1 Edge Magic Labeling for Special Class of Graphs

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1 S.Srram et. al. / Iteratoal Joural of Moder Sceces ad Egeerg Techology (IJMSET) ISSN ; Avalable at Volume, Issue 0, 05, pp Edge Magc Labelg for Specal Class of Graphs * S.Srram D.G.Vashav College, Arumbakkam, Assstat Professor, Patrca College of Arts,ad Scece, Adyar, Chea, Ida Emal:saksrram@gmal.com Dr.R.Govdaraja Assocate Professor & Head, P.G & U.G Departmet of Mathematcs, D.G. Vashav College, Arumbakkam, Chea, Ida Abstract V v, Let G= (V, be a smple graph where = { } ad E { vv, + } f : V {-,} ad f*: E { } such that for alluvî E, f *( uv) f ( u) ) =. Let = + = the the labelg s sad to be -Edge Magc Labelg. I ths paper we proved that Y-TreeY r + for r ³ 3, Coroa C (Whe s eve), P J, Graph Labelg Graphs. P J, S( P ), Globe Gl( ) eywords: 0-Edge Magc Labelg of Graphs, -Edge Magc Labelg of Graphs are -Edge Magc. INTRODUCTION: G pq, be a graph wth p vertces ad q edges. Let us cosder the graphs to be fte, Let ( ) udrected ad smple. Let the vertex set ad edge set of a graph G be deoted by V (G) ad E (G) respectvely. The cardalty of V(G) ad E(G) are respectvely called order ad sze of G. Labelg of graphs s a oe to oe mappg that carres a set of graph elemets to a set of umbers (usually tegers) called Labels.. Labelg of graphs has eormous applcato may practcal problems volved crcut desgg, commucato etwork, astroomy etc. []. The cocept of labelg s troduced by Rosa[5]. The cocept of 0-Edge magc labelg was troduced by J.Jayaprya ad.thrusagu []. The cocept of -EdgeMagc labelg of graphs was troduced by Neelam umar ad Seema Mehra [3] ad they have proved the exstece of -Edge labelg for certa graphs. The terms ad otato are used as F.Harary[4].. PRELIMINARIES Neelam umar ad Seema Mehra [3] defe the cocept of -Edge Magc Labelg of graph as follows Let G= (V, be a smple graph where V = { v, } ad E { vv, + } f : V {-,} ad f*: E { } such that for alluvî E, f *( uv) f ( u) ) =. Let = + = the the labelg s sad to be -Edge Magc labelg. Defto..: The vertex weght of a vertex v G uder a edge labelg s the sum of edge labels correspodg to all edges cdet wth v. Uder a total labelg, vertex-weght of v s defed as the sum of the label of v ad the edge labels correspodg to the etre edges cdet wth v. If all the vertces G have the same weght k, we call the labelg vertex-magc labelg or vertex- IJMSET-Advaced Scetfc Research Forum (ASRF), All Rghts Reserved IJMSET promotes research ature, Research ature erches the world s future 60

2 S.Srram et. al. / Iteratoal Joural of Moder Sceces ad Egeerg Techology (IJMSET) ISSN ; Avalable at Volume, Issue 0, 05, pp magc total labelg respectvely ad we call k a magc costat. If all the vertces G have dfferet weghts, the the labelg s called vertex-at magc edge labelg or vertex at magc total labelg. Defto..: The edge-weght of a edge e uder a vertex labelg s defed as the sum of the vertex labels, correspodg to every vertex cdet wth e, uder a total labelg, we also add the label of e. Usg edge-weght, we drve edge magc vertex or edge magc total labelg ad edge-at magc vertex or edge at magc total labelg. Defto..3: A (p,q) graph, G s sad to be (,0) edge-magc wth the commo edge cout k f there exst a bjecto f : V( G) {,... p} such that for all e= uvî E( G), ( ) ( ) It s sad to be (,0) edge at magc f for all e= uvî E( G), f ( u) + ) s dstct. f u + f v = k. Defto..4: A (p,q) graph G s sad to be (0,) vertex magc wth the commo vertex cout k f there exsts a bsecto f : E( G) {,... q} such that for each uî V ( G), f ( e) ( ) å = k, where eî E G ad e s cdet o u. It s sad to be (0,) vertex at magc f all the weghts are dstct. Defto..5: A (p,q) graph G s sad to be (,) edge-magc wth the commo edge cout k f there exst a bjecto f : V( G) È E( G) {,... p+ q} such that f ( u) ) f ( e) k e= uvî E( G). It s sad to be (,) edge at magc f f ( u) ) f ( e) e= uvî E( G). Defto..6: Let G = ( V, be a graph V = { v : } ad E { vv : + } f : V {-,} ad f * : E { 0}, such that for all e= uvî E( G), f ( uv) f ( u) ) the the labelg s sad to be 0-Edge Magc Labelg. + + = for all + + are dstct for all =. Let * = + = 0 I ths paper we proved that Y-TreeY r + for r ³ 3, Coroa Graph C (Whe s eve), P J P J S( P ), Globe ( ) Gl are -Edge Magc Labelg Graphs. 3. Ma Results Theorem.3.: The Graph Y-treeY r +, r ³ 3 s a -Edge Magc Labelg Proof: Let G = ( V, be a smple graph ad let G be Y-treeY r +, r ³ 3 Let V ( G) = {[ u, ], u '}, E( G) = é( uu ) : ( ' + - ùè uu ) Defe f : V {-,} by f ( u ') =- f ( u ) =- f º mod for = f º 0mod The the edge labelg are f * u, u = f u + f u =- + = + + f * u, u' = f u + f u' = - = { } ë û Therefore, the Y-Tree graph Y r +, r ³ 3 s a -Edge Magc Labelg. IJMSET-Advaced Scetfc Research Forum (ASRF), All Rghts Reserved IJMSET promotes research ature, Research ature erches the world s future 6

