F Geometric Mean Graphs

Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 2 (December 2015), pp. 937-952 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) F Geometrc Mean Graphs A. Dura Baskar and S. Arockaraj Department of Mathematcs Mepco Schlenk Engneerng College Mepco Engneerng College (PO) Svakas 626005 Taml Nadu, Inda Emal: a.durabaskar@gmal.com; psarockaraj@gmal.com Receved: November 1, 2014; Accepted: June 26, 2015 Abstract: In a study of traffc, the labellng problems n graph theory can be used by consderng the crowd at every juncton as the weghts of a vertex and expected average traffc n each street as the weght of the correspondng edge. If we assume the expected traffc at each street as the arthmetc mean of the weght of the end vertces, that causes mean labellng of the graph. When we consder a geometrc mean nstead of arthmetc mean n a large populaton of a cty, the rate of growth of traffc n each street wll be more accurate. The geometrc mean labellng of graphs have been defned n whch the edge labels may be assgned by ether floorng functon or celng functon. In ths, the readers wll get some confuson n fndng the edge labels whch edge s assgned by floorng functon and whch edge s assgned by celng functon. To avod ths confuson, we establsh the F-geometrc mean labellng on graphs by consderng the edge labels obtaned only from the floorng functon. An F-Geometrc mean labellng of a graph G wth q edges, s an njectve functon from the vertex set of G to {1, 2, 3,..., q +1} such that the edge labels obtaned from the floorng functon of geometrc mean of the vertex labels of the end vertces of each edge, are all dstnct and the set of edge labels s {1, 2, 3,..., q}. A graph s sad to be an F Geometrc mean graph f t admts an F Geometrc mean labellng. In ths paper, we study the F-geometrc meanness of the graphs such as cycle, star graph, complete graph, comb, ladder, trangular ladder, mddle graph of path, the graphs obtaned from duplcatng arbtrary vertex by a vertex as well as arbtrary edge by an edge n the cycle and subdvson of comb and star graph. Keywords: Labellng, F-Geometrc mean labellng, F-Geometrc mean graph MSC 2010 No.: 05C78 937

938 A. Dura Baskar and S. Arockaraj I. I. Introducton Throughout ths paper, by a graph we mean a fnte, undrected and smple graph. Let G(V, E) be a graph wth p vertces and q edges. For notatons and termnology, the readers are referred to the book of Harary (1972). For a detaled survey on graph labellng we refer the reader to the book of Gallan (2014). A path on n vertces s denoted by P n and a cycle on n vertces s denoted by C n. K 1,n s s the graph obtaned from G by attachng a called a star graph and t s denoted by S n. new pendant vertex to each vertex of G. Let G 1 and G 2 be any two graphs wth p 1 and p 2 vertces respectvely. Then the Cartesan product G1 G2 has pp 1 2 vertces whch are u, v : u G1, v G2. The edges are obtaned as follows: u1, v 1 and u2, v 2 are adjacent n G1 G2 f ether u1 u2 and v 1 and v2 are adjacent n G 2 or u 1 and u2 are adjacent n G 1 and v1 v2. The mddle graph M(G) of a graph G s the graph whose vertex v : vv( G) e : e E( G) and the edge set s set s e e : e, e E( G) and e and e 1 2 1 2 1 2 are adjacent edges of G ve : vv( G), e E( G) and e s ncdent wth v. Let G be a graph and let v be a vertex of G. The duplcate graph DG, v of G s the graph whose vertex set s V ( G) v and edge set s E( G) v x : x s the vertex adjacent to v n G. Let G be a graph and let e uv be an edge of G. The duplcate graph DG, e u v of G s the graph whose vertex set s V ( G) u, v and edge set s E( G) u x, v y : x and y arethevertces adjacent wth u and v ng respectvely. The trangular ladder TL, n 2 s a graph obtaned by completng the ladder P2 Pn by addng the edges uv 1 for 1 n 1. For a graph G the graph S(G) s obtaned by subdvdng each edge of G by a vertex. An arbtrary subdvson of a graph G s a graph obtaned from G by a sequence of elementary subdvsons formng edges nto paths through new vertces of degree 2. The study of graceful graphs and graceful labellng methods was frst ntroduced by Rosa (1967). The concept of mean labellng was frst ntroduced and developed by Somasundaram and Ponraj (2003). Further, t was studed by Vasuk et al. (2009, 2010, 2011). Vadya and Lekha Bjukumar (2010) dscussed the mean labellng of some graph operatons. Mohanaselv and Hemalatha (2014) dscussed the super geometrc mean labellng of varous classes of some graphs. n

