Radio Geometric Mean Number of Splitting Of Star and Bistar

Transcription

1 Rado Geometrc Mean Number of Splttng Of Star and Bstar V. Hemalatha 1 Dr. V. Mohanasel 2 and Dr. K. Amuthaall 3 1 Department of Mathematcs Vekanandha College of Technology for Women Truchengode Namakkal. e-mal:hemaragumaths@gmal.com 2 PG and Research Department of Mathematcs Nehru Memoral College Puthanampatt Truchrappall. e-mal:mohanasel@gmal.com 3 Department of Mathematcs Got. Arts and Scence College Vepanthatta Perambalur. e-mal:thrcka@gmal.com Abstract: A rado Geometrc Mean Labelng of a connected graph G s a one to one map f from the ertex set V(G) to the set of natural numbers N such that for two dstnct ertces u and of G d u f ( u) f ( ) 1 dam( G). The rado geometrc mean number of f rgmn( f ) s the maxmum number assgned to any ertex of G. The rado geometrc mean number of G r ( G ) s the mnmum alue of r ( ) gmn f taken oer all rado geometrc mean labelng f of G. In ths paper we determne the rado geometrc mean number of splttng graph of star and bstar. Keywords: Rado Geometrc Mean labelng Star Bstar Dameter. gmn 1. INTRODUCTION We consder fnte smple undrected graphs only. Let V(G) and E(G) respectely denote ertex set and edge set of G. Chartand et al.[1] defned the concept rado labelng of G n Rado labelng of graphs s appled n channel assgnment problem [1]. Rado number of seeral graphs determned [2759]. In ths sequence Ponra et al.[8] ntroduced the rado mean labelng n G. Here we ntroduce a new type of labelng a rado geometrc mean labelng s a one to one mappng f from V(G) to N satsfyng the condton d( u ) f ( u) f ( ) 1 dam( G) for eery u V( G). The span of a labelng f s the maxmum nteger that f maps to a ertex of graph G. The rado geometrc mean number of G r gmn (G) s the lowest span taken oer all rado geometrc mean labelng of the graph G. In ths paper we determne the rado geometrc mean number of some star lke graphs. Let x be any real number. Then x stands for smallest nteger greater than or equal to x. Terms and defntons not defned here are followed from Harary [12] and Gallan [13]. The channel assgnment to rado transmtters s one of the man obectes n setup of wreless communcaton system. A proper channel assgnment to rado transmtters whch satsfes nterference constrants wth maxmum use of spectrum s a need of wreless communcaton system. The nterference constrants between a par of transmtters s closely related wth separaton of channels and dstance between transmtters. In a network f two transmtters are closer then hgher the nterference between them and large separaton. Defnton 1.1 A Star s the complete bpartte graph K 1 n. Defnton 1.2 The graph Bstar B nn obtaned by onng the center ertces of two copes of K 1n wth an edge. Defnton 1.3 [14] For a graph G the splt graph s obtaned by addng to each ertex a new ertex ' such that ' s adacent to eery ertex that s adacent to n G. The resultant graph s denoted as Spl(G). 2. MAIN RESULTS Theorem 2.1 Rado Geometrc Mean number of Splttng of star rgmn Spl K1 n 2n 1. Proof: Let G be a Spl K 1n wth 2(n+1) ertces and 3n edges. The dameter of Spl K1 n n 1 3. Let n be the pendant ertces and be the apex ertex of K1 n n1 and 1696

