ONE MODULO THREE GEOMETRIC MEAN LABELING OF SOME FAMILIES OF GRAPHS
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1 ONE MODULO THREE GEOMETRIC MEAN LABELING OF SOME FAMILIES OF GRAPHS A.Maheswari 1, P.Padiaraj 2 1,2 Departet of Matheatics,Kaaraj College of Egieerig ad Techology, Virudhuagar (Idia) ABSTRACT A graph G is said to be oe odulo three geoetric ea graph if there is a ijective fuctio fro the vertex set of G to the set a 1 a 3q - 2 ad either a 0(od 3) or a 1(od 3) where q is the uber of edges of G ad iduces a bijectio * fro the edge set of G to a 1 a 3q - 2 ad a 1(od3) give by *(uv)= ( u) ( v) or ( u) ( v) ad the fuctio is called oe odulo three geoetric ea labelig of G. I this paper, we establish that soe failies of graphs are oe odulo three geoetric ea graphs Keywords: Mea labelig, oe odulo three ea labelig, geoetric ea labelig, oe odulo three geoetric ea labelig, oe odulo three geoetric ea graph. AMS Classificatio (2010): 05C78 I. INTRODUCTION All graphs cosidered here are siple, fiite, coected ad udirected. The vertex set ad the edge set of a graph are deoted by V(G) ad E(G) respectively. We follow the basic otatios ad teriologies of graph theory as i [1]. A graph labelig is a assiget of itegers to the vertices or edges or both, subject to certai coditios ad a detailed survey of graph labelig ca be foud i [2]. The cocept of ea labelig was itroduced by Soasudara ad Poraj [3]. A graph G ( p, q) with p vertices ad q edges is called a ea graph if there is a ijective fuctio f that aps V(G) to {0,1,2,3,..., q } such that for each edge uv, is labeled with eve ad f ( u) f ( v) 2 if f ( u) f ( v) is f ( u) f ( v) 1 if f ( u ) f ( v ) is odd. Jeyathi admaheswari itroduced the cocept of oe odulo 2 three ea labelig i [4]. A graph G is called oe odulo three ea graph if there is a ijective fuctio fro the vertex set of G to the set a 0 a 3q - 2 ad either a 0(od 3) or a 1(od 3) where q is the uber of edges of G ad iduces a bijectio * fro the edge set of G to a 1 a 3q - 2 ad either a 1(od3) give by *(uv)= u v) ad the fuctio is called P a g e
2 oe odulo three ea labelig of G. The cocept of geoetric ea labelig was due to Soasudra et al.[5], A graph G=(V,E) with p vertices ad q edges is said to be geoetric ea graph if it is possible to label the vertices xv with distict labels f(x) fro 1,2,,q+1 i such a way that whe each edge e=uv is labeled with f (e=uv)= f ( u ) f ( v ) or f ( u) f ( v) called geoetric ea labelig of G., the the resultig edge labels are all distict. I this case, the fuctio f is Motivated by the cocepts i [5] we defie a ew type of labelig called oe odulo three geoetric ea labelig as follows: A graph G is said to be oe odulo three geoetric ea graph if there is a ijective fuctio fro the vertex set of G to the set a 1 a 3q - 2 ad either a 0(od 3) or a 1(od 3) where q is the uber of edges of G ad iduces a bijectio * fro the edge set of G to a 1 a 3q - 2 ad either a 1(od3) give by *(uv)= ( u) ( v) or ( u) ( v) ad the fuctio is called oe odulo three geoetric ea labelig of G. I [6] we proved that P, k 1, ( > 2), cob, P k 2, s(p k 1 ), s(p k 2 ),c ( 5), L = P X P 2, are oe odulo three geoetric ea graphs ad also we proved that if G is a graph i which every edge lies o a triagle, the G is ot a oe odulo three geoetric ea graph. We begi with a brief suary of defiitios which are ecessary for the preset study. Defiitio1.1:Duplicatio of a edge e k =v k v k+1 of a graph G produces a ew graph G such that N v k = N v k v