AAM: Intern. J., Vol. 10, Issue 2 (December 2015) 939 A functon f s called an F-Geometrc mean labellng of a graph G(V,E) f f : V ( G) 1,2,3,..., q +1 s njectve and the nduced functon f : E( G) 1,2,3,..., q defned as ( ) ( ), for all ( ), f uv f u f v uv E G s bjectve. A graph that admts an F Geometrc mean labellng s called an F Geometrc mean graph. Somasundaram et al. (2011) defned the geometrc mean labellng as follows: A graph G ( V, E) wth p vertces and q edges s sad to be a geometrc mean graph f t possble to label the vertces xv wth dstnct labels f( x ) from 1,2,..., q+1n such a way that when each edge e uv s labelled wth f uv f ( u) f ( v) or f ( u) f ( v), labels are dstnct. then the edge Somasundaram et al. (2012) have gven the geometrc mean labellng of the graph C5 C7 as n the Fgure 1. Fgure 1. A Geometrc mean labellng of C5 C7 From the above fgure, for the edge uv, they have used floorng functon f ( u) f ( v) and for the edge vw, they have used celng functon f ( v) f ( w) for fulfllng ther requrement. To avod the confuson of assgnng the edge labels n ther defnton, we just consder the floorng functon f ( u) f ( v) for our dscusson. Based on our defnton, the F-Geometrc mean labellng of the same graph C5 C7 s gven n Fgure 2. Fgure 2. An F-Geometrc mean labellng of C5 C7 and ts edge labellng

940 A. Dura Baskar and S. Arockaraj In ths paper, we study the F-Geometrc meanness of the graphs, namely, cycle C n for n 3, the star graph S n for n 3, the complete graph K n for n 3, the comb for any postve nteger n, the ladder P2 Pn for any postve nteger n, the mddle graph M( P n), the graphs obtaned by duplcatng an arbtrary vertex as well as arbtrary edge n the cycle C n, the trangular ladder TLn for n 2, the graph and the arbtrary subdvson of S 3. 2. Man Results To study the F-geometrc meanness, some of the standard graphs, and graphs obtaned from some graph operatons are taken for dscusson. Lemma 2.1. Let G be a graph. If V ( G) E( G) 1, then G does not admt an F-Geometrc mean labellng. If V ( G) E( G) 1, then the vertex labellng wll not be njectve and hence the result follows. Theorem 2.2. The unon of any two trees s not an F-Geometrc mean graph. Let G be the unon of two trees S and T. Then If V ( G) V ( S) V ( T) and E( G) E( S) E( T) V ( S) V ( T) 2 then by Lemma 2.1, the result follows. Corollary 2.3. Any forest s not an F-Geometrc mean graph. Theorem 2.4. Every cycle s an F-Geometrc mean graph. Let v 1, v 2,..., v n be the vertces of the cycle C n. We defne

