Isomorphism on Fuzzy Hypergraphs
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1 IOSR Jounl of Mthemts (IOSRJM) ISSN: 8-8 Volume Issue (Sep-Ot. ) PP - Isomophsm on Fuzzy Hypegphs C.Rhmn C.Rhk (Assstnt Pofesso Deptment of Mthemts Kongu Ats n sene ollege oe Tml Nu In) Astt: In ths ppe the oesze n egee of the noes of the somoph fuzzyhypegphs e susse. Isomophsm etween fuzzy hypegphs s pove to e n equvlne elton. Keywos - Fuzzy hypegph o-wek somophsm wek somophsm equvlene elton I. Intouton Loft Zeh n hs lssl ppe n 9 ntoue the noton of fuzzy susets of set. He lso ntoue the onept of fuzzy eltons. Sne then the fuzzy set theoy eome we eseh e n vous splnes lke mene sol senes engneeng sttsts gph theoy mngement senes ompute senes tfl ntellgene ptten eognton expet systems eson mkng oots sgnl poessng n utomt theoy. Rosenfel ntoue n 9 the onept of fuzzy gphs n the gph theoet onepts lke pths yles n onneteness wee ntoue n fuzzy gphs. In [] P.Bhtthy ssote fuzzy gph wth fuzzy gph n the ntul wy s n utomophsm goup. The onept of wek somophsm n somophsm etween fuzzy gphs wee ntoue y K.R.Bhutn n []. Moeson n Pemhn N [] ntoue the onept of fuzzy hypegphs n sevel fuzzy nlogs of hypegph theoy. Opetons on fuzzy hypegphs wee ntoue y Bege []. In [] A.Ngoogn n J.Mlvzh susse the somophsm etween fuzzy gphs n some popetes of self omplementy n self wek omplementy fuzzygphs. In ths ppe we ntoue the somophsm of fuzzy hypegphs n some of the mpotnt popetes. Defnton[] Let X e fnte set n let supp x. lle the olleton of ege sets of H. Note II. Pelmnes e fnte fmly of non tvl fuzzy susets of X suh tht : X... s Then the p H s lle fuzzy hypegph n A fuzzy hype gph H wth unelyng set X s H : X whee : X susets : s fuzzy elton on the fuzzy susets x x... x x x... x suh tht Defnton enote s Gven fuzzy hypegph : X p x n sze of H s efne n enote s q x x... x x x... x X I n e fuzzy H wth the unelyng set X the oe of H s efne n X III. Isomophsm Bs Popetes Defnton A homomophsm of fuzzy hypegphs h : H H s mp h : X X whh stsfes x h x x X n P g e
2 Defnton A wek somophsm h : H x h x x xmple. Isomophsm On Fuzzy Hypegphs x x... x hx hx... hx x x... x X H s mp h : X X Let H : X n H : X X = { } n X = { } whee H efne y e the eges of ' Inene mtes e gven s follows: whh s jetve homomophsm tht stsfes e two fuzzy hypegphs wth unelyng sets e the eges of H n Defnton A o- wek somophsm h : tht stsfes xmple. X= { H H s mp h : X X x x... x hx hx... h x x... x X x Let H X n H X : } n X= { } : whh s jetve homomophsm e the fuzzy hypegphs wth unelyng sets P g e
3 Isomophsm On Fuzzy Hypegphs Inene mtes e gven s follows Defnton A somophsm h : H H s mp h : X X whh s jetve homomophsm tht stsfes x hx x X x x... x hx hx... hx x x... x X ' We enote the somophsm of the hypegphs H n H s H H' Remk. A wek somophsm of fuzzy hypegph peseves the weghts of the noes ut not neessly the weghts of the eges.. A o- wek somophsm peseves the weghts of the eges ut not neessly the weghts of the noes.. An somophsm peseves oth the weghts of the eges n the noes.. An enomophsm of fuzzy hypegph H s homomophsm of H to tself.. An utomophsm of fuzzy hypegph H s n somophsm of H to tself.. When the two fuzzy hypegphs H n H ' e sme the wek somophsm etween them eomes n somophsm n smlly the o-wek somophsm etween them lso eomes somophsm. In sp hype gphs when two hypegphs e somoph they e of sme oe. Also the sme s tue n the se of fuzzy hypegphs. We pove ths esult n the followng theoem. Theoem. Fo ny two somoph fuzzy hypegphs the oe n sze e sme. Poof If h : H H s n somophsm etween the fuzzy hype gphs H & H wth the unelyng sets X & X' espetvely then x hx x x x... x hx hx... h x x... x X x () p = oe H x hx oeh xx () q = Sze ( H ) = SzeH Coolly Convese of the ove theoem nee not e tue. We pove ths y n exmple. xmple. Conse the fuzzy gphs H n H wth unelyng sets X n X s P g e
4 Isomophsm On Fuzzy Hypegphs P g e X = { } X = { } espetvely. Inene mtes e gven s follows p = p = q = q = Hee oe n sze e sme ut H s not somoph to H Remk If the fuzzy hypegphs e wek somoph then the oes e sme. But the fuzzy hype gphs of sme oe nee not e wek somoph. We pove ths y the followng exmple. xmple. Conse the fuzzy hypegphs H n H wth unelyng sets X n X s X = { } n X = { } espetvely. Inene mtes e gven s follows Oe= Oe= Hee the oe s sme. But they e not wek somoph. Remk If the fuzzy hypegphs e o-wek somoph the szes e sme. But the fuzzy hype gphs of sme sze nee not e o-wek somoph. We pove ths y the followng exmple. xmple. Inene mtes e gven s follows
