Joural o Physics: Coerece Series PAPER OPEN ACCESS Mea square cordial labellig related to some acyclic graphs ad its rough approximatios To cite this article: S Dhaalakshmi ad N Parvathi 018 J. Phys.: Co. Ser. 1000 01040 Related cotet - The glite workload maagemet system P Adreetto, S Adreozzi, G Avellio et al. - Graph theory or FPGA miimum coiguratios Rua Aiwu, Li Wechag, Xiag Chuayi et al. - A example o umerical simulatio i causal set dyamics Alexey L Krugly ad Iva A Tserkovikov View the article olie or updates ad ehacemets. This cotet was dowloaded rom IP address 148.51.3.83 o 17/07/018 at 19:34
Natioal Coerece o Mathematical Techiques ad its Applicatios (NCMTA 18) IOP Publishig IOP Co. Series: Joural o Physics: Co. Series 134567890 1000 (018) 01040 doi :10.1088/174-6596/1000/1/01040 Mea square cordial labellig related to some acyclic graphs ad its rough approximatios S Dhaalakshmi 1 ad N Parvathi. 1 Departmet o mathematics,srm Istitute o Sciece &Techolgy Cheai -600089, Idia Email : parvathi.@ktr.srmuiv.ac.i Abstract. I this paper we ivestigate that the path P, comb graph P ʘK 1,-cetipede graph,cetipede graph (,) ad star S admits mea square cordial labelig. Also we proved that the iduced sub graph obtaied by the upper approximatio o ay sub graph H o the above acyclic graphs admits mea square cordial labelig. 1. Itroductio Graph labelig [1]1is oe o the most importat area i graph theory ad it has lot o applicatios i may ields like circuit desigig, commuicatio etworks,database maagemet system,astroomy etc... Here we cosider a simple, iite, coected ad udirected graph G =(V,E). For all other termiology ad otatios i graph theory, we ollow Harary[].Rough set theory [3]was proposed by Z.Pawlak i 198 ad it is deied through upper ad lower approximatios. The cocept o cordial labelig was itroduced by Cahit i the year 1987 i [4]. Mea cordial labelig o a graph deied by Poraj et al[5].a.nellaimuruga et al itroduced the mea square cordial labelig ad they have studied it or some special graphs[6].also they have studied mea square cordial labelig o sometree ad cycle related graphs[7],[8].dhalakshmi et al have discussed graceul ad eve graceul labelig o rough approximatios[9],[10]. Dhalakshmi et al have studied prime labelig o rough approximatios or some graphs[11]. I this paper we ivestigate that the graph pathp, star graph S,comb graph P ʘK 1,cetipede graph (,) ad star S admits mea square cordial labelig. Also we have proved that the iduced sub graph obtaied by the upper approximatio o ay sub graph H o the above acyclic graphs admits mea square cordial labelig. Prelimiaries Deiitio.1 Comb is a graph obtaied by joiig a sigle pedat edge to each vertex o a path. Deiitio. -Cetipede graph is a graph o vertices obtaied by joiig a sigle pedat edge to each vetex o a path Deiitio.3 Cetipede graph(,) is a graph o 3 vertices obtaied by joiig a two pedat edges which are adjacet i each vertex o a path Deiitio.4 Star graph S is a graph obtaied by addig leaves i oe iteral ode(apex vertex). Deiitio.5 Let G = (V,E) be a graph with p vertices ad q edges. A Mea Square Cordial labelig o a Graph G with vertex set V is a bi bijectio rom V to {0, 1} such that each edge uv is assiged the Cotet rom this work may be used uder the terms o the Creative Commos Attributio 3.0 licece. Ay urther distributio o this work must maitai attributio to the author(s) ad the title o the work, joural citatio ad DOI. Published uder licece by IOP Publishig Ltd 1
