PRAJÑĀ Volume 18, 2010

Transcription

1 ISSN PRAJÑĀ Volume 18, 010 Joural of Pure ad Appled Sceces SARDAR PATEL UNIVERSITY VALLABH VIDYANAGAR Guarat , INDIA

2 PRAJÑĀ - Joural of Pure ad Appled Sceces, Vol. 18: , (010) ISSN MEAN LABELING FOR SOME NEW FAMILIES OF GRAPHS S. K. Vadya 1 * ad Lekha Bukumar 1 Saurashtra Uversty, Rakot , Guarat Lekha Bukumar Shaker Sh Vaghela Bapu Isttute of Techology, Gadhagar, Guarat ABSTRACT Some ew famles of mea graphs are vestgated. We prove that the step ladder graph, total graph of path P are mea graphs. I addto to ths we derve that two copes of cycle C sharg a commo edge admts mea labelg. Key words: mea labelg, mea graphs, step ladder graph, total graphs. INTRODUCTION We beg wth smple, fte, coected ad udrected graph G = (V(G),E(G)) wth p vertces ad q edges. For all other stadard termology ad otatos we follow Harary [1]. We wll provde bref summary of deftos ad other formato whch serve as prerequstes for the preset vestgatos. Defto 1.1 Let P be a path o vertces deoted by (1,1),(1,),...,(1,) ad wth -1 edges deoted by e 1,e,...,e -1 where e s the edge og the vertces (1,) ad (1,+1). O each edge e, = 1,,, -1 we erect a ladder wth -(-1) steps cludg the edge e. The graph obtaed s called a step ladder graph ad s deoted by S(T ), where deotes the umber of vertces the base. Defto 1. The vertces ad edges of a graph are called ts elemets. Two elemets of a graph are eghbours f they are ether cdet or adacet. The total graph of a graph G s deoted by T(G) s a graph wth vertex set V( G) U E( G) ad two vertces are adacet T(G) whe ever they are eghbours G. Defto 1.3 If the vertces are assged values subect to certa codtos the t s kow as graph labelg. Graph labelg s oe of the fascatg areas of graph theory wth wde ragg applcatos. A eormous body of lterature has grow aroud graph labelg last fve decades. A systematc study of varous applcatos of graph labelg s carred out Bloom ad Golomb [3]. Accordg to Beeke ad Hegde [] graph labelg serves as a froter betwee umber theory ad structure of graphs. For detaled survey o graph labelg we refer to A Dyamc Survey of Graph Labelg by Galla [4]. Defto 1.4 A fucto f s called a mea labelg of graph G f f : V( G) {0,1,,..., q} s ectve ad the duced fucto f *: EG ( ) {1,,..., q} defed as f ( u) + f( v) f *( e= uv) = f f ( u) + f ( v) s eve f ( u) + f ( v) + 1 = f f ( u) + f ( v) s odd s bectve. The graph whch admts mea labelg s called a mea graph. *Correspodg author samrkvadya@yahoo.co. The mea labelg s troduced by Somasudaram ad Pora [5] ad they proved the graphs P, C, P Pm, Pm C etc. admt mea labelg. The same authors [6] have dscussed the mea labelg of subdvso of K for 3 whle [7] 1, they proved that the wheel W does ot admt the mea labelg for 4. Mea labelg the cotext of some graph operatos s dscussed by Vadya ad Lekha [8]. I the preset work three ew results correspodg to mea labelg ad some ew famles of mea graphs are vestgated. MAIN RESULTS Theorem-.1: The step ladder graph S(T ) s a mea graph. Proof: Let P be a path o vertces deoted by (1,1),(1,),...,(1,) ad wth -1 edges deoted by e 1,e,...,e -1 where e s the edge og the vertces (1,) ad (1,+1). The step ladder graph S(T ) has vertces deoted by (1,1),(1,),...,(1,), (,1),(,),...,(,), (3,1),(3,),...,(3,- 1),...,(,1),(,). I the ordered par (,), deotes the row (couted from bottom to top) ad deotes the colum ( from left to rght) whch the vertex occurs. Defe f : V( S( T )) {0,1,,..., q} as follows. = + 1 f (,1) ( ) ( 1) 1 f(1, ) = ( + ) ( k) [( + k) ( 1)] k= 1 k= 1 (, ) = ( + ) ( ) [( + ) ( 1)] ( 1) k= 1 k= f k k, & + f(, + ) = I vew of the above defed labelg patter f s a mea labelg for the step ladder graph S(T ). That s, S(T ) s a mea graph. v, v,..., v Proof: Let 1 be the vertces of path P wth -1 edges deoted b y e1, e,..., e 1. Accordg to the defto of total graph ad two vertces are adacet T(P ) f they are eghbours P. Defe f : VTP ( ( )) {0,1,,..., q} as follows. f ( v 1) = 0 f ( v ) = 4( ) + for f ( e ) = 4 for 1