2 u u u u n... n to u be added ertces correspondng to obtan 1n We defne the labelng f as follows f u 2( n 1) f ;1 n f 2n 1 f u n ;1 n Spl K. Now we check the rado geometrc mean condton for any two ertces t should satsfy d( u ) f ( u) f ( ) 1 dam( G) Case (): Check the par u d( u ) f ( u) f ( ) 1 2( n 1).(1) 4 Case (): Check the par u d( u ) f ( u) f ( ) 2 2( n 1).(2n 1) 8 Case (): Check the par uu d( u u) f ( u) f ( u) 3 2( n 1).( n 1) 8 Case (): Verfy the par u Subcase (): If d( u ) f ( u ) f ( ) 2 (1)( n 1) 4 Subcase (): If d( u ) f ( u ) f ( ) 2 (2)( n 1) 5 Case (): Verfy the par u u d( u u ) f ( u ) f ( u ) 2 ( n 1)( n 2) 6 Case (): Verfy the par d( ) f ( ) f ( ) 2 (1)(2) 4 Case (): Check the par d( ) f ( ) f ( ) 1 (2n 1).(1) 4 Case (): Check the par u d( u) f ( ) f ( u) 1 (2n 1).( n 1) 5 Hence gmn n Theorem 2.2 r Spl K1 2 n 1 n 1. Rado Geometrc Mean number of Splttng of bstar rgmn Spl Bn n 4n 1. Proof: Consder B nn wth the ertex set u u :1 n where u are the pendant ertces. In order to obtan nn u' ' u' ' ertces correspondng to u u where1 n. V G 4( n 1) and E G 3(2n 1). Spl B add The dameter of the splttng of bstar s 3. We defne the labelng f as follows Assgn the labels of the ertces u u be f ( u) 4( n 1) f ( ) 4n 2 f ( u ) 2 1 ; 1 n f ( ) 2 ; 1 n and the labels of the ertces u' ' u ' ' f ( u ') 4n 3 f ( ') 4n 1 f ( u ') 2n 2 ; 1 n be f ( ') 2n 2 1 ; 1 n Now we check the rado geometrc mean condton for any two ertces t should satsfy d( u ) f ( u) f ( ) 1 dam( G) Case (1): Check the par uu ' d( u u ') f ( u) f ( u ') 2 4( n 1).(4n 3) 10 Case (2): Check the par uu d( u u) f ( u) f ( u) 1 4( n 1).(1) 4 Case (3): Check the par uu ' d( u u ') f ( u) f ( u ') 1 4( n 1).(2n 2) 7 Case (4): Verfy the par u u 1697

3 d( u u ) f ( u ) f ( u ) 2 (1)(3) 4 Case (5): Verfy the par u' u ' d( u ' u ') f ( u ') f ( u ') 2 (2n 2)(2n 4) 7 Case (6): Verfy the par u u ' Subcase (): If d( u u ') f ( u ) f ( u ') 2 (1)(2 n 2) 4 Subcase (): If d( u u ') f ( u ) f ( u ') 2 (2)(2n 2) 5 Case (7): Check the par u' u d( u ' u) f ( u ') f ( u) 2 (4n 3).(1) 5 Case (8): Check the par u' u ' d( u ' u ') f ( u ') f ( u ') 1 (4n 3).(2n 2) 7A Case (9): Check the par ' d( ') f ( ) f ( ') 2 (4n 1).(4n 2) 8 Case (10): Check the par d( ) f ( ) f ( ) 1 (4n 2).(2) 5 Case (11): Check the par ' d( ') f ( ) f ( ') 1 (4n 1).(4n 2) 7 Case (12): Verfy the par d( ) f ( ) f ( ) 2 (2)(4) 5 Case (13): Verfy the par ' ' d( ' ') f ( ') f ( ') 2 (2n 1)(2n 3) 8 Case (14): Verfy the par ' Subcase (): If d( ') f ( ) f ( ') 2 (2)(2n 1) 5 Subcase (): If d( ') f ( ) f ( ') 2 (2)(2n 3) 6 Case (15): Check the par ' d( ' ) f ( ') f ( ) 2 (4n 1).(2) 6 Case (16): Check the par ' ' d( ' ') f ( ') f ( ') 1 (4n 1).(2n 1) 5 Case (17): Check the par u d( u ) f ( u) f ( ) 1 4( n 1).(4n 2) 8 Case (18): Check the par ' u d( u ') f ( u) f ( ') 1 4( n 1).(2n 1) 6 Case (19): Check the par u d( u ) f ( u) f ( ) 2 4( n 1).(2) 6 Case (20): Verfy the par u ' d( u ') f ( u) f ( ') 2 4( n 1)(4n 1) 9 Case (21): Verfy the par u' d( u ' u ') f ( u ') f ( u ') 2 (2n 2)(2n 4) 7 Case (22): Verfy the par u' ' d( u ' ') f ( u ') f ( ') 2 4( n 1).(2) 6 Case (23): Check the par u ' d( u ' u) f ( u ') f ( u) 2 (4n 3).(2) 6 Case (24): Check the par u' ' 1698