k+1 v k+1 ad N v k+1 = N v k+1 v k v k. Defiitio1.2: The tadpole graph is fored by joiig the ed poit of a path P to a cycle C. It is deoted by C. Defiitio1.3:A key graph is a graph obtaied fro K 2 by appedig oe vertex of C to oe ed poit ad cob graph P K 1 to the other ed of K It is deoted as KY(,). Theore 1.4[6]:The cob graph is aoe odulo three geoetric ea graph. Theore 1.5[6]:The cycle C is a oe odulo three geoetric ea graph for 5. Theore 1.6[6]:The path P is a oe odulo three geoetric ea graph II. ONE MODULO THREE GEOMETRIC MEAN LABELING OF FAMILIES OF GRAPHS Theore 2.1:Let G 1 p 1, q 1, G 2 p 2, q 2,, G p, q be oe odulo three geoetric ea cycles or path with q i (1 i ) ad u i, v i be the vertices of G i (1 i ) labeled with 3q i 2 ad 1. The the graph G obtaied by joiig u 1 with v 2 ad u 2 with v 3 ad u 3 with v 4 ad so o util we joi u 1 with v by a edge is a oe odulo three geoetric ea graph. The graph G has p 1 + p p vertices ad q i + ( 1) i=1 edges. 669 P a g e
3 Let φ i be oe odulo three geoetric ea labelig of G i (1 i ). Defie a vertex labelig φ: V G {1, 3,4,, 3( + ( 1)) 2} as Ifx V G 1, φ x = φ 1 (x) If x V G i, 2 i φ x = 3 (q k + 1) k=1 φ i x + 3 (q k + 1) k=1 for te lowest vertex label of G i for all oter reaiig vertices of G i The the iduced edge labels of G are 1, 4,, 3( + ( 1)) Hece φ is oe odulo three geoetric ea labelig of G. Theore 2.2: Let G 1 p 1, q 1, G 2 p 2, q 2,, G p, q be oe odulo three geoetric ea cycles with q i (1 i ) ad u i, v i be the vertices of G i (1 i ) labeled with 3q i 2 ad 1. The the graph G obtaied by idetifyig u 1 with v 2 ad u 2 with v 3 ad u 3 with v 4 ad so o util u 1 idetified with v is a oe odulo three geoetric ea graph. The graph G has p 1 + (p 2 1) + + (p 1) = (p 1 + p p 1 ) vertices ad Let φ i be oe odulo three geoetric ea labelig of G i (1 i ). Defie a vertex labelig φ: V G {1, 3,4,, 3( Ifx V G 1, φ x = φ 1 (x) If x V G i, 2 i φ x = The the iduced edge labels of G are 1, 4,, 3( of G. ) 2} as 3 k=1 q k ) 2 for te lowest vertex label of G i. φ i x + 3 k=1 q k for all oter reaiig vertices of G i edges. ) Hece φ is oe odulo three geoetric ea labelig Theore 2.3: Let G 1 p 1, q 1, G 2 p 2, q 2,, G p, q be oe odulo three geoetric ea cycles with q i (1 i ) ad e i, e i be the edges of G i (1 i ) labeled with 3q i 2 ad 1. The the graph G obtaied by idetifyig e 1 with e 2 ad e 2 with e 3 ad e 3 with e 4 ad so o util e 1 idetified with e is a oe odulo three geoetric ea graph. The graph G has p 1 + (p 2 2) + + (p 2) = (p 1 + p p 2 1 ) vertices ad edges. Let φ i be oe odulo three geoetric ea labelig of G i (1 i ). Defie a vertex labelig φ: V G {1, 3,4,, 3( ( 1)) 2} as If x V G 1, φ x = φ 1 (x) ad Ifx V G i, 2 i φ x = φ i x + 3( The the iduced edge labels of G are 1, 4,, 3( of G. k=1(q k 1 )) ( 1) ) Hece φ is oe odulo three geoetric ea labelig 670 P a g e
4 Theore 2.4:The Key graph KY(,) is a vertex equitable graph if 5. Let G 1 = P K 1 ad G 2 = C. Sice G 1 has 2-1 edges ad G 2 has edges, By Theore 1.4, 1.5 oe odulo three geoetric ea graphs. Let φ i be oe odulo three geoetric ea labelig of G i (1 i 2). Ifx V G 1, φ x = φ 1 (x) If x V G 2 φ x = 3 k=1 (q k + 1) for te lowest vertex label of G 2 φ i x + 3 k=1 (q k + 1) for all oter reaiig vertices of G 2 P K 1,C are The the iduced edge labels of G are 1, 4,, Hece φ is oe odulo three geoetric ea labelig of G. Theore 2.5:The tadpole graph C is a oe odulo three geoetric ea graph for 5. Let G 1 = C ad G 2 = P. Clearly G 1 has edges ad G 2 has -1 