V. Hemalatha, V. Mohana Selvi,

Transcription

1 Iteratoal Joural of Scetfc & Egeerg Research, Volue 6, Issue, Noveber-0 ISSN - SUPER GEOMETRIC MEAN LABELING OF SOME CYCLE RELATED GRAPHS V Healatha, V Mohaa Selv, ABSTRACT-Let G be a graph wth p vertces ad q edges Let f: VG ( ) {,,,, p+ q} be a jectve fucto For a vertex labelg f, the duced edge labelg f ( e = uv) s defed by f() e = f() u f() v ( or) f() u f() v The f s called a Super Geoetrc ea labelg f ( ) { } f VG ( ) f( e) / e EG ( ) = {,,,, p+ q} A graph whch adts Super Geoetrc ea labelg s called Super Geoetrc ea graph I ths paper, we vestgate Super geoetrc ea labelg of soe cycle related graphs Keywords: Graph, Super Geoetrc ea labelg, Super Geoetrc ea graph, Dubell Graph, Kayak Paddle (,,t), Polygoal sake INTRODUCTION Defto We beg wth sple, fte, coected ad udrected graph G (V, E) wth p vertces ad q edges For a detaled survey of graph labelg we refer to Galla [] Ters are ot defed here are used the sese of Harary [] SSoasudra ad R Poraj troduced ea labelg of graphs [], [6] RPoraj ad D Raya troduced Super ea labelg of graphs [] S Soasudra, P Vdhyara ad R Poraj troduced Geoetrc ea labelg of graphs [6] I ths paper, we vestgate Super Geoetrc ea labelg of soe graphs We ow gve the followg deftos whch are useful for the preset vestgato The graph C : s blocks of C coected wth a chord MAIN RESULTS Theore The Flag Fl graph s a super geoetrc ea graph Defto Let fvg : ( ) {,,,, p+ q} be a jectve fucto For a vertex labelg f, the duced edge labelg f ( e = uv) s defed Let { v ad 0 v : } be the vertces ad by f() e = f() u f() v ( or) f() u f() v The f s called a { e : + } be the edges of Fl Super Geoetrc ea labelg f f VG ( ) f( e) / e EG ( ) = {,,,, p+ q} A graph whch Here v 0 s a root vertex ( ) { } adts Super Geoetrc ea labelg s called Super Geoetrc ea graph Defto The Flag Fl s obtaed by jog oe vertex of C to a extra vertex s called the root Defto The Dubell graph D s obtaed by jog two dsjot cycles wth a chord Defto A Polygoal cha G, s a coected graph all of whose blocks are polygos C 6 Defto Defe f v0v+ f sodd f( e + ) = f v0v+ f s eve Defe a fucto f : V( ) {,,, p q} F + by l Kayak Paddle (,, t) s the graph obtaed by the jog C ad C by a path of legth t VHealatha, Departet of Matheatcs,Vvekaadha college of Techology for woe, Truchegodu-60 E-al: hearaguaths@galco VMohaa Selv, PG ad Research Departet of Matheatcs,Nehru eoral college, Puthaapatty- 600 E-al: vohaaselv@galco 0

2 Iteratoal Joural of Scetfc & Egeerg Research, Volue 6, Issue, Noveber-0 ISSN - ( + ) 0 = = + +, f s odd, f sodd f( v ) = &, f s eve f ( v &, f s eve + + ( + ), f s odd ( + ), f s odd + + &, &, f seve f s eve The the duced edge labels are ( + ), f s eve + +, f s odd &, f s odd f ( u &, f s eve + f( e) +, f s eve = + 6 +, f s odd + &, f s odd + &, f seve + + The the duced edge labels are Thus both vertces ad edges together get dstct labels fro, f s eve {,,, p+ q} &, f sodd Hece the Flag Fl graph s a Super Geoetrc ea graph + f ( e +, f s eve Theore + &, f sodd The graph D, s a Super Geoetrc ea graph for ay, f() e = + Let the vertces of D, be { v : } be { e : } as represeted Fg Defe f v+ u f sodd f() e = f vu f seve ad the edges of D, f ( + ), f s eve &, f sodd + e = + 6, f s eve + &, f sodd + ( ' ) Defe a fucto f : V( D ) {,,, p q} + by Thus both vertces ad edges together get dstct labels fro,,, p+ q { } Hece the graph D, s a Super Geoetrc ea graph for ay, 0 Note: If =, the graph D, s called the Dubell graph D (Fg )

