On Matrices associated with L-Fuzzy Graphs

Transcription

1 lol Jorl of Pr d Appld Mthmts ISSN olm 3 Nmr 6 7 pp Rsrh Id Pltos O Mtrs ssotd wth -Fzzy rphs Prmd Rmhdr P Dprtmt of Mthmts St Pl s Collg Klmssry Koh Krl Id K Thoms P Dprtmt of Mthmts & Rsrh Ctr Bhrt Mt Collg Thrr Koh-68 Krl Id Astrt I ths ppr w df dffrt ltt mtrs ssotd wth -fzzy grph F mly th -fzzy dgr mtrx -fzzy d mtrx d -fzzy dy mtrxw show tht th o d mt of -fzzy dy mtrs of two Fs lso yld -fzzy dy mtrs W prov tht th dgol trs of th omposto of th -fzzy dy mtrx wth tslf r xtly th mmrshp dgrs of th orrspodg vrts Flly w stlsh th rlto tw th -fzzy d mtrx d -fzzy dy mtrx of F d show tht th -fzzy dy mtrs of somorph Fs r rltd Kywords: -fzzy grphs ltt mtrx somorph Fs -fzzy dy mtrx -fzzy d mtrx Mthmts St Clssfto 5C5 5C7 INTRODUCTION rph Thory ws frst trodd to th world y ohrd Elr 736 It ws qly ptd s o of th most ovt wys to modl rltoshps tw ots d to stdy thm Th fld hs s trmdos growth

2 8 Prmd Rmhdr d K Thoms Th opt fzzy grphs ws trodd y Azrl Rosfld 975 ]Tody fzzy grphs r th s mthmtl strtr sh rs of rsrh tht ld lstrg lyss grop strtr dts thory 8] otrol systms ] d v dso thory 3]Th wor of Mordso d Ml 6] promptd th thors to trod th opt of -fzzy grphs Fs 9] Dffrt typs of somorphsms o Fs thr proprts d th omplmt of F wr stddi ] th strog prodt of Fs ws trodd d som of ts proprts wr stdd Th opt of ltt mtrs pprd frst 964 5] d thy hv stdd sg th grph thort pproh ]I ]wor hs do o th hrtrst roots of ltt mtrs Ths ppr ms to df d stdy rt ltt mtrs ssotd wth FTh oo 8] s prmry rfr ths ppr Throghot ths worw hoos to ft omplt ltt wth lst lmt d grtst lmt wth prtl ordr lss othrws sttdfor ll fdmtl rslts grph thory w rfr to 3] d for ll rslts rgrdg ltts w rfr 6]I sto of ths ppr w lst s dftos from th thory of fzzy grphs -fzzy grphs d ltt mtrsi sto 3 w df th -fzzy dgr mtrx -fzzy dy mtrx d -fzzy d mtrx of F d rrv t th m rsltsw old sto 4 wth smmry of th wor do d possl ftr wor PREIMINARIES I ths sto w rvw som s dftos d rslts of fzzy grph thory -fzzy grphs d ltt mtrs tht wll dd th sql Dfto 9] A -fzzy grph F wth th drlyg st s ompty st togthr wth pr of ftos : d : sh tht d v v v rsptvly dot th spports of d Th drlyg grph of th F s th rsp grph Throghot ths ppr w hoos -fzzy grphs whos drlyg grphs r ft smpl grphs 3] ; wthot loops or prlll dgs Dfto 9] Th F odto v v v s sd to strog f t stsfs th

3 O Mtrs ssotd wth -Fzzy rphs 8 Dfto 3 9] Th F odto v v v s sd to omplt f t stsfs th Dfto 4 9] By somorphsm of Fs w m tv mppg h : togthr wth tv mppg l : sh tht l ] h ] d l v] h h v] v Symollly w wrt Wh l oms th dtty mp d Dfto 5 9] A F H s sd to prtl fzzy sgrph of th F f v d v v v Dfto 6 9] Cosdr th F W df th ordr p d sz q of s p d q v v Dfto 7 9] Cosdr th F Th w df th dgr d of vrtx d v v v s Dfto 8 5] A ltt mtrx s mtrx whos trs r from ltt W dot th st of ll sh m mtrs y M m d y M f oly sqr mtrs of ordr r osdrda lmt of ths st s dotd y A ] m Dfto 9 5] t M th st of ll mtrs ovr tt Mtrs W shll dot y th lmt of whh stds th of A M For A ] B ] C ] M w df th try

