Further Results on Pair Sum Graphs
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1 Applid Mathmatis, 0,, Publishd Oli Marh 0 ( Furthr Rsults o Pair Sum Graphs Raja Poraj, Jyaraj Vijaya Xavir Parthipa, Rukhmoi Kala Dpartmt of Mathmatis, Sri Paramakalyai Collg, Alwarkurihi, Idia Dpartmt of Mathmatis, St. Joh s Collg, Palayamottai, Idia Dpartmt of Mathmatis, Maomaiam Sudaraar Uivrsity, Tirulvli, Idia porajmaths@idiatims.om, parthi68@rdiffmail.om, karthipyi9@yahoo.o.i Rivd Jauary, 0; rvisd Fbruary 9, 0; aptd Fbruary 6, 0 ABSTRACT Lt G b a pq, graph. A ijtiv map f : V G,,, p is alld a pair sum lablig if th idud dg futio, f : EG Z 0 dfid f uv is o-o ad f EG is ithr of th form k, k,, kq/ or k, k,, k( q)/ k( q)/ aordig as q is v or odd. A graph with a pair sum lablig is alld a pair sum graph. I this papr w ivstigat th pair sum lablig bhavior of subdivisio of som stadard graphs. Kywords: Path; Cyl; Laddr; Triagular Sak; Quadrilatral Sak. Itrodutio Th graphs osidrd hr will b fiit, udirtd ad simpl. V G ad EG will dot th vrtx st ad dg st of a graph G. Th ardiality of th vrtx st of a graph G is dotd p ad th ardiality of its dg st is dotd q. Th oroa GG of two graphs G ad G is dfid as th graph obtaid takig o opy of G (with p vrtis) ad p opis of G ad th joiig th ith vrtx of to G all th vrtis i th ith opy of G. If = uv is a dg of G ad w is a vrtx ot i G th is said to b subdividd wh it is rplad th dgs uw ad wv. Th graph obtaid subdividig ah dg of a graph G is alld th subdivisio graph of G ad it is dotd SG. Th graph P P is alld th laddr. A drago is a graph formd joiig a d vrtx of a path P m to a vrtx of th yl C. It is dotd as. C@ Pm Th triagular sak T is obtaid from th path P rplaig vry dg of a path a triagl C. Th quadrilatral sak Q is obtaid from th path P vry dg of a path is rplad a yl C. Th 4 opt of pair sum lablig has b itrodud i []. Th Pair sum lablig bhavior of som stadard graphs lik omplt graph, yl, path, bistar, ad som mor stadard graphs ar ivstigatd i [-]. That all th trs of ordr 9 ar pair sum hav b provd i [4]. Trms ot dfid hr ar usd i th ss of Harary [5]. Lt x b ay ral umbr. Th x stads for th largst itgr lss tha or qual to x ad x stads for th smallst itgr gratr tha or qual to x. Hr w ivstigat th pair sum lablig bhavior of SG, for som stadard graphs G.. Pair Sum Lablig Dfiitio.. Lt G b a pq, graph. A ijtiv map f : V G,,, p is alld a pair sum lablig if th idud dg futio, f : EG Z 0 dfid f uv f E G is ithr of th form is o-o ad or k, k,, kq / k, k,, k( q)/ k( q)/ aordig as q is v or odd. A graph with a pair sum lablig dfid o it is alld a pair sum graph. Thorm. []. Ay path is a pair sum graph. Thorm. []. Ay yl is a pair sum graph.. O Stadard Graphs Hr w ivstigat pair sum lablig bhavior of C@ P m ad K K. Thorm.. If is v, C@ P m is a pair sum graph. Proof. Lt C b th yl uuu uu ad lt Pm b th path vv v m. m 0 mod4 Cas. Copyright 0 SiRs.