Natioal Coerece o Mathematical Techiques ad its Applicatios (NCMTA 18) IOP Publishig IOP Co. Series: Joural o Physics: Co. Series 134567890 1000 (018) 01040 doi :10.1088/174-6596/1000/1/01040 x label ( ( ( u) ( v) ) / ) where (ceil( x)) is the least iteger greater tha or equal to x with the coditio that the umber o vertices labeled with 0 ad the umber o vertices labeled with 1 dier by at most 1 ad the umber o edges labeled with 0 ad the umber o edges labeled ad the umber o edges labeled with 1 dier by at most 1. The graph that admits a Mea Square Cordial Labelig is with 1 dier by at most 1. The graph that admits a mea square cordial labelig is Mea square cordial graphdeiitio.6 Let H be a sub graph o ay graph G the the eighborhood o v is N( v) { v} { u V ( G) : vu E( G)} Deiitio.7 For ay sub graph H o graph G the we deie (i)the lower approximatio operatio as V( H) { v V( H)/ N( v) V( H)} (ii)the upper approximatio operatio as V( H) { N( v)/ v V( H)} Lower ad upper approximatio H- mea square cordial labeligo G:Based o the deiitio o lower ad upper H prime labeligwe here itroduce the ollowig deiitios. Deiitio.8 Lower approximatio H mea square cordial labelig o G The iduced sub graph obtaied by the lower approximatio operatio o ay sub graph H o G has mea square cordial labelig the we ca say the subgraph is lower approximatio H mea square cordial labeligo G. Deiitio.9 Upper approximatio H mea square cordial labelig o G The iduced sub graph obtaied by the upper approximatio operatio o ay sub graph H o G has mea square cordial labeligthe we ca say the sub graph H is upper approximatio H mea square cordial labelig o G. 3. Mai results Theorem: 3.1 Path graph P, admits mea square cordial labelig. Proo:Let G = (V,E) be a path graph P.Let v 1,v,..., v be the vertices o a path graph P Cosider V(G) = {v i:1 i } ad E(G) ={(v iv i+1):1 i -1]} Deie :V(G) {0,1} Case(i) : I is odd (v i) = 1 0,1 i 3 1, i The iduced edge labelig are as ollows (v iv i+1) 1 0,1 i 1 1, i 1 Case(ii) : I is eve (v i) = 0,1 i 1, 1 i
Natioal Coerece o Mathematical Techiques ad its Applicatios (NCMTA 18) IOP Publishig IOP Co. Series: Joural o Physics: Co. Series 134567890 1000 (018) 01040 doi :10.1088/174-6596/1000/1/01040 The iduced edge labelig are as ollows (v iv i+1) 0,1 i 1, 1 i 1 I the view o abovelabelig patter, we have v ( 0) v (1) 1 ad e ( 0) e (1) 1.. Hece path graph P admits mea square cordial labelig. Illustratio 3.1: Mea square cordial labelig o a path graph P 6 o eve ad P 5 o odd verticesshow i the Figure 1adFigure Figure 1 Figure Theorem: 3.Comb graph P ʘK 1, admits mea square cordial labelig. Proo:Let G = (V,E) be a comb graph P ʘK 1. Cosider V(G) = {v i,u i:1 i } ad E(G) ={[(u iu i+1):1 i -1] [(v iv i+1):1 i -1]} Deie :V(G) {0,1} (u i) = 0, 1 i (v i) = 1, 1 i The iduced edge labelig are as ollows (u iu i+1) = 0, 1 i -1 ((u iv i) =1, 1 i I the view o abovelabelig patter, we have v ( 0) v (1) 1 ad e ( 0) e (1) 1.. Hece comb graph P ʘK 1admits mea square cordial labelig. Figure 3 Illustratio 3.:Mea square cordial labelig o a comb graphp 6ʘK 1 show i thefigure 3 3
Natioal Coerece o Mathematical Techiques ad its Applicatios (NCMTA 18) IOP Publishig IOP Co. Series: Joural o Physics: Co. Series 134567890 1000 (018) 01040 doi :10.1088/174-6596/1000/1/01040 Remark 3..1 -Cetipede graph also admits mea square cordial labelig by the above labelig patter. Illustratio 3..:Mea square cordial labelig o a 5-cetipede graph show i thefigure 4 Figure 4 Theorem: 3.3Cetipede graph (,), admits mea square cordial labelig. Proo:Let G = (V,E) be a Cetipede graph (,),. Cosider V(G) = {v i,u i,w i1 i } ad E(G) ={[(u iv i):1 i -1] [(v iv i+1):1 i -1] Deie :V(G) {0,1} by Case(i) : I is odd 1 (u i) = 3 0, i, i 1,,..., 1 (v i) =(w i) = 1 0, i, i 1,,..., [(v iw i):1 i ]} The iduced edge labelig are as ollows (u iv i) = 1 3 0, i, i 1,,..., (v iv i+1) = 1 1 0, i 1, i 1,,..., (v iw i) = 1 1 0, i, i 1,,..., 4
Natioal Coerece o Mathematical Techiques ad its Applicatios (NCMTA 18) IOP Publishig IOP Co. Series: Joural o Physics: Co. Series 134567890 1000 (018) 01040 doi :10.1088/174-6596/1000/1/01040 Case(ii) : I is eve (v i) = (v i) = (w i) = 0,1 i 1, 1 i The iduced edge labelig are as ollows (u iv i) = (v iw i) = 0, 1 i, i 1,,..., (v iv i+1) = 1 0, 1 i 1, i 1,,..., (v iw i) 1 = 1 0, i, i 1,,..., I the view o abovelabelig patter, we have v ( 0) v (1) 1 ad e ( 0) e (1) 1.. Hece Cetipede graph (,), admits mea square cordial labelig. Illustratio 3.4:Mea square cordial labelig o acetipede graph (6,) o eve ad acetipede graph (7,) odd vertices show i thefigure 5 ad Figure 6 Figure 5 5
