1(cid:0);1; bbgg
¥ ¥
(TP(a; b);IP (a; b);FP(a; b)) and(TN(a; b);IN (a; b);FN(a; b)) issaidtobethestrengthofconnectednessbetween
Z2 Z2 Z2 Z2 Z2 Z2
twoverticesxandyinG,where
( ) ( { } { } { })
¥
TP(a;b);IP(a;b);FP(a;b) = sup TP(a;b) ; inf TP(a;b) ; inf FP(a;b)
Z2 Z2 Z2 Z2 Z2 z2
( ) ( { } { } { })
¥
TN(a;b);IN(a;b); FN(a;b) = inf TN(a;b) ;sup TN(a;b) ;sup FN(a;b)
Z2 Z2 Z2 Z2 Z2 z2
( ( ) ( ) ( ) )
¥ ¥ ¥
If TP(a; b)(cid:21) TP(a; b) ; IP (a; b)(cid:20) IP (a; b) ; FP(a; b)(cid:20) FP(a; b) and
(Z2 (Z2 ) Z2 (Z2 ) Z2 (Z2 ) )
¥ ¥ ¥
TN(a; b)(cid:21) TN(a; b) ; IN (a; b)(cid:20) IN (a; b) ; FN(a; b)(cid:20) FN(a; b) then the arc ab in G is called a
Z2 Z2 Z2 Z2 Z2 Z2
strongarc.Apatha-bisastrongpathifallarcsonthepatharestrong.
3 BSVN detour distance
Explanation3.1
BSVNdetourdistanceisdefinedasthelengthofa(cid:0)b.Thestrongpathbetweenaandbisifthereisnootherpathlonger
than P between a and b and we denote this by BSND(a; b). Any a-b strong path whose length is BSND (a; b) is called
ana(cid:0)bBSVNdetourpath.Theeccentricity(EccBND(G))ofanodeisthedistancefromanodetothefurthestnodeinthe
bipolarsingle-valuedneutrosophicgraphG.Theradius(radBNG(G))ofabipolarsingle-valuedneutrosophicgraphGisthe
minimumamongalleccentricityofnodes.Theperipheryofthegraphistheeccentricityequaltothediameterofthepath.The
diameter(diamBND(G))ofabipolarsinglevaluedneutrosophicgraphGisthemaximumamongalleccentricityofnodes.
4 BSVN detour periphery (Per (G)) and BSVN detour eccentric graph (Ecc (G))
BND BND
Theorem4.1ABSVNgraphGisaBSVNdetourself-centeredifandonlyifeverynodeofGisaBSVNdetoureccentric.
Proof.SupposeGisaBSVNdetourself-centeredBSVNgraphandletbbeanodeinG.
Let a 2 b(cid:3)BND. So EccBND(b) = BNDG(a; b): Since G is a BSVN detour self-centered BSVN graph, EccBND(a) =
EccBND(b)=BNDG(a; b)andthisimpliesthatb2a(cid:3)BND.HencebisaBSVNdetoureccentricnodeofG.
Conversely,leteverynodeofGisaBSVNdetoureccentricnode.Ifpossible,letGbenotBSVNdetourselfcentradBSVN
graph.ThenradBND(G)̸=diamBND(G)andthereexistnoder2GsuchthatEccBND(r)=diamBND(G).Also,letP2rB(cid:3)ND.
LetU ber(cid:0)pBSVNdetourinG.SotheremusthaveanodeqonUforwhichthenodeqisnotaBSVNdetoureccentric
node ofU. Also, q cannot be a BSVN detour eccentric node of every other node. Again if q be a BSVN detour eccentric
nodeofanodea(say),thismeansq2a(cid:3) .Thenthereexistsanextensionofa (cid:0) qBSVNdetouruptororupto p.But
BND
thiscontradictsthefactsthatq2a(cid:3)BND.Hence,radBND(G) = diamBND(G)andGisaBSVNdetourself-centeredBSVNgraph.
Theorem4.2. IfGisaBSVNdetourself-centeredBSVNgraph,thenradBND (G)= diamBND (G)=n(cid:0)1,wherenisthe
numberofnodesofG.
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Proof.LetGbeaBSVNdetourself-centeredBSVNgraph.
Ifpossible,let diamBND(G) = l < n (cid:0) 1.LetU1andU2betwodistinctBSVNdetourperipheralpath.Letp2U1; q2U2.