3 S.Srram et. al. / Iteratoal Joural of Moder Sceces ad Egeerg Techology (IJMSET) ISSN ; Avalable at Volume, Issue 0, 05, pp Illustrato u (-) u(-) u() u3(-) u4() u5(-) u6() u (-) Fg: 3..: Y 5 + Y-Tree graph u(-) u() u3(-) u4() u5(-) Fg:3..: Y 4 + Y-Tree graph Theorem.3.: Coroa Graph C a -Edge Magc Labelg whe s eve Proof : Let G ( V, Let V( C ) { u, u,... u } = be a smple graph ad let G be C, for s eve = ad let V ( C ) { u v w } =,, : ad ( ) {,,, : ; = + j j j j - } E C uu uu uv uw j Defe f : V {-,} by f ( u ) =- f º mod for = f º 0mod j ) = f j º mod for j = - f j º 0mod f ( w j ) = f j º mod for j = - f j º 0mod The the edge labelg are f * uu = f u + f u = + + ( j) ( ) ( j) f * uv = f u + f v = IJMSET-Advaced Scetfc Research Forum (ASRF), All Rghts Reserved IJMSET promotes research ature, Research ature erches the world s future 6

4 S.Srram et. al. / Iteratoal Joural of Moder Sceces ad Egeerg Techology (IJMSET) ISSN ; Avalable at Volume, Issue 0, 05, pp f * uw = f u + f w = ( j) ( ) ( j) Hece Coroa Graph C a -Edge Magc Labelg whe s eve Illustrato : w() v(-) w(-) v() w6(-) u(-) u() v3() v6(-) u6() u3(-) w3() u5(-) u4() w5() v5() w4(-) v4(-) Fg. 3.. C6 graph Theorem.3.3: The Graph P J s -Edge Magc Labelg Proof: Let G ( V, = s a smple graph. Let G = P J ad let ( ) { :, :, j : } V G = u v w j - ; { j j } {( j j+ ) } ( ) ( ) é ( ) { } E G = u w : j È w u : j - È uv : ù ë û Defe f : V {-,} By f ( u ) =- for ( j ) f w = for j - ) = for The the edge labelg are ( j j) ( j) ( j) f * u w = f u + f w = for j - ( j j+ ) ( j) ( j+ ) f * w u = f w + f u = for j - f * uv = f u + f v = for Hece the Graph P J s -Edge Magc Labelg IJMSET-Advaced Scetfc Research Forum (ASRF), All Rghts Reserved IJMSET promotes research ature, Research ature erches the world s future 63

5 S.Srram et. al. / Iteratoal Joural of Moder Sceces ad Egeerg Techology (IJMSET) ISSN ; Avalable at Volume, Issue 0, 05, pp Illustrato: u(-) w() u(-) w() u3(-) w3() u4(-) w4() u5(-) v() v() v3() v4() v5() Fg : P5J Graph u(-) w() u(-) w() u3(-) w3() u4(-) v() v() v3() v4() Fg : P4J Graph Theorem :3.4 : The graph P J s -Edge Magc Labelg Proof: Let G ( V, = s a smple graph. Let G = P J ad let V ( G) = { u, v, v, wj :, j -} ( ) = { j j, j j+, j j+,, :, -} E G u w wu wu uv uv j Defe f : V {-,} By f ( u ) =- for ( j ) f w = for j - ) = for ) = for The the edge labelg are ( j j) ( j) ( j) f * u w = f u + f w = for j - ( j j+ ) ( j) ( j+ ) f * w u = f w + f u = for j - f * uv = f u + f v = IJMSET-Advaced Scetfc Research Forum (ASRF), All Rghts Reserved IJMSET promotes research ature, Research ature erches the world s future 64

6 S.Srram et. al. / Iteratoal Joural of Moder Sceces ad Egeerg Techology (IJMSET) ISSN ; Avalable at Volume, Issue 0, 05, pp f * uv = f u + f v = Hece the graph P J s -Edge Magc Labelg Illustrato: u(-) w() u(-) w() u3(-) w3() u4(-) w4() u5(-) V() v() v() v() v3() v3() v4() v4() v5() v5() Fg : P5J Graph Theorem.3.5: The graph S( P ) s -Edge Magc Labelg Proof: Let G = ( V, s a smple graph. Let G S( P ) V( P ) = { u, v : }. Let x ( ) ( ) ad y j ( j - ) be the vertex whch dvdes the edge uu + ( ) V S( P ) = u v x y j - ( ) {,,, j :, } ( ) { } f : V S P -, Defe ( ) By f ( u ) =- for f ( x ) = for ) =- for ( j ) f y = for j - The the edge labelg are ( j j) ( j) ( j) f * u y = f u + f y = for j - ( j j+ ) ( j) ( j+ ) f * y, u = f y + f u = for j - f * u x = f u + f x = for f * xv = f x + f v = for S P s -Edge Magc Labelg. Hece ( ) = ad let be the vertex whch dvdes the edge uv -. The IJMSET-Advaced Scetfc Research Forum (ASRF), All Rghts Reserved IJMSET promotes research ature, Research ature erches the world s future 65

7 S.Srram et. al. / Iteratoal Joural of Moder Sceces ad Egeerg Techology (IJMSET) ISSN ; Avalable at Volume, Issue 0, 05, pp Illustrato: u(-) y() u(-) y() u3(-) y3() u4(-) y4() u5(-) x() x() x3() x4() x5() v(-) v(-) v3(-) v4(-) v5(-) Fg : S( P5 ) graph Theorem.3.6: Globe Gl( ) s -Edge Magc Labelg graph {,, : } Proof : Let G = ( V, be a smple graph ad let G be Gl( ). Let ( ) ( ) ad let E( G) = é{ ( uw) : } È {( vw) : } Defe ( ) f ( u ) =- ) =- ë ( ) { } f : V Gl -, by f ( w ) =, Hece the edge labelg are f * uw = f u + f w = f * vw = f v + f w = Therefore the Globe Gl( ) s a -Edge Magc Labelg graph ù û V G = u v w Illustrato: u(-) w() w () w3 () w4() v(-) Fg.3.6. : Gl( 4) graph IJMSET-Advaced Scetfc Research Forum (ASRF), All Rghts Reserved IJMSET promotes research ature, Research ature erches the world s future 66

8 S.Srram et. al. / Iteratoal Joural of Moder Sceces ad Egeerg Techology (IJMSET) ISSN ; Avalable at Volume, Issue 0, 05, pp u(-) w() w() w3() w4() w5() v(-) Fg.3.6.: Gl5 ( ) graph 4. CONCLUSION: I ths paper we have vestgated some graphs Y-TreeY r + for r ³ 3, Coroa GraphC (Whe s eve), P J are -Edge Magc Labelg Graphs P J S( P ), Globe ( ) Gl.ad proved that they 5. ACNOWLEGEMENTS: Authors are thakful to the revewers for ther valuable commets whch mprove the stadard of the paper. 6. REFERENCES [].Galla J.A, A Dyamc Survey of Graph Labelg, The Electroc Joural of Combatorcs 6(00)#DS6 [].J.Jayaprya ad.thrusagu, 0-Edge Magc Labelg for Some class of graphs, Ida Joural of Computer Sceces ad Egeerg 3(0),45- [3].Neelam umar ad Seema Mehra,-Edge Magc Labelg for Some class of graphs, Iteratoal Joural of Computer Egeerg & Sceces, Nove.03,pp [4].F.Harary, Graph Theory,Addso-Wesley Publshg Compay Ic, USA,969 [5].A.Rosa, o certa valuatos of the vertces of a graph, Theory of Graphs (Iterat,Symposum, Rome, July 966), Gordo ad Breach, N.Y ad Duod Pars,(967), AUTHOR S BRIEF BIOGRAPHY: Dr.R.Govdaraja: He s a Assocate Professor & Head, P.G & U.G Departmet of Mathematcs, D.G.Vashav College, Arumbakkam, Chea. Mr.S.Srram: He s a Assstat Professor Mathematcs, Patrca College of Arts ad Scece, Adyar, Chea-0 ad he s dog part tme Ph.D uder the able gudace of Dr.R.Govdaraja IJMSET-Advaced Scetfc Research Forum (ASRF), All Rghts Reserved IJMSET promotes research ature, Research ature erches the world s future 67