AT MOST EDGE 3 - SUM CORDIAL LABELING FOR SOME GRAPHS THE STANDARD

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1 Iteratoal Joural o Research Egeerg ad Appled Sceces IJREAS) Avalable ole at Vol. x Issue x, July 6, pp. 86~96 ISSNO): , ISSNP) : Impact Factor: Thomso Reuters ID: L AT MOST EDGE 3 - SUM CORDIAL LABELING FOR SOME GRAPHS THE STANDARD S. Pethaach Selvam* S. Padmash** Artcle hstory: Receved July 9 th, 6; Approved July 5 th, 6; Avalable ole: Aug 5 th, 6. Keywords: Edge 3-sum, At most edge 3-sum cordal graph. Abstract Let G V, E) be a graph o order ad sze m. A edge labelg ucto : E {,,} duces a vertex labelg ucto : V {,, } m e, where e E ad as, v ) ) mod3) each,,,..., ). deed e s cdet to v, or The the map s called Edge 3-sum cord al labelg v ) v ad e ) e or ad, {,, }, where v x) ad x) e deote the umber o vertces ad edges label ed wth x, x {,,}.A edge 3-sum cordal graph whch admts a edge sum cordal labelg s called at most edge 3-sum cordal labelg. A graph havg at most edge 3-sum cordal labelg s called at most edge 3-sum cordal graph. I ths paper, we prove that some stadard graphs ad specal graphs lke Gear graph, Double star ad Fredshp graph are at most edge 3-sum cordal graphs. Author Correspodece S. Pethaach Selvam* Dept. o Mathematcs, The Stadard Freworks Raaratam College or Wome, Svakas Tamladu) Copyrght 6 The Author. Publshed by Iteratoal Joural o Egeerg ad Appled Scece. Ths s a ope access artcle uder the All rghts reserved.. Itroducto Throughout ths paper we cosder te, smple ad udrected graphs. Let G V, E) be a graph, where V ad E respectvely deote the vertex set ad edge set o G. Also, or Iteratoal Joural o Research Egeerg & Appled Sceces A ope access scholarly, ole, peer-revewed, terdscplary, mothly, ad ully reereed ourals 86

2 Iteratoal Joural o Research Egeerg ad Appled Sceces IJREAS) Vol. 6 Issue 7, July 6 ISSNO): , ISSNP) : Impact Factor: a graph G, p ad q deote the umber o vertces ad edges respectvely. I the vertces or edges or both o the graphs are assged values subect to certa codtos, t s kow as graph labelg. A detaled study o varety o applcatos o graph labelg s carred out Bloom ad Golomb[]. Cordal graphs were rst troduced by Caht[] as a weaker verso o both graceul ad harmoous graphs. The product cordal labelg ad k-product cordal labelg o graphs was troduced by Pora.R, Svakumar.M ad Sudaram.M [4,6], whch they proved that some stadard ad specal graphs admt k-product cordal labelg. The cocept o sum cordal labelg was troduced by J.Shama [5] whch she proved that some graphs lke path P, cyclec, star K,,etc., are sum cordal graphs. Motvated by these papers, we dee edge 3-sum ad at most edge 3-sum cordal labelg o G.. Edge Sum ad At most Edge 3-sum Cordal Labelg Let G V, E) be a graph o order ad sze m. A edge labelg ucto : E {,} duces a vertex labelg ucto : V {, } deed as, m v ) e ) mod ), where e E ad e s cdet to v, or each,,,..., ). The the map s called Edge sum cordal labelg v ) v ad e ) e or ad, {, }, where v x) ad x) deote the umber o vertces ad edges labeled wth x, x {, }. e Dept. o Mathematcs,The Stadard Freworks Raaratam College or Wome, Svakas Tamladu)* Dept. o Mathematcs, Sr S.R.N.M College, Sattur Tamladu)** Iteratoal Joural o Research Egeerg & Appled Sceces A ope access scholarly, ole, peer-revewed, terdscplary, mothly, ad ully reereed ourals 87

3 Iteratoal Joural o Research Egeerg ad Appled Sceces IJREAS) Vol. 6 Issue 7, July 6 ISSNO): , ISSNP) : Impact Factor: Let G V, E) be a graph o order ad sze m. A edge labelg ucto : E {,,} duces a vertex labelg ucto : V {,, } deed as, m v ) e ) mod 3), where e E ad e s cdet to v, or each,,,..., ). The the map s called Edge 3-sum cordal labelg v ) v ad e ) e or ad, {,, }, where v x) ad x) deote the umber o vertces ad edges labeled wth x, x {,, }. A edge 3-sum cordal graph whch admts a edge sum cordal labelg s called at most edge 3-sum cordal labelg. A graph havg at most edge 3-sum cordal labelg s called at most edge 3-sum cordal graph. Note: Every at most edge 3-sum cordal graph s edge 3-sum cordal graph. But the coverse eed e ot be true. For example, cosder the 5-pa graph. It s a edge 3-sum cordal graph. But t s ot at most edge 3-sum cordal graph, as t does ot admt edge sum cordal labelg. 3. Ma Results We prove that some stadard graphs P, C ad specal graphs lke Gear graph G, double star ad Fredshp graph F are at most Edge 3-sum cordal graphs. K,, Iteratoal Joural o Research Egeerg & Appled Sceces A ope access scholarly, ole, peer-revewed, terdscplary, mothly, ad ully reereed ourals 88