3 116 PRAJÑĀ Joural of Pure ad Appled Sceces 18: December f ( e ) = 4 1 for = 1 Thus f provdes a mea labelg for T(P ). That s, T(P ) s a mea graph. Illustrato.: The Fgure 1 shows the labelg patter for S(T 6 ). f ( v ) = ( ) + 4 for f ( v ) = ( ) + 3 for 3 The the above defed fucto f provdes mea labelg for two copes of cycle sharg a commo edge. Illustrato.6: The Fgure 3 shows the mea labelg patter for two copes of C 10 sharg a edge. Fg. 1 Illustrato.4: The labelg patter of T(P 5 ) s gve Fgure. Fg. Theorem-.5: Two copes of cycle C sharg a commo edge admt mea labelg. Proof: Let v1, v,..., v be the vertces of cycle C. Cosder two copes of cy cle C.. Let G be the graph for two copes of cycle sharg a commo edge whch v1, v,..., v- s a spag path. The VG ( ) = - ad EG ( ) = -1. To defe f : VG ( ) {0,1,,..., q} the followg two cases are to be cosdered. Case 1: s odd. Wthout loss of geeralty assumes that e = v v be the commo edge betwee two copes of C. f( v ) ( 1) for = f ( v ) = 1 for 3 f ( v ) = ( ) + 4 for f ( v ) = ( ) + 3 for 3 Case : s eve. Wthout loss of geeralty assume that e v v commo edge betwee two copes of C. f( v ) ( 1) for = + 1 f ( v ) = 1 for = + 3 be the Fg. 3 CONCLUDING REMARKS AND FURTHER SCOPE As all graphs are ot mea graphs t s very terestg to vestgate graphs whch admt mea labelg. Here we cotrbute three ew famles of mea graphs. It s possble to vestgate smlar results for other graph famles ad the cotext of dfferet labelg techques. ACKNOWLEDGEMENT Authors are hghly thakful to aoymous referee for valuable commets ad kd suggestos. REFERENCES [1] Harary, F. (197) Graph Theory, Addso Wesley, Readg, Massachusetts. [] Beeke, L. W. ad Hegde, S. M. (001), Strogly multplcatve graphs, Dscuss. Math. Graph Theory, 1: [3] Bloom, G. S. ad Golomb, S. W. (1977) Applcatos of umbered udrected graphs, Proceedgs of IEEE, 65(4): pp [4] Galla, J. A. (009) A dyamc survey of graph labelg, The Electrocs Joural of Combatorcs, 16 #DS6. [5] Somasudaram, S. ad Pora, R. (003) Some results o mea graphs, Pure ad Appled Mathematcal Sceces, 58: [6] Somasudaram, S. ad Pora, R. (004) O mea graphs of order < 5, J. Decso ad Mathematcal Sceces, 9: [7] Somasudaram, S. ad Pora, R. (003) No exstece of mea labelg for a wheel, Bullet of Pure ad Appled Sceces, E: [8] Vadya, S. K. ad Lekha, B. (010) Mea labelg the cotext of some graph operatos, It. Joural of Algorthms, Comp. ad Math. 3(1): 1-8.