3 Iteratoal Joural of Scetfc & Egeerg Research, Volue 6, Issue, Noveber-0 ISSN - Theore, f s eve The Kayak Paddle KP(,,t) s a Super Geoetrc ea graph &, f sodd for, ad t + f ( e ), = + f s eve + Let { v : }, { u : } ad { w : t} be the &, f sodd vertces of C, C ad P t respectvely Let { e : + t } ad { e : + + t } are the edges of the gve graph as represeted Fg ( + t) + ( ), f s eve v = + w f sodd & Defe ad, f sodd wt = u v = w f s eve ( ' + f e ( + t) +, f s eve Defe a fucto f : V ( KP(,, t) ) {,,, p + q } by + &, f sodd ( + t) = +, f sodd Thus both vertces ad edges together get dstct labels fro f ( v &, f s eve {,,, p+ q} Hece Kayak paddle KP(,,t ) s a Super Geoetrc ea + ( + ), f s odd graph for, ad t + &, f seve Theore f(w ) = ( + ) t ( + t ) + ( + t) +, f s odd f ( u &, f s eve + ( + t) + +, f s odd + &, f seve The the duced edge labels are The graph Polygoal sake G, s a Super Geoetrc ea graph Let { vj : ad j } { ej : ad j } be the vertces ad be the edges of the polygoal sake where ad Defe v = v f sodd + + ( + ) v = v f s eve ( + ) Defe a fucto f : V( ) {,,, p q } G, + by 0

4 Iteratoal Joural of Scetfc & Egeerg Research, Volue 6, Issue, Noveber-0 ISSN - f ( vj ) ( )( ) +,, j = + ( )( ) + j,, j, f s odd & j, f seve = ( )( ) + ( j+ ),, + j, f sodd + & j, f s eve The the duced edge labels are Defe a fucto f : V( : ) {,,, p q } C + by f + (+ )( ), j = + ( + )( ) + j, j, f s odd v = & j, f s eve + ( + )( ) + ( j + ), j, f s odd + & j, f seve ( j ) The the duced edge labels are ( )( ) + j, ( + )( ) + j, j, f s eve j, f seve & j, f sodd & j, f sodd (+ )( ) + j+, f ( ej f ( ej ( )( ) + j+, + j, f s eve + j, f s eve + & j, f sodd + & j, f sodd (+ )( ) +, j = ' ( )( ) +, j = f( e ) = (+ ) Thus both vertces ad edges together get dstct labels fro {,,, p+ q} Thus both vertces ad edges together get dstct labels fro Hece the graph Polygoal sake G, s a Super Geoetrc {,,, p+ q} ea graph Hece the graph C : where ad s a Super Geoetrc ea graph Theore REFERENCES The graph C : where ad s a Super Geoetrc ea graph [] JA Galla, A Dyac Survey of Graph Labelg, The Electroc Joural of Cobatorcs DS, 0 [] FHarary, Graph Theory, Narosa Publshg House Readg, New Delh Let { vj : ad j } be the vertces ad [] VMohaaselv ad VHealatha, O Super Geoetrc Mea Labelg of Soe Graphs, Asa Pacfc Joural ' { ej : ad j } ad { e : } be the edges of of Research, Vol: I Issue XIV, Jue 0, - the graph C : where ad [] R Poraj ad D Raya, Super ea labelg of graphs, Preprt [] SSoasudara ad R Poraj, Mea labelg of graphs, Natoal acadey scece letters, 6 (00), f v v ( + ) f s odd + 0- ' Defe f( e ) = [6] SSoasudara ad R Poraj, Soe Results o Mea graphs, Pure ad Appled Matheatcs, f v v ( + ) f s eve + (00), - [] S Soasudara, PVdhyara ad R Poraj, Geoetrc Mea Labelg of Graphs, Bullet of Pure ad Appled Sceces, 0E (0), -60 0

5 Iteratoal Joural of Scetfc & Egeerg Research, Volue 6, Issue, Noveber-0 6 ISSN v e + e v e e v v o Fg : Fl wth ordary labelg Fg : Super Geoetrc ea labelg of Fl 6 ad Fl v u v e e e v e v u e e u Fg : D, wth ordary labelg u e u Fg : Super Geoetrc ea labelg of D,0 0

6 Iteratoal Joural of Scetfc & Egeerg Research, Volue 6, Issue, Noveber-0 ISSN Fg : Super Geoetrc ea labelg of D v v e u e e v e + e +t- e u v e w w w t e 6 u Fg 6 : KP(,,t) wth ordary labelg Fg : Super Geoetrc ea labelg of KP (, 6, ) 0

7 Iteratoal Joural of Scetfc & Egeerg Research, Volue 6, Issue, Noveber-0 ISSN - v v v v e e e v e v e v v e - e e v v Fg : G, wth ordary labelg Fg : Super Geoetrc ea labelg of G 6, v v e e v v e e e - v v v e e e e v v v Fg 0: C : wth ordary labelg 0

8 Iteratoal Joural of Scetfc & Egeerg Research, Volue 6, Issue, Noveber-0 ISSN Fg : Super Geoetrc ea labelg of C : 0