AAM: Intern. J., Vol. 10, Issue 2 (December 2015) 941 as follows: f : V( C ) 1,2,3,..., n 1 n, 1 n+1 1, f( v ) 1, n+1 n. The nduced edges labellng s as follows: and f, 1 n+1 1, ( vv1 ) 1, n+1 n 1, f ( v v ) n+1. 1 n Hence, f s an F-Geometrc mean labellng of the cycle F-Geometrc mean graph. C n. Thus the cycle C n s an An F-Geometrc mean labellng of C 6 s shown n Fgure 3. Fgure 3. An F-Geometrc mean labellng of C 6 and ts edge labellng Theorem 2.5. The star graph S n s an F-Geometrc mean graph f and only f n 3. The number of vertces and edges of S n are n 1 and n respectvely. If f s an F-geometrc mean labellng of S, then t s a bjectve functon from VS to 1,2,3,..., n 1 n n. As we

942 A. Dura Baskar and S. Arockaraj have to label 1 for an edge, the vertex labels of ts par of adjacent vertces are ether 1 and 2 or 1 and 3. So, the central vertex of S n s labelled as ether 1 or 2 or 3. 1 cannot be a label for the central vertex n the case of n 2, snce two of the pendant vertces of S n are to be labelled as 2 and 3. When n 3, 2 cannot be the label for the central vertex, snce two of ts pendant vertces havng the labels 3 and 4. When n 4, the pendant vertces are labelled to be 4 and 5 f the label of central vertex s 3. The F-Geometrc mean labellng of S n, n 3 are gven n Fgure 4. Theorem 2.6. Fgure 4. The F-Geometrc mean labellng of S n, n 3 and ts edge labellng Every complete graph s not an F-Geometrc mean graph. To get the edge label q, q and q +1 should be the vertex labels for two of the vertces of K n, say x and y. Also to obtan the edge label 1, 1 s to be a vertex label of a vertex of K n, say v. Snce q nc2 n Kn q vy are one and the same. Hence and 1 n 1 2 the F-geometrc mean labellng of for n 4, the edge labels of the edges vx and K n s not an F-Geometrc mean graph. Whle n =2 and 3, K n are gven n Fgure 5. Fgure 5. The F-Geometrc mean labellng of K2 and 3 Theorem 2.7. Every comb graph s an F-Geometrc mean graph. K and ts edge labellng

AAM: Intern. J., Vol. 10, Issue 2 (December 2015) 943 Let be a comb graph for any postve nteger n havng 2n vertces and 2n 1 edges. Let u1, u2,..., u n be the vertces of the path P n and v be the pendant vertces attached at each u, for 1 n. Then, the edge set of G s uu 1 ; 1 n 1 u v n. ;1 We defne f : V( G) 1,2,3,...,2 n as follows: The nduced edge labellng s as follows: f ( u ) 2, for 1 n and f ( v ) 2 1, for 1 n. 1. f ( u u ) 2, for 1 n 1 and f ( u v ) 2 1, for 1 n Hence, f s an F-Geometrc mean labellng of the comb. Thus, the comb s an F-Geometrc mean graph for any postve nteger n. An F-Geometrc mean labellng of s shown n Fgure 6. Fgure 6. An F-Geometrc mean labellng of and ts edge labellng Theorem 2.8. Every ladder graph s an F-Geometrc mean graph. Let G P2 Pn be a ladder graph for any postve nteger n havng 2n vertces and 3n 2 edges. Let u1, u2,..., u n and v 1, v 2,..., v n be the vertces of G. Then the edge set of G s, ; 1 1 u v ;1 n uu 1 vv 1 n We defne f : V( G) 1,2,3,...,3 n-1 as follows:. f ( u ) 3 1, for 1 n and f ( v ) 3 2, for 1 n.