2 68 R. PONRAJ ET AL. Dfi m f : V P,,, m i,i m m i i,i m 4 m i i,i m 4 m i i m i,i im i,i. 7,, m, 5,, m m 4, m 8,, m 4 m 4, m 8,, m 4,. f E C P m Thrfor f is a pair sum lablig. Cas. m mod4 Dfi m f : V P,,, m i,i m 4 m i i,i m 4 m i i,i m m i i m i,i i m i,i Hr Figur. A pair sum lablig of P. 8 5, 7,, m, m m, 0,, f E C P m,5, 7,, m m m m, m0,, m ( ),. H f is a pair sum lablig. Cas. m mod4 Labl th vrtx ui i, vi i m as i Cas. Th labl m to vm. Cas 4. m mod4 Assig th labl m to v m ad assig th labl to th rmaiig vrtis as i Cas. Illustratio. A pair sum lablig of P 8 9 is show i Figur. Thorm.. K K is pair sum graph if is v. Proof: Lt u, u,, u b th vrtis of K ad u, v, w, z. b th vrtis i K. Lt V K K V K V K ad EK,,,,, : K uv wz uui vui wui zui i. Dfi f : V K K,,, 4 i i, i i i,i,, f z Hr f E K K,, 5,,,, 5,,,, 5,,,, 5,, 4, 8,,, 4, 8,,,,,,,,,,,,. Copyright 0 SiRs.
3 R. PONRAJ ET AL. 69 Thrfor f is a pair sum lablig. Illustratio. A pair sum lablig of K8 K is show i Figur. 4. O Subdivisio Graph Hr w ivstigat th pair sum lablig bhavior of SG for som stadard graphs G. Thorm 4.. SL is a pair sum graph, whr L is a laddr o vrtis. Proof. Lt Lt i, i, i, j, j :, V S L u v w a b i j i i, i i : E S L u w wv i ua au vbbv :i. i i, i i, i i, i i Cas : is v. Wh =, th proof follows from th Thorm.. For >, Dfi f : VSL,,,, 5, i0, i 0, 5, 5 i0, i, i0, i0, i0 5, i0, i0, i i i i i i i 0i, i i i i i f a f a i i f a i i f b f b i i f b 0i 5, i. Wh = 4,, 4,5,8,0,6, 0, 4, 8, 4, 5, 8, 0, 6, 0, 4, 8. f E S L For > 4, f Figur. A pair sum lablig of K +K. 8 ESL f ESL 4 6,6,40,44,48,8, 6, 6, 40, 44, 48, 8, 46,56,60,64,68,58, 46, 56, 60, 64, 68, 58,, 04,04,00, 06,0,0, 04, 04, 00, 06, 0, 0. Thrfor f is a pair sum lablig. Cas. is odd. Clarly SL P ad h graph Thorm.. For >, Dfi SL is a pair sum 5 f : V S L,,, f a 6,, 9 0 0, i i i f a i i 0i 0, i 6 0 i, i Copyright 0 SiRs.
4 70 R. PONRAJ ET AL. 60 i,i i 0i, i i 0i, i i, f v 0 0, f b 6 f b 4, f w 4 8, f w 8 f a 0i, i i f a 0i, i i f b 0i4, i i f b 0i4, i. i ad Thrfor f E S L,, 4, 6, 9,8, 0,,, 4, 6, 9, 8, 0 5 f E S L f E S L 4,0,4,8,4,6, 4, 0, 4, 8, 4, 6 wh > 5, f ESL f ESL5 40,50,54,58,6,5, 40, 50, 54, 58, 6, 5, 60,70,74,78,8,7, 60, 70, 74, 78, 8, 7,, 00,00,06, 0,08,08, 00, 00, 06, 0, 08, 08. Th f is a pair sum lablig. Illustratio. A pair sum lablig of 7 i Figur. Thorm 4.. Proof. Lt S C K is a pair sum graph S L is show ui :i wi, vi :i V S C K Lt Figur. A pair sum lablig of i i: u w i vw i E S C K uu i : :. i i i i Cas. is v. Dfi f : S C K,,, 4 i i,i i,i i, i i i i i,i, i i i i i,i Hr 7 S L. 4,8,,, 4 4 4, 8,,, 4 4,8,4,,54, 8, 4,, 5 4 5,5 6,5 0,,7 f E 5, 5 6, 5 0,, 7. Th f is pair sum lablig. Cas. is odd. Dfi Copyright 0 SiRs.
