On the Reformulated Zagreb Indices of Certain Nanostructures

Transcription

1 lobal Joural o Pur ad Applid Mathmatics. ISSN Volum, Numbr 07, pp Rsarch Idia Publicatios O th Rormulatd Zagrb Idics o Crtai Naostructurs Krthi. Mirajar ad Priyaa Y.B Dpartmt o Mathmatics, Karata Uivrsity s Karata Arts Collg, Dharwad , Karataa, Idia. Corrspodc Author: Krthi. Mirajar Abstract Th Zagrb idics ar th bst ow ad widly usd topological idics or a molcular graph i chmical graph thory, th irst ad scod rormulatd Zagrb idics o a graph ar obtaid rom th Zagrb idics by rplacig vrtx dgr with dg dgr did as, dg ad dg dg ` E E, whr dg dots th dgr o th dg ad mas that dgs ad ar adjact. I this articl, w giv xplicit ormula or th irst ad scod rormulatd Zagrb idics or th aostructurs. AMS Subjct Classiicatio: 05C99 Kywords: Rormulatd Zagrb idics, Naostar Ddrimr, D-Lattic, Naotub, Naotorus.. INTRODUCTION Lt =V,E b a coctd simpl graph with = V vrtics ad m = E dgs. Th dgr o a vrtx v is dotd as dgv or dv. W ollow[5] or trmiologis ad otatios. A chmical graph or a molcular graph is a graph rlatd to th structur o a chmical compoud. Each vrtx o this graph rprst a atom o th molcul ad covalt bods btw atoms ar rprstd by dgs btw th corrspodig vrtics. I thortical chmistry, th physio chmical proprtis o chmical compouds ar ot modld by th molcular graph basd molcular

2 88 Krthi. Mirajar ad Priyaa Y.B structur dscriptors which ar also rrd to as topological idics[4]. Such amog th varity o thos idics, Zagrb idics ar vry importat ad thy hav promit rol i chmistry, spcially i QSPR ad QSAR study. Th irst Zagrb idx M uals to th sum o suars o th vrtx dgrs ad th scod Zagrb idx M uals to th sum o product o dgrs o pairs o adjact vrtics[4]. M M M d v vv M d u d v uve Proprtis o th Zagrb idics may b oud i th umbr o rports[,,,8,,4,5]. I 004 Miličvič, Niolic ad Triajstić rormulatd th Zagrb idics i trms o dg dgr istad o vrtx dgrs[0]. ad d E d d E Whr dg dotd th dgr o th dg i ad ~ implis that th dgs ad ar adjact ad shar a commo vrtx i. Dgr o a dg is did by dg = dgu + dgv - with = uv ad shar a commo vrtx i. Whr is th irst rormulatd Zagrb idx ad is th scod rormulatd Zagrb idx.. NANOSTRUCTURE I this sctio w comput rormulatd Zagrb idics or a iiit amily o aostar ddrimr D[], D-lattics, aotub ad aotorus o T C [ 4C8 p, ]. Ddrimrs ar hypr-brachd macromolculs with a rigorously tailord architctur ad ar o o th mai objcts o Nao biotchology. Thy ca b sythsizd, i a cotrolld mar, ithr by a divrgt or a covrgt procdur. Ddrimrs hav gaid a wid rag o applicatio i supra-molcular chmistry, particularly i host gust ractios ad sl assmbly procss. Thir applicatios i chmistry, biology ad aoscic ar ulimitd. I this articl w dot th th growth o aostar ddrimr by D[], N. It has VD[] = + umbr o vrtics or atoms bcaus i th structur o ddrimr aostar D[]thr ar vrtics or atoms with dgr, + vrtics o D[] with dgr ad 5 8 vrtics with dgr. Ad also thr ar ED[] = [ ] = 4 + dgs or bods i th Naostar ddrimr

3 O th Rormulatd Zagrb Idics o Crtai Naostructurs 89 D[]. Rctly, th topological idics o som ddrimr aostars hav b ivstigatd i [7,9,]. Figur. Naostar ddrimr D[] From th structur o ddrimr aostar D[] i th th growth o aostar ddrimr i igur, w s that a lmt as igur calld La is addd to D[-]i th th growth o aostar ddrimr. Figur : La o Naostar Ddrimr D[] Not. I D[], th lavs ar coctd with vrtics/atoms N, to costruct th proo w cosidr ctral vrtx/atom as N ad th rmaiig vrtics/atoms as N'. Rctly Nadm t.al.[] ad S.M.Hosamai t.al.[6] obtaid xprssios or crtai topological idics o aostructurs. I this papr w study th D-Lattic, aotub ad aotorus o T C [ 4C8 p, ], whr p ad dot th umbr o suars i a row ad th umbr o rows o suars, rspctivly.