4 d( u ' ') f ( u ') f ( ') 3 (4n 3).(4n 1) 7 Case (25): Verfy the par u' ' Subcase (): If d( u ' ') f ( u ') f ( ') 3 (2n 2)(2n 1) 7 Subcase (): If d( u ' ') f ( u ') f ( ') 3 (2n 2)(2n 3) 10 Case (26): Verfy the par u ' d( u ' ) f ( u ') f ( ) 2 (2n 2)(4n 2) 7 Case (27): Verfy the par u' Subcase (): If d( u ' ) f ( u ') f ( ) 3 (2n 2)(2) 5 Subcase (): If d( u ' ) f ( u ') f ( ) 3 (2n 2)(4) 8 Case (28): Verfy the par u ' ' d( u ' ') f ( u ') f ( ') 2 (2n 2)(4n 1) 7 Case (29): Verfy the par u ' Subcase (): If d( u ') f ( u ) f ( ') 3 (1)(2 n 1) 5 Subcase (): If d( u ') f ( u ) f ( ') 3 (1)(2 n 3) 6 Case (30): Verfy the par u d( u ) f ( u) f ( ) 2 (1)(4 n 2) 5 Case (31): Verfy the par u ' d( u ') f ( u) f ( ') 2 (1)(4 n 1) 5 Case (32): Verfy the par u Subcase (): If d( u ) f ( u ) f ( ) 3 (1)(2) 5 Subcase (): If d( u ) f ( u ) f ( ) 3 (3)(2) 6 Hence eery par of ertces satsfes the rado geometrc mean condton. Thus rgmn Spl Bn n n 4 1. REFERENCES [1] Gray Chartrand Dad Erwn Png Zhang Frank Harary Rado labelng of graphs Bull. Inst. Combn. Appl. 33(2001) [2] R. Kchkech M. Khennoufa O. Togn Lnear and cyclc rado k-labelngs of trees Dscuss. Math. Graph Theory 130 (3) (2007) [3] R. Kchkech M. Khennoufa O. Togn Rado k- labelngs for Cartesan products of graphs Dscuss. Math. Graph Theory 28 (1) (2008) [4] M. Khennoufa O. Togn The rado antpodal and rado numbers of the hypercube Ars Combn. 102 (2011) [5] D. Lu Rado number for trees Dscrete Math. 308 (7) (2008) [6] D. Lu X. Zhu Multleel dstance labelng for paths and cycles SIAM J. Dscrete Math. 19 (3) (2005) [7] D. Lu M. Xe Rado number for square of cycles Congr. Numer. 169 (2004) [8] D. Lu M. Xe Rado number for square of paths Ars Combn. 90 (2009) [9] D. Lu R. K. Yeh On dstance two labelng of graphs Ars Combn. 47 (1997) [10] R.Ponra S. Sathsh Narayanan R. Kala Rado mean labelng of a graph AKCE Internatonal ournal of graphs and Combnatorcs 12 (2015) [11] R.Ponra S. Sathsh Narayanan R. Kala On Rado Mean Number of Graphs Internatonal J. Math. Combn. Vol. 3(2014) [12] F.Harary graph Theory Addson Wesley New Delh [13] J.A. Gallan A Dynamc Surey of graph labelng Electron. J. Combn. 19 (2012) #DS6. [14] P. Selara P. Balaganesan J Renuka Path and Star Related graphs on Een Sequental Harmonous Graceful Odd Graceful and Felctous Labelng Internatonal Journal of Pure and Appled Mathematcs Vol. 87 No

5 Illustratons: u 1 u 2... u n n u 18 Spl (K 1n ) General Graph Labelng RGML of Spl (K 18 ) RGML of Spl (B 55 ) 1700