4 8 Prmd Rmhdr d K Thoms A B Th o A B C Th mt A B C d Th omposto A B C T Th trspos A C C C f Th omplmt A C provdd s omplmtd g Th dtty I ] f f 3 THE MATRICES ASSOCIATED WITH FS I ths sto vrtx of th drlyg grph s dotd y d th dg tw d y W g y trodg th ltt mtrs ssotd wth Fs d s som of thr proprts Cosdr th F wth drlyg grph sh tht d m W lst th lmts of s d thos of s m Dfto 3 Th -fzzy dgr mtrx of s th ltt mtrx wth rows d olms orrspodg to It s dotd y ' D d ] whr d f othrws Dfto 3 Th -fzzy d mtrx of s th ltt mtrx wth rows orrspodg to d olms orrspodg to m It s dotd y ' C ] whr m f th th dg hs o d othrws

5 O Mtrs ssotd wth -Fzzy rphs 83 Dfto 33 Th -fzzy dy mtrx of s th ltt mtrx wth rows d olms orrspodg to It s dotd y ' A ] whr f f W ot tht ths mtrs hg o hg th llg of th vrts or dgs Illstrtd low fgrs d r ltt d ssotd F rsptvlyth lls of th lmts th ltt r dtd wth thm I th F th mmrshp dgrs of th vrts r dtd wth thm d thos of th dgs y thr sds: Fgr Fgr t th dgs rsptvly lld

6 84 Prmd Rmhdr d K Thoms Th th ssotd - fzzy mtrs r: D C d A As drt osq of ths dftos w hv th followg osrvtos: Osrvto 34 t A C D d rsptvly th -fzzy dgr mtrx th -fzzy d mtrx d th -fzzy dy mtrx of th F Th D d A r symmtr mtrs Eh olm of C hv xtly two ql o zro trs- th mmrshp dgr of th dg th rows orrspodg to ts d vrts 3 A olm D or A wth ll trs dts soltd vrtx 4 Th o of ll th trs olm or row of A gvs th dgr of th orrspodg vrtx 5 Th o of ll lmts A gvs th sz of th F 6 Th o of th dgol trs of D gvs th ordr of th F 7 v y sqr symmtr ltt mtrx of ordr '' wth dgol trs w ostrt F whos - fzzy dy mtrx s th sm Nxt w df th trsto d o of two Fs W rll th dfto of o d trsto of two fzzy sts from 8] Dfto 35 t d y two Fs W df th -fzzy trsto d th -fzzy o of ths Fs s follows: Th trsto Th o

7 O Mtrs ssotd wth -Fzzy rphs 85 Th followg thorm shows tht th o d trsto r lso Fs: Thorm 36 Th trsto d o of two Fs s dfd ov r lso Fs Proof W shll show tht th odtos for th - fzzy sts to F r stsfd for h d of dg: Cosdr th drlyg grph E E of Th for h E E w hv ] ] y th dfto of Fs ] ] y th ommttvty d ssotvty of ltt lmts ] ] H ] ] E E Ths th trsto of two Fs s lso F Nxtosdr th drlyg grph E E of Th for h E E w hv d y th proprty of Fs H Ths Smlrly Togthr y th proprty of ltts Ths th o of two Fs s F

