INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY

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1 [Mjuh, : Jury, 0] ISSN: -96 Scefc Jourl Impc Fcr: 9 ISRA, Impc Fcr: IJESRT INTERNATIONAL JOURNAL OF ENINEERIN SCIENCES & RESEARCH TECHNOLOY HAMILTONIAN LACEABILITY IN MIDDLE RAPHS Mjuh*, MurlR, B Shmukh *Deprme of Mhemcs,opl college of Egeerg d Mgeme, Bglore, Id Deprme of Mhemcs, DrAmbedkr Isue of Techology, Bglore, Id Deprme of Mhemcs, PES College of Egeerg, Mdy,Id ABSTRACT Ths pper s med dscuss Hml lcebly he coex of he Mddle grph of grph We explore lcebly properes of he Mddle grph of he er grph, F grph, Wheel grph, Ph d Cycle KEYWORDS: Coeced grph, Mddle grph, er grph, F grph, Hml- Hml--lcebly umber, * -Coeced grph INTRODUCTION All grphs cosdered hs pper re fe, smple, coeced d udreced Le d be wo verces grph The dsce bewee u d v deoed by d u, v s he legh of shores u - v u ph s Hml- -lceble f here exss Hml ph bewee every pr verces u d v wh he propery d u, v, dm, where s posve eger If hs propery s cheved for les oe pr of verces u d v, he grph s ermed Hml- -lceble Vrous resuls o Hml lcebly properes grphs re vlble [], [], [8], [9], [0], [], [] d [] v I hs pper we gve some resuls reled he Lcebly properes of he Mddle grph of he er grph, F grph F, Wheel grph W, Ph grph P, d Cycle C, The verex se of d he edge se of re deoed respecvely by V d E respecvely Terms o defed here c be foud [] -lceble grph, Fgure : Hml Lceble grph Defo Le P be ph from he verces j grph d le P be ph from k The he ph P P s he ph obed by exedg he ph P from j k hrough he commo verex e, f d P: j k, he P P: j k Fgure below llusres Hml--lceble grph d Hml-*-lceble grph j j hp: // wwwjesrcom Ierol Jourl of Egeerg Sceces & Reserch Techology [0]

2 [Mjuh, : Jury, 0] ISSN: -96 Scefc Jourl Impc Fcr: 9 ISRA, Impc Fcr: Defo The er grph s wheel grph W, wh verex dded bewee ech pr of djce verces of he ouer cycle Fgure below shows he grph 6 Fgure : A Hml--lceble grph d Hml-*-lceble rph Defo Le d j be y wo dsc verces coeced grph Le be y mml se of edges o d P be ph such h s Hml Ph from d P E j The E ' hp: // wwwjesrcom E ' s clled he of d he edges E, -lcebly umber re clled he j lcebly edges wh respec, Defo A grph s -coeced f s Hml- - lceble for ll dm Defo The Mddle grph of deoed by M s defed s follows The verex se of M s V E Two verces he verex se of M re djce M cse oe of he followg holds: re E d re djce s V, s E d, x y re cde Fgure below llusres he Mddle grph of cubc grph Fgure : A Cubc rph d s Mddle rph - Defo 6 A F grph Km P where verces d Fgure : er rph F m, Ierol Jourl of Egeerg Sceces & Reserch Techology [0] P The F grph K m 6 s defed s he grph jo s he empy grph o s he ph grph o F, verces s llusred Fgure below Fgure : F rph F, RESULTS Theorem : The rph M, s -coeced wh Proof: Le V { } {,,,, } d E { e : } { e : } { e } where e he edge s, s he edge d m e e s he