Natioal Coerece o Mathematical Techiques ad its Applicatios (NCMTA 18) IOP Publishig IOP Co. Series: Joural o Physics: Co. Series 134567890 1000 (018) 01040 doi :10.1088/174-6596/1000/1/01040 Figure 6 admits mea square cordial labelig.proo:let G = (V,E) be a star Theorem: 3.4Star graphs, graph. Let v be the apex vertex, v 1, v,...,v be the pedet vertices Cosider V(G) = {(v, v 1,v,..., v ):1 i } ad E(G) ={[(vv i):1 i ]} Deie :V(G) {0,1} by (v) = 0, 1 i (v i) = 1, i 1 mod 0,i 0 mod, The iduced edge labelig are as ollows (uv i) = 1, i 1 mod 0,i 0 mod I the view o abovelabelig patter, we have v ( 0) v (1) 1 ad e ( 0) e (1) 1. Hece star graphs admits mea square cordial labelig. Illustratio 3.4Mea square cordial labelig o astar graph S 5show i thefigure7 Figure 7 4.Mea square cordial labelig o a rough approximatio or the above acyclic graphs : Theorem 4.1: Ay sub graph o a path graph P, has upper approximatio H mea square cordial labelig o G. 6
Natioal Coerece o Mathematical Techiques ad its Applicatios (NCMTA 18) IOP Publishig IOP Co. Series: Joural o Physics: Co. Series 134567890 1000 (018) 01040 doi :10.1088/174-6596/1000/1/01040 Proo:Let the path graph be G ie.,v(g) = {v 1,v,v 3,.v } ad H be ay sub graph o G ie.,{v 1,v,v 3,.v k} V(H),. The the upper approximatio o V(H) is = {v 1,v,v 3,.v k,v k+1} is also a path graph ad by theorem 3.1,the upperapproximatio is a path graph admits mea square cordial labelighece the upper approximatio o ay sub graph H o a path graph admits H measquare cordial labelig o G. has upper approximatio H mea square Theorem 4.:Ay sub graph o a comb graph P ʘK 1, cordiallabelig o G. Proo:Let the comb graph be G ie.,v(g) = {u 1,u,u 3,.u,v,v 3,.v } ad H be ay subgraph o G.Based o the sub graph o G we have some cases o upper approximatio ad itslabelig patter which is give below Case (i):cosider the sub graph H as the pedet vertices o G, ie., {v 1,v,v 3,.v k} V(H). The the upper approximatio o V(H) is = { {u 1,u,u 3,.u k,v 1,v,v 3,.v k } is a comb graph ad by theorem 3.,the upper approximatio admits mea square cordial labelig. Case (ii): Cosider the sub graph H as the vertices o the pathie., { u 1,u,u 3,.u k,v 1,v,.v k } V(H). The the upper approximatio o V(H) is ={ u 1,u,u 3,.u k,u k+1,v 1,v 3,.,v k+1,v k } is a comb graph ad by theorem 3., the upper approximatio is a comb graph admits mea square cordial labelig. Case (iii): Cosider the subgraph as the vertices o the path,{ u 1,u,u 3,.u k } V(H).The the upper approximatio o V(H) is = { u 1,u,u 3,.u k,u k+1,v 1,v,.v k,v k+1}, a comb graph ad bytheorem 3.,the upper approximatio admits mea square cordial labelig. Hece the upper approximatio o ay sub graph H o a comb graph admits mea square cordial labellig. Theorem 4.3:Ay sub graph o a cetipede graph (,), has upper approximatio H mea squarecordiallabellig o G. Proo:Let the cetipede graph (,) be G. ie.,v(g) = {u 1,u,u 3,.u,v,v 3, v,w 1,. w,} ad H be ay sub graph o G.Based o the sub graph o G we have some cases o upper approximatio ad its labelig patter which is give below Case (i)cosider the sub graph H as the vertices o the path, ie., {v 1,v,v 3,.v k} V(H)The the upper approximatio o V(H) is = { u 1,u,u 3,.u k,u k+1,v 1,v,..v k+1,w 1,w,.w k+1} } is a path graph ad bytheorem 3.4the upper approximatio admits mea square cordial labelig.case (ii)cosider the sub graph H as the pedet vertices ie., { u 1,u,u 3,.u k } V(H).The the upper approximatio o V(H) is ={ u 1,u,u 3,.u k,u k+1,v 1,v,v 3,.,v k,v k+1 } is a (k+1)-cetipede graph ad byremark o theorem 3. admits mea square cordial labeligcase (iii)cosider the sub graph H as aother set o pedet vertices i G, ie., {w 1,w,.w k } V(H)The the upper approximatio o V(H) = { v 1,v,v 3,.,v k,v k+1,w 1,w,.,w k,w k+1 }is a comb graph admits mea square cordial labelig.case (iv)cosider the sub graph H as the vertices o the path, ie., {u 1,u,u 3,.u k v 1,v,v 3,.v k} V(H).The the upper approximatio o V(H) is = { u 1,u,u 3,.u k,u k+1,v 1,v,..v k+1,w 1,w,.w k+1} ad by the labelig patter o case (i),we ca say that upper approximatio admits mea square cordiallabelig.case (v) Cosider the sub graph H as the 7