5 R. PONRAJ ET AL. 7 Hr, 4 f : V S C K,, 4, i i i i 4i,i, i i i i,i i i i,i i,i i f, 4,, 7. E S C K 8,8 6, 4 0, , 86,, 4 0, 46,,, 6,,,, 6,,, 4,, 7 Th f is pair sum lablig. Illustratio 4. A pair sum lablig of SC7K is show i Figur 4. Figur 4. A pair sum lablig of S CK 7. Lt Thorm 4.. S PK is a pair sum graph. Proof. Lt V S P K u :i w v :i i i, i i i: E S P K uu i u w : i vw : i. i i i i Cas. is v. Wh =, th proof follows from Thorm.. For >, Dfi f : VSP K,,, 4, Hr i i i 5i,i (5i), i 5i,i i 5i,i 4, 5 i i 6, 6 i i i i 5i 4, i 5i 4, i 5 5, 5i 5, i 4,,, 9,5,7,9,,, 9, 5, 7, 9. f E S PK For > 4, 4 0, 5, 7, 9 0, 5, 7, ,5 5,5,5 f E S P K f E S PK 0, 5, 7, 9 0, 5,, 9,, 50, 55, 5, 5. Th f is pair sum lablig. Cas. is odd. Si SPK P, whih is a pair sum graph Thorm.. For >, Dfi Copyright 0 SiRs.
6 7 R. PONRAJ ET AL. 4 f : V S PK,,,, 8 8 0i, i i 5i5,i i 0i, i i i 5i5,i, f w 5 6 5i7,i i i 5i7,i 9, 9 Figur 5. A pair sum lablig of Thorm 4.4. ST 5i8,i i i 0i6, i i 0i4, i 5i8,i i i 0i6, i Hr i 0i4, i f ESPK,,,,,, 4, 4,5, 5, 5 fespk 5 fespk 4, 5 8,,, 5, 8,,, 5 i 0i, i Wh > 5, i 0i, i fespk 6,,,5 i 0i, i 6,,, 56,4,4,45 i 0i, i 6, 4, 4, 45,, 59,54,5,5, 59, 54, 5, 5. i 0 i, i Th f is pair sum lablig. i 0 i, i Illustratio 5. A pair sum lablig of SPK 8 is show i Figur 5. For = 4, 8 S PK. is a pair sum graph whr is a triagular sak with triagl. Proof. Lt V ST ui :i vi :i wi :i ad E S T uu :i i i viwi, viwi : i u v, u v : i. i i i i Cas. is v. Wh =, Dfi f(u ) = 7, f(u ) = 6, f(u) =, f( u 4 ) = 6, f(u 5 ) = 7, f(v ) = 5, f(v ) =, f(v ) = 4, f(v 4 ) = 5, f(w ) =, f(w ) =. Wh >, Dfi f : V S T,,, 5, 6 7, 6 7 T Copyright 0 SiRs.
7 R. PONRAJ ET AL. 7,5,7,8,,,5,8,,, 6,0 f E S T, 5, 7, 8,,, 5, 8,,, 6, 0. For > 4 fest fest4,8,40, 4, 46,50,, 8, 40, 4, 46, 50, 5, 58, 60, 6, 66, 70, 5, 58, 60, 6, 66, 70,, 08,0,00,08,04,00, 08, 0, 00, 08, 04, 00. Th f is pair sum lablig. Cas. is odd. Clarly ST C6, ad h ST is a pair sum graph Thorm.. For >, Dfi 6 f : V S T,,, 5 C 8,, 5 5, i i i i i i i i i i i i 5, 7 4, 7, i i i i i i i5 i, i i i 4 5, 0 4, 4 5, 4 5 5, 5, 5, i 5, 9 i 8,, i 5 i, i Hr =,,,4,5,7,8,,7,9,, 4, 5, 7 8,, 7, 9. f E S T 5 i, i Figur 6. A pair sum lablig of S T. For >, fest fest,, 4, 5, 8, 9,,, 4, 5, 8, 9, 56,58,59,50,5,54, 56, 58, 59, 50, 5, 54. 4, 4, 44, 45, 48, 49, 4, 4, 44, 45, 48, 49,, Th f is pair sum lablig. Illustratio 6. A pair sum lablig of ST 5 i Figur 6. Thorm 4.5. SQ is a pair sum graph. Proof. Lt i : V S Q u i vi :i wi :i ad ES( Q) uu i i :i ui wi, v iwi :i ui wi, wi vi :i vv i i :i vivi : i. is show Cas. is v. Wh =, Dfi f(u ) =, f(u ) = 6, f(u ) =, f(u 4 ) = 6, f(u 5 ) =, f(w ) = 9, f(w ) =, f(w ) = 4, f(w 4 ) = 9, f(v ) = 7, f(v ) = 5, f(v ) =, f(v 4 ) =, f(v 5 ) = 5, f(v 6 ) = 7. Wh >, Dfi 7 f : V S Q,,,, 6 i 8, 6, 8 4i 8, i i 4i6, i 5 Copyright 0 SiRs.