3 [Mjuh, : Jury, 0] ISSN: -96 Scefc Jourl Impc Fcr: 9 ISRA, Impc Fcr: edge v v hp: // wwwjesrcom, d by he defo of Mddle grph M Fgure 6: Mddle grph of er rph Cse : For d d he ph I he grph,, e,, e e e e, ' ', e,, e e, e 8, 8, e 8 e,, e 6 e, 6 6, e 6 e,, e e,, e e,, e e,, e e, e e, e e, e e, e 6, e, 6 e e8 e,, e ph from Where, e e e e s Hml s he lcebly edge Hece s Hml-*- lceble Cse : For I he grph, P d d he ph :,, e e, e e,, e e, e e, e e, e e, e 6 e, e 6 e, e e, v ' e,, e e, e, e e,, e 8 V M V E { } { : }, e : } { e : } respecvely, Clerly e 6 e, e e, 8 8 e,, e , e e,, e,, e e, e, e, e s Hml ph from Where e s he lcebly edge Hece s Hml-*-lceble Cse : For I he grph Ierol Jourl of Egeerg Sceces & Reserch Techology [0] d,,, e e, e, e, e,, e e,, e 8 e, d he ph e,, e e e 8, 8 e, e 6 e, 6 6, e 6 e, e e,, e e, e e, e e, e e, e e,, e, e e,, e e, Where, e, e e, s Hml ph from b s he lcebly edge Hece Hml-*-lceble Cse v: For I he grph d, e e,,,, e, e e, e e, e, 6, e 6 6 e, 6 e, 8, e 8 8 e,, e, e e, e e, e e, e e, e e, e,, d he ph s e,, e, e, e e,, e e, e e, e e s Hml ph from Where e s he lcebly edge Hece s Hml-*-lceble Hece he proof

4 [Mjuh, : Jury, 0] ISSN: -96 Scefc Jourl Impc Fcr: 9 ISRA, Impc Fcr: Theorem The rph M, s V F,, e F, -coeced Proof: Cosder he F grph deoe he, verces b = { } {,,,,, } { b, b, b, b, b} d E F { e : } F, Where s he edge by he defo of mddle grph V M F V F E F =,,, { : } { b : } { : } Clerly e d I he rph, d, d he ph, e e, e 6,, b b, b, e e, e e, e e,, b, b b, b,, b b, s Hml ph from s Hml--lceble Hece he proof Theorem The rph M, -coeced Proof: Cosder Wheel grph w, W,, b Hece s deoe he verces { } {,,,, } { b, b, b, b} V =d E W { e : } { e } W, Where he edge, e s he edge, e s, by he defo of mddle grph V M W, V W, E W, = { : } { e : } Clerly d Fgure : Mddle rph of F rph M hp: // wwwjesrcom F, Cse : For I he rph, d, e d he ph, e e, e, e, e e e e, e u b, b,, b b,, b, b b,, e e, e 6 e,,, b b, b, e Hml ph from Hece s Hml--lceble Cse : For s Fgure 8: Mddle rph of Wheel rph M W, Cse : For I he rph, d, b d he ph {, e e,, e e, e e, e e, e, 6 e b Ierol Jourl of Egeerg Sceces & Reserch Techology [06]

5 [Mjuh, : Jury, 0] ISSN: -96 Scefc Jourl Impc Fcr: 9 ISRA, Impc Fcr: b,, b b, b, e, b, b b,, b b,, b b,, b b, s Hml ph from s Hml-*-lceble, e b Hece where edge b s he verex of of P Clerly M P d correspodg Cse : For I he rph, d, b d he ph d he ph, e, b e, b b hp: // wwwjesrcom, e b, e e, e,, b b, e e,, b b, e e, e, b, e e, b, e e,, b b,, e e, e s Hml ph from b b Hece s Hml-*-lceble Cse : For I he rph, d, d he ph, e e b,, b, b, e b,,, e, e e, e 6 e, b b,, b, b b, b, b, b, e e, e e, b e, b, v, b e b s Hml ph from b Hece s Hml-*-lceble Theorem : The rph M P, s Hml- -lceble for,, wh Proof: Cosder he grph c be obed by he Mddle grph M P Sce M P cos les oe cycle of legh, we c coclude h M P Le V P {,,,, } d V M P { : } { b : } Fgure 9: Mddle rph of Ph M P Cse : For I he grph, d, b, d he ph, b,,, b b b, b, b, Hml ph from he lcebly edge Hece lceble Cse : For b,, b, Ierol Jourl of Egeerg Sceces & Reserch Techology [0] b b Where s s s Hml-*- I he grph, d,, d he ph, b, b,, b b b b,, b u, v b, b,, s Hml ph from, b Where, s he lcebly edge Hece s Hml-*-lceble Cse : For I he grph, d,, d he ph, b b,, b b,, b b,, b,, b b, b b,