Natioal Coerece o Mathematical Techiques ad its Applicatios (NCMTA 18) IOP Publishig IOP Co. Series: Joural o Physics: Co. Series 134567890 1000 (018) 01040 doi :10.1088/174-6596/1000/1/01040 vertices{v 1,v, v k,w 1,..w k} V(H)The the upper approximatio o V(H) is = { u 1,u,u 3,.u k,u k+1,v 1,v,..v k+1,w 1,w,.w k+1} ad by the labelig patter o case (i),we ca say that upper approximatio admits mea square cordial labelig.hece the upper approximatio o ay sub graph H o a cetipede graph(,) admitsh mea square cordial labelig o G Theorem 4.4: Ay sub graph o a star graph S, has upper approximatio H mea square cordial labelig o G. Proo:Let the star graph be G ie.,v(g) = {v,v 1,v,v 3,.v } ad H be ay sub graph o G. Based o the sub graph o G we have some cases o upper approximatio ad its labelig patter which is give below.. Case (i) :Let the sub graph be the set o all pedet vertices,ie., {v 1,v,v 3,.v k} V(H)The the upper approximatio o V(H) is = {v,v 1,v,v 3,.v k } is also a star graph ad by theorem 3.4 the upperapproximatio is a star graph admits mea square cordial labelig.case (ii) :Let the sub graph be the apex vertex,ie., {v} V(H)The the upper approximatio o V(H) is The the upper approximatio o V(H) is = {v,v 1,v,v 3,.v k } is also a star graphad by theorem 3.4,the upper approximatio is a star graph admits mea square cordial labelig o G. Hece the upper approximatio o ay sub graph H o a star graph admits H mea square cordial labelig o G 5.Cocludig RemarksThe ivestigatio olabelled graph isvery importat due toitsvarious applicatios i diverse ields. It is very iterestig to study the various types o graphs which admitsmea square cordial labelig.the above proved results o mea square cordial labelig ad also its rough approximatios o some acyclic graphs are demostrated by meas o illustratios which is helpul or your better uderstadig. It is a ope area o research to discuss some more similar results or various graphs. Reereces [1] Gallia 011, A Dyamic Survey o Graph Labelig, Electroic Joural o Combiatorics, Vol. 18, pp. 1-19 [] Harary Graph Theory Narosa Publishig House, New Delhi, [3] Pawlak Rough sets Iteratioal joural o computer ad iormatio sciece.11(198),341-356 [4] Cahit Cordial Graphs(1987) A Weaker Versio o Graceul ad Harmoious Graphs ArsCombiatoria, Vol. 3, No. 3, pp. 01-07. [5] Poraj, M.Sivkumar ad M.Sudaram(01), Mea cordial labelig o graphs,ope joural o Discrete Mathematics, (4), 145-14 [6] NellaiMuruga, Heerajoh(01) Special Class o Mea Square Cordial Graphs,Iteratioal Joural o Applied Research ; 1(11): 18-131 [7] NellaiMuruga, S.HeerajohTree Related Mea Square Cordial Graphs,outreach IX 016 16-131,A multidiscipliary reereed joural [8] NellaiMuruga, S.HeerajohCycle related o Mea Square Cordial Graphs, Iteratioal Joural o research ad developmet orgaizatio. [9] S.Dhaalakshmi ad N.Parvathi 017, Eve Graceul Labelig o RoughApproximatios For P ad Star Related Graphs Iteratioal Joural o Pure ad Applied Mathematics, 114 [10] Dhaalakshmi ad N.Parvathi(017) Prime labelig o Rough approximatios or some special graphs, Idia joural o sciece ad techology, 10 8
Natioal Coerece o Mathematical Techiques ad its Applicatios (NCMTA 18) IOP Publishig IOP Co. Series: Joural o Physics: Co. Series 134567890 1000 (018) 01040 doi :10.1088/174-6596/1000/1/01040 [11] Dhaalakshmi ad N.Parvathi 016, Lower ad upper approximatio H-graceul or some classes o graphs, Global joural o pure ad applied mathematics,issn 0973-1768 1 9