8 74 R. PONRAJ ET AL. Hr i 4i6, i i 4i8, i, 9 4, 9 i 4i4, i i 4i4, i i 4i4, i i 4i4, i, 5 7, 5, 7 i 4i, i i 4 i, i i 4i, i i 4i, i i i i 4, i 4i, i f E S Q,, 5, 5, 7, 7,8, 8,,,4, 46, 6,7, 7 8,, 6,8,0,4, 40,4 8,, 6, 8, 0, 4, 40, 4 46,50,54,56,58,6,68,70 46, 50, 54, 56, 58, 6, 68, 70 74, 78,8,86,84, 90, 96, 98 74, 78, 8, 86, 84, 90, 96, 98,, 48,44,40,48, 46,4,46,44 48, 44, 40, 48, 46, 4, 46, 44. Th f is pair sum lablig. Cas. is odd. SQ is a pair sum graph follows from Thorm..Wh >. Dfi 7 f : V S Q,,, 6, 7, 8,, 8 4 i 4i0, i i 4i, i i 4i0, i 4i 4i, i 5, 4, 0 5,, 0 i 4i0, i i 8 4 i, i i 4i0, i i 8 4 i, i 6, f v 5 8, f 7 v 7 f 5 v 7 0, 6 4, i i 4 6, 7 i i i i 4i, i 4i 4, i 5 i i 4i, i 5 4i 4, i 7 4i 6, i 9 For =,,,6,8,9,,,5,0,4,8, 4 4 f E S Q,, 6, 8, 9,,, 5, 0,, 8, 4. >, Copyright 0 SiRs.
9 R. PONRAJ ET AL. 75 Figur 7. A pair sum lablig of 4 S Q. i Figur Akwldgmts W thak th rfrs for thir valuabl ommts ad suggstios. fes( Q f ESQ 46,50,54,56,58,6,68,70, 6, 74,78,8,84,86,90,96,98, 4,48,4,4 44, 40, 4 46, 50, 54, 5 58, 6, 68, 70, 74, 78, 8, 84, 86, 90, 96, 98,, 4 4,4 0,4 6,4 4, 6, 4 4, 4, 48, 4, 4. Th f is pair sum lablig Illustratio 7. A pair sum lablig of 4 S Q is show REFERENCES [] R. Poraj ad J. V. X. Parthipa, Pair Sum Lablig of Graphs, Th Joural of Idia Aadmy of Mathmatis, Vol., No., 00, pp [] R. Poraj, J. V. X. Parthipa ad R. Kala, Som Rsults o Pair Sum Lablig, Itratioal Joural of Math- 4, 00, pp matial Combiatoris, Vol. [] R. Poraj, J. V. X. Parthipa ad R. Kala, A Not o Pair Sum Graphs, Joural of Sitifi Rsarh, Vol., No., 0, pp. -9. [4] R. Poraj ad J. V. X. Parthipa, Furthr Rsults o Pair Sum Lablig of Trs, Applid Mathmatis, Vol., No. 0, 0, pp doi:0.46/am [5] F. Harary, Graph Thory, Narosa Publishig Hous, Nw Dlhi, 998. Copyright 0 SiRs.
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