6 [Mjuh, : Jury, 0] ISSN: -96 Scefc Jourl Impc Fcr: 9 ISRA, Impc Fcr: s Hml ph from Where, Theorem : The rph M C, s he lcebly edge Hece s Hml- *-lceble Hece he proof Hml- Hml- wh -lceble for s -lceble for d, Proof: V C {,,, }, d V M C {,,, }, { b, b, b,, b } where, b correspodg he edges C s he verex of of Cse : For I he grph, b, b, b, b, d d he ph,, b,, b b b, b, b, b, b, b, b, b b, b, b,, b, b s Hml ph from Hece s Hml-*-lceble Cse : For, b, b, d, d he ph I he grph, b b,, b,, b b, b,, b b, b,, b b,, b b, b b, v, u b, b Where, from edge Hece b s Hml ph b s he lcebly s Hml-*-lceble Fgure 0: Mddle rph of hp: // wwwjesrcom M C Cse : For I he grph, d, b d he ph, b b,, b b, b,, b, b b, b,, b b, b,, b b, b,, b b, Hml ph from b Hml-*-lceble ACKNOWLEDEMENTS The frs uhor s hkful he Mgeme d Sff of he Deprme of Mhemcs, opl College of Egeerg d Mgeme, Bglore for her suppor d ecourgeme The uhors re lso hkful he Mgeme d R & D Cere, Deprme of Mhemcs, Dr Ambedkr Isue of Techology, Bglore d he Deprme of Mhemcs, PES College of Egeerrg, Mdy REFERENCES F Hrry, rph Theory, Nros Publshg 969 D Mchlk, O Mddle d Tol grphs wh Corseess, b Number equl, Lecure oe, b s s Mhemcs, Volume 08, rph heory, Sprger- Hece s Verlg, Berl,9-098 J Akym d Tksh Hmd d IYoshmur, O Ierol Jourl of Egeerg Sceces & Reserch Techology [08]

7 [Mjuh, : Jury, 0] ISSN: -96 Scefc Jourl Impc Fcr: 9 ISRA, Impc Fcr: Chrecerzo of mddle g rphs, True Mhemcs 9,pp -9 Thmmrju S N d Murl R, Lcebly O Clss of Regulr rphs, Iero Jourl of Compu ol Scece d Mhem cs,00::9-06 Verold Vv J d K Thlgvh, O Hrmoous Colourg of Le rph of Cerl rph of Phs, Appld MhemclmSc eces Vol,009, No, 0-6 S K Vdy d P L Vhol, Crdl Lbelg for Mddle rph of Some rphs,dscree Mhemcs E lxr DsMh C rsh A d R Murl, -Hml Lcebly Some Produc rphs, Iero Jourl of Compuol Scece d Mhemcs, ISSN 09-89, Volume No 0,pp rsh A d R Murl, Hml Lcebly Some Clsses of Sr rphs ISSN 9-96 IJESIT Volume, Issue, My 0 9 Mjuh d R Murl, Hml Lcebly Brck Produc C+,, r,advces Appled Mhemcl Bosce ces, ISSN 8-998, Volume, No0, pp- 0 Mjuh, R Murl d S N Thmmrju, Hml Lcebly Modfed Brck Produc of Odd Cycles, IJMS, Acceped Mjuh d R Murl, Hml*-lcebly Jump rphs of Dmeer wo, IOSR Jourl of Mhemcs,e-8-008,p- ISSN:9-66Volume0,Issue VerIIIMy Ju0, pp--6 Mjuh,RMurl d rsh A, Hml Lcebly Le rphs, Ierol Jourl of Compuer Applco IJCAVolume 98, Number, pp- Mjuh, R Murl d Rjedr S K, Hml Lcebly Tol rphs, Ierol Jourl of Mhem cs d Reserch, Volume, Issue Dec 0, pp-8 ISSN:0-6 hp: // wwwjesrcom Ierol Jourl of Egeerg Sceces & Reserch Techology [09]