edges, By Theore 1.5, 1.6 C ad P are oe odulo three geoetric ea graphs. Let G = C, the G has + vertices ad + edges. By Theore 2.1 the tadpole graph C is a oe odulo three geoetric ea graph. Theore 2.6: The graph obtaied by duplicatio of a arbitrary edge i c ( 4) adits a oe odulo three geoetric ea labellig. u, u,... u, u. Let C be the cycle Let G be the graph obtaied by duplicatig a arbitrary edge of C. With out of loss of geerality let this edge be e = u 1 u 2 ad the ewly added edge be e = u 1 u 2. The V G = u 1, u 2,, u, u 1 u 2, E G = {E C, e 1, e, e } where e 2 = u 2 u 3 ad e 1 = u u 1. The V(G) = +2 ad E(G) = +3. Defie φ: V G {1, 3, 4,, 3q 2} by Case i: If is odd, 9 φ u 1 = 1, φ u 2 = 3, φ u 3 = 21,, φ u 1 = 16.φ u i = i 4, 4 i φ u 2 +1 = 3 + 6,φ u = 12φ u i = i 1, 1 i 2 3. Case ii : If is eve, P a g e
5 φ u 1 = 1, φ u 2 = 3, φ u 3 = 21, φ u 1 = 16, φ u i = i 4, 4 i φ u 2 +2 = 3 + 6,φ u = 12,φ u 1 = 24, φ u i = i 2, 2 i 2 3. The the iduced edge labels of G are 1, 4,,3 + 3 Hece φ is oe odulo three geoetric ea labelig of G. If = 4, we defie the labelig as φ u 1 = 1, φ u 2 = 3, φ u 3 = 13,φ u 4 = 18, φ u 1 = 9, φ u 2 = 19. If = 5, we defie the labelig as φ u 1 = 1, φ u 2 = 3, φ u 3 = 15, φ u 4 = 10, φ u 5 = 12, φ u 1 = 21, φ u 2 = 2 If = 6, we defie the labelig as φ u 1 = 1, φ u 2 = 3, φ u 3 = 21, φf u 4 = 16, φ u 5 = 12, φ u 6 = 10,φ u 1 = 24, φ u 2 = 25. If = 7, we defie the labelig as φ u 1 = 1, φ u 2 = 3, φ u 3 = 21, φ u 4 = 28, φ u 5 = 27, φ u 6 = 19,φ u 7 = 12, φ u 1 = 16. If = 8, we defie the labelig as φ u 1 = 1, φ u 2 = 3, φ u 3 = 21, φ u 4 = 25, φ u 5 = 31, φ u 6 = 30,φ u 7 = 22,φ u 8 = 12,φ u 1 = 16. Theore 2.7:The graph c k 1 is a oe odulo three geoetric ea graph for > 3.. Let u 1, u 2,, u be the vertices of C. Let v i be the pedat vertices attached at each u i for 1 i. The V G = u 1, u 2,, u, v 1, v 2,, v, E G = E C e i = u i v i 1 i The V(G) = 2 ad E(G) = 2. Defie φ: V G {1, 3, 4,, 3(2) 2} by Whe = 4, we defie the labelig as φ u 1 = 3, φ u 2 = 7, φ u 3 = 22, φ u 4 = 21, φ v 1 = 1, φ v 2 = 12, φ v 3 = 18, φ v 4 = 13. Whe = 5, we defie the labelig as φ u 1 = 3, φ u 2 = 6, φ u 3 = 28, φ u 4 = 27, φ u 5 = 13, φ v 1 = 1, φ v 2 = 15, φ v 3 = 21, φ v 4 = 19, φ v 5 = 18, Case i : If is odd, 7 φ u 1 = 3, φ u 2 = 13, φ u i = i 3, 3 i φ u 2 +1 = 6 3,φ u = 7, φ u 1 = 21, φ u i = i 2, 2 i 5 φ v 1 = 1, φ v 2 = 18, φ v i = i 3, 3 i φ v 2 +1 = 12, φ v 1 = 22, φ v 2 = 25, φ v i = i 3, 3 i 3 25 = , φ v = Case ii : If is eve, 6 φ u 1 = 3, φ u 2 = 7, φ u 3 = 24, φ u i = i 4, 4 i φ u 2 +2 = 6 3,φ u = 30, φ u i = i 1, 1 i P a g e
6 φ v 1 = 1, φ v 2 = 6, φ v 3 = 10, φ v 4 = 18, φ v i = i 5, 5 i 2 + 1, φ v 15 = = 27 = ,φ v = 12, φ v i = i 1, 1 i 4 The the iduced edge labels of G are 1, 4,, 2 Hece φ is oe odulo three geoetric ea labelig of G. REFERENCES [1]. F.Harary, Graph theory, Addiso Wesley, Massachusetts, (1972). [2]. A.Gallia,A dyaic survey of graph labelig, The Electroic Joural of Cobiatorics, 17 (2016). #DS6, [3]. S. Soasudara ad R. Poraj, Mea labeligs of graphs, Natioal acadey Sciece letters, 26 (2003), [4]. P.Jeyathi ad A.Maheswari, Oe Modulo Three Mea Labelig of Graphs,Aerica Joural of Applied Matheatics ad Statistics,Vol 2(2014),No.5, [5]. S. Soasudara, P.Vidhyarai ad R. Poraj, Geoetric Mea Labeligs of Graphs, Bulleti of Pure ad Applied Scieces, 30E (2011), [6]. P.Jeyathi ad A.Maheswari, P. Padiaraj Oe Modulo Three Geoetric Mea Labelig of Graphs (Preprit). 673 P a g e
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