Sothereexistastrongpathbetween pandq,becauseoftheconnectednessofG.ThenthereexistnodesonU1andU2,whose
eccentricity > l, but this is impossible, because diamBND(G) = l. HenceU1 andU2 are not distinct. SinceU1 andU2 are
arbitrary,sothereexistnoderinGsuchthatriscommoninallBSVNdetourperipheralpaths.SoEccBND(r)<l,whichis
impossible,becauseGisaBSVNdetourself-centered.Hence,diamBND(G) = n (cid:0) 1 = radBND(G):
Theorem4.3. ForaconnectedBSVNgraphG,PerBND(G)=GifandonlyiftheBSVNdetoureccentricityofeachnodeofG
isn(cid:0)1,n=numberofnodesinG.
Proof.LetPerBND(G)=G.ThenEccBND(p) =diamBND(G),8p2G.SoeverynodeofGisaBSVNdetourperipherynode
ofG.ThereforeGisaself-centeredBSVNgraphandradBND (G)= diamBND (G)= n(cid:0)1.SotheBSVNdetoureccentricity
ofeachnodeofGis n(cid:0)1.
Conversely,lettheBSVNdetoureccentricityofeachnodeofGisn(cid:0)1.So,radBND (G)= diamBND (G)= n(cid:0)1.Allnodes
ofGareBSVNdetourperipheralnodesandhencePerBND(G) = G:
Theorem4.4. ForaconnectedBSVNgraphG,EccBND(G)=GifandonlyiftheBSVNdetoureccentricityofeachnodeofG
isn(cid:0)1,n=numberofnodesinG.
Proof.Let,EccBND(G)=G,SoallnodesofGareBSVNdetoureccentricnode.Therefore,Gisaself-centeredBSVNgraph
anddiamBND(G) = n (cid:0) 1.Hence,theBSVNdetoureccentricityofeachnodeofGisn (cid:0) 1:
Conversely,lettheBSVNdetoureccentricityofeachnodeofGisn(cid:0)1.SoradBND(G)=diamBND(G)=n(cid:0)1.Soallnodes
ofGareBSVNdetourperipheralnodesaswellasBSVNdetoureccentricnodes.Hence,EccBND(G) =G.
Theorem4.5.In a connected BSVN graph G, a node b is a BSVN detour eccentric node if and only if b is a BSVN detour
peripheralnode.
Proof.LetbbeaBSVNdetoureccentricnodeofGandletb2a(cid:3) .LetxandybetwoBSVNdetourperipheralnodes,
BND
thenBND(x; y)=diamBND(G)=k(say).LetP1andP2beanyx (cid:0) yanda (cid:0) bBSVNdetourinGrespectively.Therearise
twocases.
Case1:When b is not an internal node in G i.e, there is only one node, say c w∩hich is adjacent to b. Soc2P2. Since G
is connected, c is connected to a node of P1, say c′. So either c′ 2P2 or c′ 2(P1 P2). Thus, in any case the path from a
′
to x or a to y through c and c is longer thanP2. But it is impossible since b is a BSVN detour eccentric node of a: Hence
EccBND(a)=diamBND(G)i.e,bisaBSVNdetourperipheralnodeofG.
Case2:WhenbisaninternalnodeinG,thenthereexistsaconnectionbetweenbtoxandbtoy;becauseoftheconnectedness
ofG.Thena (cid:0) bBSVNdetourcanbeextendedtoxory.ThisisimpossiblebecausebisaBSVNdetoureccentricnodeofa.
HenceEccBND(a)=diamBND(G)i.e,bisaBSVNdetourperipheralnodeofG.
Conversely, we assume that b be a BSVN detour peripheral node G. So there exists a BSVN detour peripheral node say
a(distinctfromb).ThereforebisaBSVNdetoureccentricnodeofa.
5 Conclusions
Inthisstudy,weextendedthepartofBSVNsetstoBSVNgraphs.ByTheorem4.1,ifradBND(G)isequaltodiamBND(G)and
alsoBSVNgraphisaBSVNdetourisself-centeredBSVNgraph.Theorem4.2indicatesradBNDofGanddiamBNDofGisequal
toonelessthanthenumberofnodesofG.Fromtheorem4.3andtheorem4.4BSVNdetoureccentricityisn(cid:0)1;wherenisa
numberofnodes.Finally,Theorem4.5provesthatanynodeisBSVNdetoureccentricnodebasedonBSVNdetourperipheral
node.
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