8 86 Prmd Rmhdr d K Thoms Ths lds s to: Proposto 37 t d two Fs wth th sm drlyg grph wth dy mtrs t thr - fzzy dy mtrs A d A rsptvly Th A A d A A r oth th dy mtrs of Fs Proof t A ]da ] Th y dfto A A ] d A A Cosdr th Fs d By thorm 3 ths Fs hv th dy mtrs A A d A A rsptvly ] Rmr 38 hs th sm drlyg grph s d If s rglr ltt ; th lso hs th sm drlyg grph Hl th ordry fzzy s whr th sm or dffr of dy mtrs do ot yld dy mtrs th o d mt of -fzzy dy mtrs yld -fzzy dy mtrs Nxt w osdr A B whr A d B r -fzzy dy mtrs Ovosly ths omposto d ot v symmtr so t dos ot slly rprst th - fzzy dy mtrx of FHowvr w do hv: Thorm 39 t d y two Fs wth th sm drlyg grph E t thr -fzzy dy mtrs A d A rsptvly Th f ostt d othr ostt th C A I s th - fzzy dy mtrx of A Proof W hv A d A

9 O Mtrs ssotd wth -Fzzy rphs 87 Th A A d C I A A Now osdr th F Th E w hv Ovosly ths F hs th -fzzy dy mtrx C I A A Thorm 3 t hv th -fzzy dy mtrx A Th th dgol trs of A A gv th dgrs of th orrspodg vrts Proof t A Th A A ] whr Now y symmtry

10 88 Prmd Rmhdr d K Thoms y dfto d y dfto y dfto Flly w hv th followg thorm: Thorm 3 t d y two Fs wth somorph drlyg grphs t thr -fzzy dy mtrs A d A rsptvly Th f th Fs r somorph o ltt mtrx otd from th othr y prmttos of rows d olms Proof Sppos d r somorph Th y dfto thr xsts to ' h tw d tht prsrvs th mmrshp dgrs of th vrts d dgsh A d A r of th sm ordr t th vrts of lld Th th row/olm orrspodg to A s dtl to th row/olm orrspodg to h A Frthr th to srs tht ths orrspod s xhstv d q Ths A otd from A y prmttos of rows d olms Rmr 3 Th ovrs of th ov thorm d ot trfor v f A otd from A y prmttos of rows d olms th Fs dffr th mmrshp dgrs of th vrts thogh ot th dgs Ths s llstrtd th followg xmpl: t th ltt ' s fgr Th th Fs fgr 4 d fgr 5 hv th sm - fzzy dy mtrx A Howvrt s ot possl to ostrt somorphsm tw th Fs

11 O Mtrs ssotd wth -Fzzy rphs 89 Fgr 4 Fgr 5 4 CONCUSION W hv dfd ths ppr th -fzzy dgr mtrx -fzzy d mtrx d -fzzy dy mtrx of F Crt proprts of ths mtrs hv provdusg ths s s ldg losw td to prs ths l of wor to m stds o th rgy of F d ts sptrm Aowldgmt Th frst thor s thfl to th UC for th wrd of Thr Fllowshp dr th XII Pl REFERENCES ] A Bglrg W SdghM Ml O th sslty/otrolllty of Fzzy Cotrol Systms Iformto Ss ] K Chlrov Powrs of mtrs ovr dstrtv ltts - rvwfzzy Sts d Systms ] J ClrDA Holto A Frst oo t rph Thory Alld Plshrs mtd 99 4] M J tor AllF Hrrr Itrprtlty of lgst fzzy rl-sd systms: ovrvw of trprtlty msrs Iformto Ss ] Y vo tt Mtrs Iformto Cotrol ] rtzr rl tt Thory Sod Edto Brhsr Bsl 998 7] J N MordsoD S MlFzzy Commttv Algr World Stf Plshg Compy 998 8] J N Mordso P S Nr Fzzy rphs d Fzzy Hyprgrphs Phys rlg

12 8 Prmd Rmhdr d K Thoms 9] P Rmhdr K Thoms O Isomorphsms of -fzzy grphs Als of Fzzy Mthmts d Iformtsolm No: Frry ] P Rmhdr K Thoms Th strog prodt of -fzzy grphsbllt of Krl Mthmts Assoto olm 3 J ] A Rosfld Fzzy rphs Fzzy Sts d thr Appltos to Cogtv d Dso Prosss ds A Zdh K S F d M Shmr Ad Prss Nw Yor ] K Thoms Joy A stdy o hrtrst roots of ltt mtrs Jorl of Mthmts olm 66Artl ID pgs