944 A. Dura Baskar and S. Arockaraj The nduced edge labellng s as follows: 1 1 f ( u u ) 3, for 1 n 1, f ( v v ) 3 1, for 1 n 1, and f ( u v ) 3 2, for 1 n. Hence, f s an F-Geometrc mean labellng of the ladder P2 Pn. Thus, the ladder P2 Pn s an F-Geometrc mean graph for any postve nteger n. An F-Geometrc mean labellng of P2 P6 s shown n Fgure 7. Fgure 7. An F-Geometrc mean labellng of P2 P6 Theorem 2.9. The mddle graph of a path s an F-Geometrc mean graph. Let V( P ) v, v,..., v and E( P ) e v v ; 1 n 1 n 1 2 n set of the path P n. Then, We defne f V M P and ts edge labellng n 1 be the vertex set and edge V M Pn v1, v2,..., vn, e1, e2,..., en 1 and, ; 1 n 1 ; 1 2 E M Pn ve ev 1 ee 1 n. n n : 1,2,3,...,3 3 as follows: f ( v ) 3 2, for 1 n 1, f ( v ) 3n 3 and f ( e ) 3 1, for 1 n 1. n The nduced edge labellng s as follows: f ( v e ) 3 2, for 1 n 1, f ( e v ) 3 1, for 1 n 1 1 and f ( e e ) 3, for 1 n 2. 1

AAM: Intern. J., Vol. 10, Issue 2 (December 2015) 945 Hence, f s an F-Geometrc mean labellng of the mddle graph MP n. Thus, the mddle graph MP n s an F-Geometrc mean graph. An F-Geometrc mean labellng of MP 6 s shown n Fgure 8. Theorem 2.10. Fgure 8. An F-Geometrc mean labellng of MP6 and ts edge labellng For any vertex v of the cycle graph, for n 3. C, the duplcate graph n D Cn, v s an F-Geometrc mean Let v 1, v 2,..., v n be the vertces of the cycle C n and let v v1 and ts duplcated vertex s Case (). n 5 We defne n f : V D C, v 1,2,3,..., n 3 as follows: v 1. 1 1 2 3 f ( v ) n 1, f ( v ) n 1, f ( v ) n 2, f ( v ) n 3, and The nduced edge labellng s as follows: 3, 4 n+3 2, f( v ) 2, n+3 3 n. 1 2 1 n 1 2 1 n 2 3 f v v n, f v v n 2, f v v n 1, f v v n 1, f v v n 2, 3, 4 n+3 2, f v3v4 n+3 and f ( vv1 ) 2, n+3 3 n 1. Hence, f s an F-Geometrc mean labellng of the graph DCn, v. Case (). n 3,4 The F-Geometrc mean labellng of DC3, v 1 and 4, 1 D C v are gven n Fgure 9.

946 A. Dura Baskar and S. Arockaraj Fgure 9. The F-Geometrc mean labellng of DC3, v 1 and 4, 1 D C v and ts edge labellng An F-geometrc mean labellng of the graph G obtaned by duplcatng the vertex v 1 of the cycle C 8 s shown n Fgure 10. Theorem 2.11. Fgure 10. An F-Geometrc mean labellng of 8, 1 D C v and ts edge labellng For any edge e of the cycle graph, for n 3. C, the duplcate graph n D Cn, e s an F-Geometrc mean Let v1, v2,..., v n be the vertces of the cycle C n and let e v12 v and ts duplcated edge s e v 12 v.

AAM: Intern. J., Vol. 10, Issue 2 (December 2015) 947 Case (). n 6 We defne n f : V D C, e 1,2,3,..., n+4 as follows: 1 1 2 2 f ( v ) n 1, f ( v ) n 1, f ( v ) n 2, f ( v ) n 3, f ( v ) n 4 and The nduced edge labellng s as follows: 3 3, 4 n+4 2, f( v ) 2, n+4 3 n. 1 2 1 n 1 n 1 2 f v2v3 n f v2v3 n f v3v4 n f v v n, f v v n 2, f v v n 1, f v v n 1, 3, 2, +4 3, 4 n+4 2, and f ( vv1 ) 2, n+4 3 n 1. Hence, f s an F-Geometrc mean labellng of the graph DCn, e. Case (). n 3,4,5 The F-Geometrc mean labellng of DC3, v1v 2, DC4, v1v 2 and 5, 1 2 n Fgure 11. D C v v are gven Fgure 11. An F-Geometrc mean labellng of DC3, v1v 2, 4, 1 2 DC5, v1v 2 and ts edge labellng D C v v and An F-geometrc mean labellng of the graph G obtaned by duplcatng an edge vv 12 of the cycle C 9 s shown n Fgure 12.