4 167 GUIDELINES FOR CONTRIBUTORS The Edtoral Board of PRAJNA Joural of Pure ad Appled Sceces vtes Orgal Research Papers the felds of Basc ad Appled Sceces (Bosceces, Chemstry, Computer Scece, Electrocs Scece, Home Scece, Materals Scece, Mathematcs, Physcs ad Statstcs) for the Next Volume of PRAJNA (December 011), publshed by Sardar Patel Uversty, Vallabh Vdyaagar, Guarat 38810, INDIA. The soft copes of regular (full-legth) research papers (ot exceedg 15 typed pages), prepared as per the fle format show below may be submtted for publcato through e-mal to Prof. T. V. Ramaa Rao, Maagg Edtor OR to a Member of the Edtoral Board who represets the author s broad research area wth a cc to the Maagg Edtor latest by August 31, 011. Each mauscrpt must be accompaed by a statemet that t has ot bee publshed elsewhere ad that t has ot bee submtted smultaeously for publcato elsewhere. Revew process: Submtted papers are peer-revewed by two to three depedet revewers after approval by the Edtoral Board. Authors are ecouraged to suggest three ames of expert revewers wth ther e-mal IDs, but selecto remas the prerogatve of the Edtoral Board. Artcles of the followg categores are also cosdered for publcato PRAJNA: Short Commucatos are lmted to a maxmum of two fgures ad oe table. They should preset a complete study that s more lmted scope tha s foud full-legth papers. The tems of mauscrpt preparato lsted above apply to Short Commucatos wth the followg dffereces: (1) Abstracts are lmted to 100 words () stead of a separate Materals ad Methods secto, expermetal procedures may be corporated to Fgure Legeds ad Table foototes (3) Results ad Dscusso should be combed to a sgle secto. Revew Artcles teded to provde cocse -depth revews of both establshed ad ew areas ad summarze recet sghts specfc research areas wth the scope of PRAJNA are solcted by the Edtoral Board from leadg researchers. The mauscrpt of ths category should be lmted to 5,000 words wth a abstract of o more tha 50 words, a maxmum of 5 tables ad fgures (total), ad up to 50 refereces. Word cout cludes oly the ma body of text (.e., ot tables, fgures, abstracts or refereces). Commetares call atteto to papers of partcular ote ad are wrtte at the vtato of the Edtoral Board. Perspectves preset a vewpot o a mportat area of research ad are wrtte oly at the vtato of the Edtoral Board. Perspectves focus o a specfc feld or subfeld wth a larger dscple ad dscuss curret advaces ad future drectos. Perspectves are of broad terest for o-specalsts ad may add persoal sght to a feld. Letters are bref commets that cotrbute to the dscusso of a research artcle publshed the last ssue of PRAJNA. Letters may ot clude requests to cte the letter wrter's work, accusatos of mscoduct, or persoal commets to a author. Letters are lmted to 500 words ad o more tha fve refereces. Letters must be submtted wth 3 moths of the publcato date of the subect artcle. Also aoucemet of forthcomg Semars / Cofereces / Symposa / Workshops etc. wll be cosdered for publcato PRAJNA. Fle format for soft copes: Texts (should be of Tmes New Roma wth 9 pot for Abstract ad 11 pot for other matter) ad Tables, f ay, must be saved *.doc (Word) or *.rtf (rch text) format, graphs Excel ad for llustratos (dagrams, maps, drawgs, etc.), the TIF format (300 dp mmal resoluto) s the most approprate (*.TIF or *.JPEG exteso). Istructos for preparato of mauscrpts: 1. The paper should be wrtte Eglsh ad eatly typed wth double spacg.. The ttle of the paper ad the ame(s) of the author(s) be captal letters. The ame of the sttuto be gve small letters below the ame (s) of the author(s). 3. The Abstract of the paper, ot more tha 150 words, should be provded o a separate page alog wth 4-6 keywords. 4. The sub-ttles, e.g. INTRODUCTION, should be wrtte captal letters.

5 5. Dsplayed formulae, mathematcal equatos ad expressos should be umbered serally. Table should be wth a ttle addto to a seral umber for t. 6. Photographs / Fgures should be orgal wth good cotrast so as to be a form sutable for drect reproducto / scag. 7. Foototes are ot ormally allowed, except to detfy the author for correspodece. 8. All fgures must be umbered serally as they appear the text, ad ther legeds / captos should ecessarly be provded. 9. Refereces should be umbered brackets [ ] the order of appearace the text. All the refereces the bblographc lst must correspod to -text refereces ad vce versa. Abbrevated perodcal ttles should follow stadard subect Abstracts. Names whch are ot lsted by ay stadard subect dexg orgazatos should be spelled out full. 10. All refereces should be clear ad follow the examples below: Perodcal artcles [] Sadqu, M., Fushma, D. ad Muoz, V. (006) Atom by atom aalyss of global dowhll prote foldg. Nature, 44: Books [16] Stebbs, G. L. (1974) Flowerg plats: Evoluto above the speces level, Arold Press, Lodo, pp Chapters from a book [19] Schafer, H. ad Muyzer, G. (001) Deaturg gradet gel electrophoress mare mcrobal ecology. I Methods Mcrobology (Ed. Paul, J. H.), Academc Press, Lodo, Vol. 30, pp Thess or other dplomas [1] Nayaka, S. (004) The vsoary studes o the lche geus Lecaora sesu lato Ida. Ph. D. Thess, Dr. R. M. L. Avadh Uversty, Fazabad, Ida. Coferece proceedgs [4] Mohapatra, G. C. (1981) Evromet ad culture of early ma the valley of rvers Cheab ad Rav, wester sub-hmalayas. I Proceedgs X Cogress of IUPPS, Mexco, pp Ole documetato [9] Kog, R. E. (1994). Home Page for Ross Kog. Retreved from Plat Physology Iformato Webste: Note: Mauscrpts prepared fathfully accordace wth the structos wll accelerate ther processg towards publcato otherwse t would be delayed vew of ther expected re-submsso. 168 For ad o behalf of Edtoral Board, PRAJNA NOTE: Ths formato may be kdly crculated amog your colleagues. Prof. T. V. Ramaa Rao Maagg Edtor, PRAJNA B R Dosh School of Bosceces, Satellte Campus, Vadtal Road, Sardar Patel Uversty, VALLABH VIDYANAGAR Guarat Phoe: (Lab): Ext. 111 Moble: Fax: /36475 e-mal: spu.praa@gmal.com Webste: