Iteratoal Joral of Scece ad Research (IJSR) ISSN (Ole): -0 Idex Copercs Vale (0):. Impact Factor (0):. (k,d)mea Labelg of Some Famly of Trees B. Gayathr, V. Slochaa PG ad Research Departmet of Mathematcs, Peryar E.V.R. College, Trchrappall, Ida Departmet of Mathematcs, Seethalakshm Ramaswam College, Trchrappall, Ida Abstract: Mea labelg of graphs was dscssed [-] ad the cocept of odd mea labelg was trodced []. kodd mea labelg ad (k,d)odd mea labelg are trodced ad dscssed [,-8]. kmea, keve mea ad (k,d)eve mea labelg are trodced ad dscssed [-]. I ths paper, we trodce (k,d)mea labelg ad we have obtaed reslts for some famly of trees. Keywords: (k,d)mea labelg, (k,d)mea graph. Itrodcto All graphs ths paper are fte, smple ad drected. Terms ot defed here are sed the sese of Harary [0]. E G wll deote the vertex set The symbols VG ad ad edge set of a graph G. A graph labelg s a assgmet of tegers to the vertces or edges or both sbject to certa codtos. If the doma of the mappg s the set of vertces (or edges) the the labelg s called a vertex labelg (or a edge labelg). Graph labelg was frst trodced the late 0 s. May stdes graph labelg refer to Rosa s research []. Labeled graphs serve as sefl models for a broad rage of applcatos sch as X-ray, crystallography, radar, codg theory, astroomy, crct desg ad commcato etwork addressg. Partclarly terestg applcatos of graph labelg ca be fod [-]. Mea labelg of graphs was dscssed [,]. Vadya ad et al. [8-] have vestgated several ew famles of mea graphs. Nagaraja ad et al. [] have fod some ew reslts o mea graphs. Poraj, Jayath ad Ramya exteded the oto of mea labelg to sper mea labelg []. Gayathr ad Tamlselv [-,] exteded sper mea labelg to ksper mea, (k,d)sper mea, ksper edge mea ad (k,d)sper edge mea labelg. Mackam ad Marda [] trodced the cocept of odd mea graphs. Gayathr ad Amthavall [,-8] exteded ths cocept to kodd mea ad (k,d)odd mea graphs. Gayathr ad Gop [-] exteded ths cocept to kmea, keve mea ad (k,d)eve mea graphs. I ths paper, we exted kmea graphs to (k,d)mea graphs sce there are graphs whch are (k,d)mea for all k ad d bt ot (k,)mea for ay k. Here, we have fod (k,d)mea labelg of some famly of trees. Throghot ths paper, k ad d deote ay postve teger greater tha or eqal to. For brevty, we se (k,d)ml for (k,d)mea labelg ad (k,d)mg for (k,d)mea graph.. Ma Reslts Defto. A pq, graph G s sad to have a mea labelg f there s a jectve fcto f from the vertces of G to {0,,,,q} sch that the dced map f defed o E by f f v f v s a bjecto from E to {,,,q}. A graph that admts a mea labelg s called a mea graph. Defto. A pq, graph G s sad to have a kmea labelg f there s a jectve fcto f from the vertces of G to {0,,,, k + q } sch that the dced map o E by f v f defed f f v s a bjecto from E to k, k, k,, k q. A graph that admts a kmea labelg s called a kmea graph. Observato. Every mea labelg s a mea labelg. Defto. A pq, graph G s sad to have a (k,d)mea labelg f there exsts a jectve fcto f from the vertces of G to 0,,,, k q d sch that the dced map defed o E by f f f v f v s a bjecto from E to,,,, k k d k d k q d. A graph that admts a (k,d)mea labelg s called a (k,d)mea graph. Observato. ) Every (k,)mea labelg s a kmea labelg ) Every (,)mea labelg s a mea labelg. Volme Isse, Jaary 0 www.jsr.et Paper ID: NOV Lcesed Uder Creatve Commos Attrbto CC BY
Iteratoal Joral of Scece ad Research (IJSR) ISSN (Ole): -0 Idex Copercs Vale (0):. Impact Factor (0):. Theorem. The path graph P s a (k,d)mea graph for all k ad d, whe s eve. V P Let ={v, v,, v } ad E P =v v, be deoted as the Fgre. v v v v v v v v Fgre.: Ordary labelg of P Frst we label the vertces as follows: Defe : 0,,,., f v = k + d( ), f s odd f v = k + d( ), f s eve The the dced edge labels are f v v = k + d( ), for. The above defed fcto f provdes (k,d)mea labelg of So, the path graph P s a (k,d)mea graph for all k ad d, whe s eve. (k,d) mea labelg of P for dfferet cases of k ad d whe s eve are show llstrato.. Illstrato. (00, 0)mea labelg of the graph P 0 s show Fgre. 00 0 0 0 0 00 0 0 0 00 0 0 0 0 Fgre.: (00, 0)ML of P 0 ad EP =, ad, be deoted as the Fgre.. Fgre.: Ordary labelg of Frst we label the vertces as follows: Defe : 0,,,, f = k f = k f = k + (q )d f = k + (q )d P f = k + d( ), for f = k + d( ),for The the dced edge labels are f = k + d( ), for f = k + d( ), for The above defed fcto f provdes (k,d)mea labelg of So, the graph P s a (k,d)mea graph for all k ad d. (k,d)mea labelg of P for dfferet cases of k ad d are show llstrato.0 Illstrato.0 (,)mea labelg of the graph P s show Fgre. (, )mea labelg of the graph P s show Fgre. 0 Fgre.: (, )ML of P (, )mea labelg of the graph P 8 s show Fgre. 88 8 Fgre.: (, )ML of P 8 (0, )mea labelg of the graph P 0 s show Fgre. 0 8 8 0 8 8 8 Fgre.: (0, )ML of P 0 Defto.8 A comb graph P s a tree obtaed from a path by attachg exactly oe pedat edge to each vertex of the path. Theorem. The comb graph P s a (k,d)mea graph for all k ad d. Let VP =,,,,,,, 8 0 0 8 0 Fgre.: (,)ML of P (,)mea labelg of the graph P s show Fgre.8 0 8 Fgre.8: (,)ML of P (,8)mea labelg of the graph P 8 s show Fgre. 8 0 8 8 8 Fgre.: (,8)ML of P 8 0 0 0 Volme Isse, Jaary 0 www.jsr.et Paper ID: NOV Lcesed Uder Creatve Commos Attrbto CC BY
Iteratoal Joral of Scece ad Research (IJSR) ISSN (Ole): -0 Idex Copercs Vale (0):. Impact Factor (0):. (,)mea labelg of the graph P s show Fgre.0 8 0 0 88 0 0 88 0 Fgre.0: (,)ML of P Defto. A twg s a tree obtaed from a path by attachg exactly two pedat edges to each teral vertex of the path. Theorem. The twg graph T s a (k,d)mea graph for all k ad d, whe s eve. V T = {v, 0,, w, } Let ad ET = {v, v w, ad v v +, 0 } be deoted as the Fgre.. v 0 v v v v v w w w w Fgre.: Ordary labelg of T Frst we label the vertces as follows: f : V 0,,,, k q d by Defe k d, f s eve f v = k d, f s odd k d, f s eve f = k d, f s odd k d, f s eve f w = k d, f s odd The the dced edge labels are f v v = k + d, for 0 f v = k + d( ), for f v w = k + d( ), for The above defed fcto f provdes (k,d)mea labelg of 0 0 8 0 0 8 8 0 08 0 0 0 Fgre.: (,)ML of T 8 (,)mea labelg of the graph T s show Fgre. 8 Fgre.: (,)ML of T (,)mea labelg of the graph T s show Fgre. 0 8 0 0 8 Fgre.: (,)ML of T (8,)mea labelg of the graph T 0 s show Fgre. 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 08 8 8 88 8 Fgre.: (8,)ML of T 0 08 0 0 Theorem. The star K, ( ) s a (k,d)mea graph for all k ad for all d satsfyg (q )d k + except whe s odd ad d s eve. Let VK, = {, v, v,, v } ad, E K = {v, } be deoted as Fgre.. So, the twg graph T s a (k,d)mea graph for all k ad d, whe s eve. (k,d)mea labelg of T for dfferet cases of k ad d whe s eve are show llstrato.. Illstrato. (,)mea labelg of the graph T 8 s show Fgre. v v v v v v Fgre.: Ordary labelg of K, Frst we label the vertces as follows: Volme Isse, Jaary 0 www.jsr.et Paper ID: NOV Lcesed Uder Creatve Commos Attrbto CC BY
Defe : 0,,,.., Case (): whe s eve Sbcase (): d s odd f = k + (q )d f v = k (q )d +, for f v = k, for Iteratoal Joral of Scece ad Research (IJSR) ISSN (Ole): -0 Idex Copercs Vale (0):. Impact Factor (0):. f v = k + (q )d + ( ), for f v = k + (q )d Sbcase (): d s eve f = k + (q )d f v = k (q )d +, for f v = k, for f v = k + (q )d + ( ), for f v = k + (q )d The the dced edge labels are f v = k + d( ), for Case (): whe s odd sbcase (): d s odd f = k + (q )d f v = k (q )d +, for f v = k + d, for f v = k + (q )d + ( ), for f v = k + (q )d The the dced edge labels are f v = k + d( ), for The above defed fcto f provdes (k,d)mea labelg of 8 0 0 8 0 Fgre.: (,)ML of K,8 (,)mea labelg of the graph K, s show Fgre. 0 Fgre.: (,)ML of K, (,)mea labelg of the graph K, s show Fgre. 8 0 8 0 Fgre.: (,)ML of K, (,)mea labelg of the graph K, s show Fgre.0 So, the star K, s a (k,d)mea graph for all k ad for all d satsfyg (q )d k + except whe s odd ad d s eve. (k,d)mea labelg of K, for dfferet cases of k ad d except whe s odd ad d s eve are show llstrato.. Illstrato. (,)mea labelg of the graph K,8 s show Fgre. 8 0 0 Fgre.0: (,)ML of K, Defto. A bstar B m, s a tree obtaed by jog the ceter vertces of the copes of K,m ad K, wth a edge. Volme Isse, Jaary 0 www.jsr.et Theorem. The Bstar B, ( ) s a (k,d)mea graph for all k ad d. V B = {, v,,,,, v, v,, v } Let, Paper ID: NOV Lcesed Uder Creatve Commos Attrbto CC BY
E B = {v,, vv, } ad, be deoted as the Fgre.. Iteratoal Joral of Scece ad Research (IJSR) ISSN (Ole): -0 Idex Copercs Vale (0):. Impact Factor (0):. Fgre.: Ordary labelg of B, Frst we label the vertces as follows: Defe : 0,,,, f = k + (q )d f = k + (q )d f v = k f = k + d, for f v = k + d( ), for The the dced edge labels are f vv = k + d( ), for f v = k + d f = k + ( + )d, for v v v v v The above defed fcto f provdes (k,d)mea labelg of So, the graph B, s a (k,d)mea graph for all k ad d. (k,d)mea labelg of B, for dfferet cases of k ad d are show llstrato.. Illstrato. (,)mea labelg of the graph B, s show Fgre.. 8 0 8 0 0 0 Fgre.: (,)ML of B, (,)mea labelg of the graph B, s show Fgre. 8 8 Fgre.: (,)ML of B, 0 8 v 0 0 Volme Isse, Jaary 0 www.jsr.et (,)mea labelg of the graph B, s show Fgre. Paper ID: NOV Lcesed Uder Creatve Commos Attrbto CC BY 0 88 8 8 8 8 Fgre.: (,)ML of B, (,)mea labelg of the graph B 8,8 s show Fgre. 0 0 8 8 Fgre.: (,)ML of B 8,8 Theorem. The Bstar B,+ ( ) s a (k,d)mea graph for all k ad for all d k +. V B = {, v,,,,, v, v,, v + } Let, ad, 0 8 0 E B = {v,,, vv, + } be deoted as the Fgre.. Fgre.: Ordary labelg of B,+ Frst we label the vertces as follows: f : V 0,,,,, k q d by Defe f = k d + f = k + d f v = k + (q )d v 8 v v v v v + f = k + d( ), for f v = k + d( ), for + The the dced edge labels are f = k + d( ), for f v = k + d f vv = k + ( + )d, for + 0 8 0
Iteratoal Joral of Scece ad Research (IJSR) ISSN (Ole): -0 Idex Copercs Vale (0):. Impact Factor (0):. The above defed fcto f provdes (k,d)mea labelg of So, the graph B,+ s a (k,d)mea graph for all k ad for all d k +. (k,d)mea labelg of B,+ for dfferet cases of k ad d k + are show llstrato.0. Illstrato.0 0 0 8 0 8 8 0 (,)mea labelg of the graph B, s show Fgre. 8 8 0 Fgre.: (,)ML of B, (,)mea labelg of the graph B, s show Fgre.8 0 8 0 Fgre.8: (,)ML of B, (,)mea labelg of the graph B, s show Fgre. Fgre.: (,)ML of B, (,)mea labelg of the graph B, s show Fgre.0 0 0 Refereces Fgre.0: (,)ML of B, [] K. Amthavall, Graph labelg ad ts Applcatos some geeralzatos of odd mea labelg, Ph.D. Thess, Mother Theresa Wome s Uversty, Kodakaal, Jly (00). [] G.S. Bloom, S.W. Golomb, Applcatos of mbered drected graphs, Proc. IEEE, (), -0. [] G.S. Bloom, S.W. Golomb, Nmbered complete graphs sal rlers ad assorted applcatos, Theory ad Applcatos of Graphs-Lectre otes