948 A. Dura Baskar and S. Arockaraj Theorem 2.12. The trangular ladder Fgure 12. An F-Geometrc mean labellng of 9, 1 2 D C v v and ts edge labellng TL n s an F-Geometrc mean graph, for n 2. Let u1, u2,..., un, v1, v2,..., vnbe the vertex set of TL n and let uu 1, vv 1, uv 1; 1 n 1 u v ; 1 n be the edge set of TL n. Then TL n have 2n vertces and 4n 3 edges. We defne f : V TL 1,2,3,...,4 n 2 n as follows: f ( u ) 4 1, for 1 n 1, f ( u ) 4n 2 and f ( v ) 4 3, for 1 n. n The nduced edge labellng s as follows: 1 f ( u u ) 4, for 1 n 1, f ( u v ) 4 3, for 1 n, 1 and 1 f ( u v ) 4 1, for 1 n 1 Hence, f s an F-Geometrc mean labellng of the F-Geometrc mean graph, for n 2. An F-Geometrc mean labellng of TL 8 s shown n Fgure 13. f ( v v ) 4 2, for 1 n 1. TL n. Thus the trangular ladder TL n s an

AAM: Intern. J., Vol. 10, Issue 2 (December 2015) 949 Theorem 2.13. Fgure 13. An F-Geometrc mean labellng of TL 8 and ts edge labellng s an F-Geometrc mean graph, for n 2. Let u, v ; 1 n and u u n u v n Let x be the vertex whch dvdes the edge 1 ; 1 1 ; 1. uv, for 1nand y be the vertex whch dvdes the edge uu 1, for 1 n 1. Then u, v, x, y j ; 1 n, 1 j n1 and u x v x n u y y u n, ; 1, 1 ; 1 1. We defne as follows: f ( u ) 4 1, for 1 n, f ( y ) 4 1, for 1 n 1, f ( x ) 4 2, for 1 n and f v The nduced edge labellng s as follows: 1, 1, 4 4, 2 n. f u y 4 1, for 1 n 1, f y u 4 1, for 1 n 1, 1 1, 1, f ux 4 2, for 1 n and f vx 4 4, 2 n. Hence, f s an F-Geometrc mean labellng of. Thus, the graph s an F-Geometrc mean graph, for n 2. An F-Geometrc mean labellng of s shown n Fgure 14. Fgure 14. An F-Geometrc mean labellng of and ts edge labellng

950 A. Dura Baskar and S. Arockaraj Theorem 2.14. Any arbtrary subdvson of S 3 s an F-Geometrc mean graph. Let v0, v1, v2, v 3 be the vertces of S 3 n whch v 0 s the central vertex and v1, v2 and v 3 are the pendant vertces of S 3. Let the edges v0v1, v0v2 and v0v 3 of S 3 be subdvded by p, p and p number of new vertces respectvely. Let G be a graph of arbtrary subdvson 1 2 3 of S 3. (1) (1) (1) (1) (2) (2) (2) (2) 0, 1, 2, 3,..., 1 1, 0, 1, 2, 3,..., 1 2 and p p 1 2 (3) (3) (3) () 2, 3,..., 1 p 3 be the vertces of G and v0 v 0, for 1 3. 3 Let v v v v v v v v v v v v v v v v (3) v0, v 1, ( ) ( ) ( ) Let e j v j1 v j, 1 j p 1 and 1 3 be the edges of G and G has vertces and p p p 3 edges wth p p p. 1 2 3 1 2 3 We defne f : V G 1,2,3,..., p p p 4 as follows: 1 2 3 p p p 4 1 2 3 () 1 0 1 2 1 2 f ( v ) p p 3, f v p p 4 2, for 1 p 1, p p 3 2, 1 p 1, p 2, p 2 p 1 and p p (2) 1 2 1 (3) f v p p p 1 2 3 f v 2 1 2 1 2 and 3, for 1 1. 1 The nduced edge labellng s as follows: (1) (1) 1 f ( v v ) p p 2 2, for 1 p, 1 2 1 p p 1 2, 1 p, p 1, p 1 p and p p, 2 1 2 1 2 (2) (2) 1 2 1 1 f v v (3) (3) (1) 1 0 1 f ( v v ) p p 3, for 1 p, f ( v v ) p p 2, (1) 0 2 1 2 3 1 2 f ( v v ) p p 1 and 1 2 (1) 0 3 f ( v v ) p p 3. 1 2 Hence, f s an F-Geometrc mean labellng of G. Thus, the arbtrary subdvson of S 3 s an F-Geometrc mean graph. An F-Geometrc mean labellng of G wth 15. p 6, p 9 and p 10 s as shown n Fgure 1 2 3