5 Isomophsm On Fuzzy Hypegphs Sze H = Sze H = Hee sze of H n H e sme. But they e not o-wek somoph. Defnton Let H : X x x... x fo x x x e fuzzy hypegph. The egee of vetex s efne s x... Theoem. If H n H e somoph fuzzy hypegphs then the egees of the noes e peseve. X X e n somophsm of fuzzy hypegphs H onto H. By the efnton of Poof: Let h : somophsm x x... x h x h x... h x x... x x x x x... x X x x h x hx... hx h x Coolly. Convese of the ove theoem nee not e tue. Conse the followng neny mtes. Hee egees of the noes e peseve ut eges e not peseve. Remk The egee of vetex s mesue only y ng the weghts of the eges nent wth tht vetex. But fuzzy hypegphs pesevng the egee of the vetes nee not e o-wek somoph. 8 P g e
6 xmple. Isomophsm On Fuzzy Hypegphs In the ove two hypegphs eh vetex s of egee no wek somoph hypegphs. Theoem. Isomophsm etween fuzzy hypegphs s n equvlene elton. Poof Let X H : X H : X :. But those two hypegphs e nethe o-wek H e fuzzy hypegphs wth unelyng sets X X n X espetvely. () Reflexve: Conse the entty mp h : Ths h s jetve mp stsfyng. x hx x X n x x... x h x h x... h X X hx x x X x x... x X x Hene h s n somophsm of the fuzzy hypegph to tself. Theefoe t stsfes eflexve elton. () Symmet: Let h : X X e n somophsm of H n H then h s jetve mp hx x x X... () Then h s jetve mp stsfyng x hx x X n x x... x hx hx... h x x... x X x Sne h s jetve y () h x x h x x x X x X n.. () h x h x... h x x x Hene we get - onto mp () Tnstve: H H' H' H x h : X X x x... x } X { whh s n somophsm fom H to H.. () 9 P g e
7 Isomophsm On Fuzzy Hypegphs Let h : X X n g : X X e n somophsm of fuzzy hypegphs H onto H n H onto H espetvely. Then g h s - onto mp fom. As h : s n somophsm x X x... x hx hx... hx x x x X X X whee (g h)(x) = g(h(x)) x X h( x ) = x x X x h x x... X x x x ' x x... x x x x As g X X g ( x ) = x x X x x... x X... : s n somophsm x gx x X x x... x gx gx... g..() ()..() x { x x x X... } Fom () n () n usng h( x ) = x x X x x g g h x Fom ( ) n () x x... x x = x X x X..() x x... x { x x... x} X = g x g x... g x x x... x X = g hx ghx... ghx x x... x X Theefoe g h s n somophsm etween H n H. Hene somophsm etween fuzzy hypegphs s n equvlene elton. Theoem. Wek somophsm etween fuzzy hypegphs stsfes the ptl oe elton. Poof: Let H : X H : X : X X X espetvely. () Reflexve: Conse the entty mp h : Ths h s jetve mp stsfyng x h x H e fuzzy hypegphs wth unelyng sets X X X suh tht h( x ) = x fo ll x X. x X x x... x hx hx... h x x... x X x Hene h s wek somophsm of the fuzzy hypegph to tself. Theefoe H s wek somoph to tself. () Ant symmet: Let h e wek somophsm etween H n H n g e wek somophsm etween H n H..e. h : X X s jetve mp h ( x ) = x stsfyng x hx x X n P g e
8 x x... x hx hx... h x x... x X g : x s jetve mp stsfyng g ( x ) = x x stsfyng x g(x) x x x... x g x g x... g Isomophsm On Fuzzy Hypegphs..(8) n x x... x X x..(9) The nequltes (8) n (9) hols goo on the fnte sets X n X only when H n H hve the sme nume of eges n the oesponng eges hve sme weght. Hene H n H e entl. () Tnstve: Let h : X X n g : X X e wek somophsms of the fuzzy hypegphs X onto X n X onto X espetvely. Then g h s - onto mp fom X to X whee (g h)(x)= g(h(x)) x X Gven h s wek somophsm h(x) = x x hx x X x x... x hx hx... h x x... x X x x X.() Smlly s g s wek somophsm fom X ' to X '' we hve g( x ) = x x x gx x X.() x x... x gx gx... g x x... x X x Fom the ove we hve x x gx x X x X g h x x X Fom ()()()&() x x... x x x... x x x gx gx gx... g h x g h x... g h... x X.() x x... x X x x... x X x Theefoe g h s wek somophsm etween H n H.e. wek somophsm stsfes tnstvty. Hene wek somophsm etween fuzzy hype gphs s ptl oe elton. IV. Conluson In ths ppe somophsm etween fuzzy hypegphs s pove to e n equvlene elton n wek somophsm s pove to e ptl oe elton. Smlly t s expete tht o-wek somophsm n lso e pove to e ptl oe elton. Refeenes []. Moeson J.N n P.S. N Fuzzy Gphs n Fuzzy Hypegphs Phys velg Heeleg 998 ; Seon ton. []. A.Ngoogn n J.Mlvzh. Isomophsm on Fuzzy Gphs. []. C.Bege. Gphes et Hypegphes. Duno Ps(Gphs n Hypegphs Noth-HollnAmstem 9 evse tnslton)9. []. Bhutn K.R. On Automophsm of Fuzzy gphs Ptten Reognton Lette 9: []. Bhtthy P Some Remks on fuzzy gphs ptten Reognton Lette : P g e
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