Math., Sprger Verlag, New York, (8), -. [] G.S. Bloom, D.F. Hs, O gracefl dgraphs ad a problem etwork addressg, Cogresss Nmeratm, (8) -0. [] J.A. Galla, A dyamc srvey of graph labelg, Electroc Joral of Combatorcs, (0) # DS. [] B. Gayathr ad K. Amthavall, kodd mea labelg of crow graphs, Iteratoal Joral of Mathematcs ad Compter Scece, () (00) -. [] B. Gayathr ad K. Amthavall, (k,d)odd mea labelg of some graphs, Bllet of Pre ad Appled Sceces, E() (00) -. [8] B. Gayathr ad K. Amthavall, k-odd mea labelg K, K, Acta Ceca Idca, () (008) 8- of,, m 8. [] B. Gayathr ad R. Gop, keve mea labelg of D C, Iteratoal Joral of Egeerg m, @ Scece, Advaced Comptg ad Bo-Techology, Vol., No., Jly-September 00, -. [0] B. Gayathr ad R. Gop, keve mea labelg of D m,, Acta Ceca Idca, Vol. XXXVII, No., - 00, 0. [] B. Gayathr ad R. Gop, keve mea labelg of C P m, Elxr Iteratoal Joral of Appled Sceces, No., 0, 0-0. [] B. Gayathr ad R. Gop, keve mea labelg of T,m,t, Iteratoal Joral of Egeerg Scece, Advaced Comptg ad Bo-Techology, Vol., No., Aprl-Je 0, -8. [] B. Gayathr ad R. Gop, Iteratoal Coferece o Mathematcs ad Compter Scece, k-eve mea labelg of Cme K, Loyola College, Chea, Jaary -8, 0, Proc. -. 8 Volme Isse, Jaary 0 www.jsr.et Paper ID: NOV Lcesed Uder Creatve Commos Attrbto CC BY
Iteratoal Joral of Scece ad Research (IJSR) ISSN (Ole): -0 Idex Copercs Vale (0):. Impact Factor (0):. [] B. Gayathr ad R. Gop, (k,d)eve mea labelg of Pm e K, Iteratoal Joral of Mathematcs ad soft comptg, Vol., No., -, Agst 0. [] B. Gayathr ad R. Gop, keve mea labelg of some graphs, Heber Iteratoal Coferece o Applcatos of Mathematcs ad Statstcs, Bshop Heber College (Atoomos), Jaary -, 0 Proc. 0-. [] B. Gayathr ad R. Gop, Iteratoal Coferece o Mathematcs Egeerg ad Bsess Maagemet, keve mea labelg of some trees, Stella Mars College (Atoomos), Chea, March -0, 0, Proc. -0. [] R. Gop, A Stdy o Dfferet kds of Mea Labelg, Ph.D. Thess, Bharathdasa Uversty, Trchy, Febrary (0). [] B. Gayathr, M. Tamlselv ad M. Drasamy, (k,d)sper mea labelg of some graphs, Iteratoal Joral of Mathematcs ad Compter Scece, () (00). [] B. Gayathr ad M. Tamlselv, ksper mea labelg of some trees ad cycle related graphs, Bllet of Pre ad Appled Sceces, Volme E() (00) 0-. [0] F. Harary Graph Theory, Addso-Wesley, Readg Masaachsetts,. [] R. Poraj, Jayath ad D. Ramya, O sper mea graphs of order, Bllet of Pre ad Appled Sceces, E (00) -. [] K. Mackam ad M. Marda, Odd mea labelg of graphs, Bllet of Pre ad Appled Sceces, E() (00) -. [] Rosa, O certa valatos of the vertces of a graph Theory of Graphs (Iteret Symposm, Rome, Jly ), Gordo ad Breach, N.Y. ad Dhod, Pars () -. [] S. Somasdaram ad R. Poraj, Mea labelg of graphs, Natoal Academy Scece Letter, (-8) (00), 0-. [] S. Somasdaram ad R. Poraj, No-exstece of mea labelg for a wheel, Bll. Pre ad Appl. Sceces (Mathematcs & Statstcs), E (00) 0-. [] M. Tamlselv, A stdy Graph Theory- Geeralzato of sper mea labelg, Ph.D. Thess, Vayaka Msso Uversty, Salem, Agst (0). [] R. Vask ad A. Nagaraja, Frther reslts o mea graphs, Sceta Maga, () (00) -. [8] S.K. Vadya ad Lekha Bjkmar, Mea labelg for some ew famles of graphs, Joral of Pre ad Appled Sceces, () (00) -. [] S.K. Vadya ad Lekha Bjkmar, Some ew famles of mea graphs, Joral of Mathematcs Research, () (00) -. [0] S.K. Vadya ad Lekha Bjkmar, New mea graphs, Iteratoal J. Math. Comb., (0) 0-. [] S.K. Vadya ad Kaa, Some ew mea graphs, Iteratoal Joral of Iformato Scece ad Compter Mathematcs, () (00) -80. Volme Isse, Jaary 0 www.jsr.et Paper ID: NOV 8 Lcesed Uder Creatve Commos Attrbto CC BY