AAM: Intern. J., Vol. 10, Issue 2 (December 2015) 951 Fgure 15. An F-Geometrc mean labellng of arbtrary subdvson of S 3 and ts edge labellng 3. Concluson In ths paper, we analysed the F-Geometrc meanness of some standard graphs. We propose the followng open problems to the readers for further research work. Open Problem 1. Fnd a sub graph of a graph n whch the graph s not an F-Geometrc mean graph. Open Problem 2. Fnd a necessary condton for a graph to be an F-Geometrc mean graph. By Theorem 2.6, we observe that G + e s not necessarly an F-geometrc mean graph when G s an F-geometrc mean graph and e s an addtonal edge. Also from Theorem 2.2, G - e s not necessarly an F-geometrc mean graph when e s a cut edge and G s an F-geometrc mean graph. So t s possble to dscuss the remanng case.

952 A. Dura Baskar and S. Arockaraj Open Problem 3. For a non-cut edge e, characterze the F-geometrc mean graph G n whch G - e s also an F-geometrc mean graph. Acknowledgement The authors would lke to thank the edtor and three anonymous revewers for helpful suggestons whch mproved the presentaton of the paper. REFERENCES Gallan, J.A. (2014). A dynamc survey of graph labellng, The Electronc Journal of Combnatorcs, 17. Harary, F. (1972). Graph theory, Addson Wesely, Readng Mass. Mohanaselv, V. and Hemalatha, V. (2014). On super geometrc mean labellng of graphs, Asa Pasfc Journal of Research, 1(14): 79-87. Nagarajan, A. and Vasuk, R. (2011). On the meanness of arbtrary path super subdvson of paths, Australas. J. Combn., 51: 41 48. Rosa, A. (1967). On certan valuaton of the vertces of graph, Internatonal Symposum, Rome, Gordon and Breach, N.Y. and Dunod Pars: 349-355. Selvam, A. and Vasuk, R. (2009). Some results on mean graphs, Ultra Scentst of Physcal Scences, 21(1): 273 284. Somasundaram, S. and Ponraj, R. (2003). Mean labellng of graphs, Natonal Academy Scence Letter, 26: 210-213. Somasundaram, S. and Ponraj, R. (2003). Some results on mean graphs, Pure and Appled Mathematka Scences, 58: 29-35. Somasundaram, S., Vdhyaran, P. and Ponraj, R. (2011). Geometrc mean labellng of graphs, Bullentn of pure and appled scences, 30E: 153-160. Somasundaram, S., Vdhyaran, P. and Sandhya, S.S. (2012). Some results on Geometrc mean graphs, Internatonal Mathematcal Form, 7: 1381-1391. Vadya, S.K. and Lekha Bjukumar (2010). Mean labellng n the context of some graph operatons, Internatonal Journal of Algorthms, Computng and Mathematcs,3(1):1-8. Vasuk, R. and Nagarajan, A. (2010). Further results on mean graphs, Scenta Magna, 6(3):1-14.