4 Iteratoal Joural o Research Egeerg ad Appled Sceces IJREAS) Vol. 6 Issue 7, July 6 ISSNO): , ISSNP) : Impact Factor: Theorem 3. Ay cycle C, 3) s at most Edge 3-sum cordal graph. Proo Let G C be a cycle. LetV C ) v /, E C ) e /. We kow that ay cycle C s a edge sum cordal graph. Dee : E,, Case ):. Cosder the ollowg three cases. Let mod3) ad mod3). The 3k ad 3k, where k Assg the edge labelg as ollows. For each,,,..., k, we have e 3 ), e ) ad e ). Case ): Let mod3), where s odd. The 3k, where k s odd 3 For each,6,,...,3 k ), assg edge labelg as ollows: 3 ), ), ), 3), 4) ad 5). e Case ): e e Let mod3), where s eve. The 3k, where k s eve Fx e ), e ). For,6,,...,3 k ), assg the edge labelg as ollows. e ), ), ), 3), 4) ad 5). e e From the above cases, we have e e e e e e v ) v & e ) e, ad, {,, } ad hece the theorem. Illustrato: Iteratoal Joural o Research Egeerg & Appled Sceces A ope access scholarly, ole, peer-revewed, terdscplary, mothly, ad ully reereed ourals 89

5 Iteratoal Joural o Research Egeerg ad Appled Sceces IJREAS) Vol. 6 Issue 7, July 6 ISSNO): , ISSNP) : Impact Factor: Theorem 3. Proo: Ay path, 3) s at most edge 3-sum cordal graph. P Let G P be a graph wth V P ) v / ad E P ) e /. We kow that ay path P 3) Dee : E,,. We cosder the ollowg three cases. Case ): Let mod3) ad mod3) s a edge sum cordal graph. e, 3k ad 3k where k. For each,,,..., k ), assg e3 ), e3 ), e ) 3. Iteratoal Joural o Research Egeerg & Appled Sceces A ope access scholarly, ole, peer-revewed, terdscplary, mothly, ad ully reereed ourals 9

6 Iteratoal Joural o Research Egeerg ad Appled Sceces IJREAS) Vol. 6 Issue 7, July 6 ISSNO): , ISSNP) : Impact Factor: Case ): Let mod3) where s odd. e, 3k where k s odd. Fx ) ad or each,,,..., k ), assg the edge labelg as ollows. e e 3 ), Case ): e ) 3, e ) 3. Let mod3), where s eve. e, = 3k +, whe k s eve Fx ). Assg the edge labelg as ollows. e e3 ), e3 ), e3 ), or each,,,..., k ), From all the above cases, we have v ) v Illustrato: & e ) e or ad,,,. P 3 : Iteratoal Joural o Research Egeerg & Appled Sceces A ope access scholarly, ole, peer-revewed, terdscplary, mothly, ad ully reereed ourals 9

7 Iteratoal Joural o Research Egeerg ad Appled Sceces IJREAS) Vol. 6 Issue 7, July 6 ISSNO): , ISSNP) : Impact Factor: Theorem 3.3 Proo: Ay Gear graph G, where 3s at most edge 3-sum cordal graph. Let G be a gear graph havg p vertces ad q 3 edges. G Let e, e,..., e be the er edges ad e v,e v,...,e3 v be the outer edges. : E {,} duce vertex labelg ucto or the gear graph whch s cordal. Clearly, G s a edge sum cardal graph. Dee : E,, Cosder the ollowg cases. Case ): Let mod3). For each,3,6,...,3 ), we ca assg labelg to the er edges as e ), e ), e ) ad cotug the labelg to oe o the outer edges whch s adacet to the last er edge. Case ): Let mod3). For each =, 3, 6,.., 3 ), we assg labelg as ollows. Case ): e), e ), e ) ad cotue labelg as case ) Whe mod3), let us start labelg rom the er edges. For each,3,6,...,3 ), assg e ), e ), e ) ad cotug the labelg to oe o the outer edges whch s adacet to the last er edge.from, all the above cases, we have v ) v ad e ) e )} or ad,,,. Thus G admts at most edge 3-sum cordal Iteratoal Joural o Research Egeerg & Appled Sceces A ope access scholarly, ole, peer-revewed, terdscplary, mothly, ad ully reereed ourals 9

8 Iteratoal Joural o Research Egeerg ad Appled Sceces IJREAS) Vol. 6 Issue 7, July 6 ISSNO): , ISSNP) : Impact Factor: labelg. Illusrato: G 4 G 5 Theorem 3.4 G 3 Ay double star K,, s at most edge 3-sum cordal graph. Proo: Let G K,,, ) be a graph. Let e, e,..., e be the rst layer edges whch each edge havg degree. e v, e v,..., e v be the secod layer edges o the double star, where e s ot adacet to e. The V & E. : E {, } K,, duces vertex labelg or the double star graph whch s cordal. Clearly, K,, s a edge sum cordal graph. Dee : E,, Case ): Let mod3). The. We have two cases. 3k where k. Assg labelg rom the rst layer edges. For each,3,6,...,3k ), we have e ), e ), e ) ad cotue the labelg to the secod layer edges also. Iteratoal Joural o Research Egeerg & Appled Sceces A ope access scholarly, ole, peer-revewed, terdscplary, mothly, ad ully reereed ourals 93

9 Iteratoal Joural o Research Egeerg ad Appled Sceces IJREAS) Vol. 6 Issue 7, July 6 ISSNO): , ISSNP) : Impact Factor: Case ): Let mod3). The 3k, where k Let us assg labelg to the rst edges as ollows. e ), e ), e ) or each,3,6,...,6k), ad cotue labelg to the secod layer edges as case ). Case ): Let mod3).e., 3k, where k. For each =,3,6,,6k), assg labelg as case ). From the above cases, we have v ) v e ) e or graph. ad,,, ad hece K,, s at most edge 3-sum cordal Illustrato: K,3,3 K,4,4 K,5,5 Theorem 3.5 The redshp graph Fl 3) s at most edge 3-sum cordal graph. Proo: Let G Fl be a redshp graph. Iteratoal Joural o Research Egeerg & Appled Sceces A ope access scholarly, ole, peer-revewed, terdscplary, mothly, ad ully reereed ourals 94

10 Iteratoal Joural o Research Egeerg ad Appled Sceces IJREAS) Vol. 6 Issue 7, July 6 ISSNO): , ISSNP) : Impact Factor: Let e, e,..., e be the er edges ad e v, e v, e v3,..., e3 v be the outer edges. The V & E 3. : E {, } duces vertex labelg or the redshp graph whch s cordal. Clearly, Fs Dee : E,,. edge sum cordal graph. We prove the ollowg cases. Case ): Let mod 3) & mod 3) Start edge labelg rom the er edges by e ), e ), e ), or each,3,6,9,...,3 ). The outer edges are labeled by the remag umber whch s ot labeled each lea. Case ): Whe mod3), start the edge labelg rom the er edges by e ), e ), e ), or,3,6,9,...,3 ). The cotue labelg as case ). From the above two cases, we have v ) v, e ) e or ad,,,. Illustrato: F 3 F4 F 7 Iteratoal Joural o Research Egeerg & Appled Sceces A ope access scholarly, ole, peer-revewed, terdscplary, mothly, ad ully reereed ourals 95

11 Iteratoal Joural o Research Egeerg ad Appled Sceces IJREAS) Vol. 6 Issue 7, July 6 ISSNO): , ISSNP) : Impact Factor: Reereces [] Bloom.G.S, Golomb.S.W., 977, Applcatos o umbered udrected graph, Proceedg o IEEE, 654), pp [] Caht.I., 987, Cordal Graphs: A weaker verso o graceul ad harmoous graphs, Ars Combatora, Vol.3, pp. 7. [3] Frak Harrary.,, Graph Theory, Narosa Publshg House. [4] Pora.R, Svakumar.M ad Sudaram.M.,, k-product cordal labelg o graphs, It. J. Cotemp. Math. Sceces, 7 5), [5] Shama.J.,, Sum cordal labelg or some graphs, IJMA-39). [6] Sudara.m Pora.M ad Soma Sudaram.S., 4, Product cordal labelg o graphs, Bull Pure ad Appled Sceces Mathematcs ad Statstc), Vol.3E, pp Vadya, Da N.A,, Some ew product cordal graphs, Joural o App. Comp. Sc. Math., 8 4), pp AT MOST EDGE 3 - SUM CORDIAL LABELING FOR SOME GRAPHS The Stadard S. Pethaach Selvam* S. Padmash** Iteratoal Joural o Research Egeerg & Appled Sceces A ope access scholarly, ole, peer-revewed, terdscplary, mothly, ad ully reereed ourals 96