4 80 Krthi. Mirajar ad Priyaa Y.B Figur : a D Lattic, b Naotub, C Naotorus o T C C p, ] I igur, D-lattic, aotub ad aotorus o T C C p, ] ar dpictd. Th ordr ad siz o D-lattic, aotub ad aotorus o T C [ 4C8 p, ] ar 4p ad 6p p, 4p ad 6p, 4p ad 6p rspctivly.. RESULTS Thorm.. Lt b a graph o aostar ddrimr D[], th th irst rormulatd Zagrb idx is, 5 Proo. Lt = D[] b a graph o aostar ddrimr,. W costruct th proo cosidrig th ollowig stps. Stp I. Cosidr th dgr o th dgs i D[] which ar icidt with ctral vrtx N which is 4. Th th irst rormulatd Zagrb idx is 4 =6 =48. Ad th irst rormulatd Zagrb idx o rmaiig dgs icidt with N' is, E N dg 44 Stp II. Cosidr th dgr o th dgs i lavs o D[] which ar ot adjact with dgs icidt with ctral vrtx N is, th th irst rormulatd Zagrb idx is =4. Ad th irst rormulatd Zagrb idx o rmaiig dgs o lavs i D[] which ar adjact to th vrtx N' is, E N dg 4

5 O th Rormulatd Zagrb Idics o Crtai Naostructurs 8 Stp III. Now w cosidr th dgr o th dgs i lavs which ar adjact to th dgs icidt with ctral vrtx N which is, th th irst rormulatd Zagrb idx is, 9 =08. Ad th irst rormulatd Zagrb idx o rmaiig dgs o lavs i D[] which ar adjact to th vrtx N' is, E N dg 08 Stp IV. Cosidr th dgr o th dgs cotaiig pdt vrtx i D[] which is, th th irst rormulatd Zagrb idx is =. From th abov stps I, II, III ad IV, th irst rormulatd Zagrb idx o D[] is, D E N [ ] dg Thorm.. Lt b a graph o aostar ddrimr D[], th th scod rormulatd Zagrb idx is, Proo. Lt = D[] b a graph o aostar ddrimr,. W us th ollowig stps to prov th thorm. Stp I. Lt us cosidr th structur o D[]with ctral vrtx N ad th lavs which ar icidt with ctral vrtx N. Thr ar thr dgs icidt with ach vrtx N. Cosidr th lavs, i ach la, thr ar 4-dgs adjact with dgs o dgr ad -dgs adjact with dgs o dgr. Thror th scod rormulatd Zagrb idx o this structur is, d E d Stp II. Sic thr ar umbr o N' vrtics/atoms cotaiig -pairs o mutually adjact dgs with dgr 4. Th th scod rormulatd Zagrb idx is,

6 8 Krthi. Mirajar ad Priyaa Y.B E ~ dg dg Stp III. Now w cosidr 4 pairs o adjact dgs with dgr ad dgr 4 o all th lavs i D[]. Th th scod rormulatd Zagrb idx is, E ~ ` dg dg Stp IV. Now w cosidr th 4 - pair o adjact dgs o ach la with dgr ad dgr which is tims. Th th scod rormulatd Zagrb idx is, E ~ dg dg Stp V. Fially cosidr th pdt dgs with dgr which ar adjact with dgs o dgr. Sic thr ar pdt dgs i D[], th th scod rormulatd Zagrb idx is, E 6 dg dg ~ From th abov stps I, II, III, IV ad V, th scod rormulatd Zagrb idx o D[] is, dg dg ] [ ~ E D Thorm.. Lt b a graph o D-lattic o ], 8 [ 4 p C C T th th irst rormulatd Zagrb idx is, p p Proo. Lt b D-lattic o ], 8 [ 4 p C C T graph with 6p p - umbr o dgs.

7 O th Rormulatd Zagrb Idics o Crtai Naostructurs 8 I D-lattic o T C [ 4C8 p, ], 4-dgs ar o dgr, [p--4] dgs ar o dgr ad rmaiig 6p-5p++4 dgs ar o dgr 4. Th th irst rormulatd Zagrb idx o D-lattic is, D E N [ ] dg 4 4 p 4 6 p 5 p 4 96 p 44 p 8 Thorm.4. Lt b a graph o D-lattic o T C [ 4C8 p, ] th th scod rormulatd Zagrb idx is, 6p[8 p p 7] Proo. Lt b D-lattic o T C C p, ] graph with dgs. W us th ollowig stps to costruct th proo. Stp I. I D-lattic o T C [ 4C8 p, ], 8-pairs o dgs dgr ad dgr, thror th scod rormulatd Zagrb idx is, E dg dg 8 48 Stp II. Sic thr ar 4[p+-] pairs o dgs with dgr ad dgr 4 i a graph o D-lattic o T C C p, ]. Thror th scod rormulatd Zagrb idx is, E dg dg 4 4[ p ] 96 p Stp III. Similarly, thr ar 8[p-p-+] pairs o dgs with dgr 4 i a graph o D-lattic o T C C p, ]. Thror th scod rormulatd Zagrb idx is, E dg dg 4 Thror, rom th stps I, II ad III, w gt E 8 [ p p ] 8[ p p ] dg dg 48 96[ p ] 8[ p p ] 6 8 p p.

8 84 Krthi. Mirajar ad Priyaa Y.B Thorm.5. Lt b a graph o aotub o T C [ 4C8 p, ], th th irst rormulatd Zagrb idx is, 4p[4 ] Proo. Lt b aotub o T C C p, ] graph with 6p-p umbr o dgs. Th dgr o all th dgs o aotub o T C [ 4C8 p, ] ar ual to 4 xcpt 4p dgs which ar o dgr. Thror, th irst rormulatd Zagrb idx o aotub o T C C p, ] is, E N dg 4 p 4 [6 p p 4 p] 6 p 6 p 5p 4 p[4 ]. Thorm.6. Lt b a graph o aotub o T C [ 4C8 p, ], th th scod rormulatd Zagrb idx is, p[64 5] Proo. Lt b a graph o aotub o T C C p, ] with 6p-p umbr o dgs. Stp I. T C C W us th ollowig stps to costruct th proo. Thr ar p-pairs o dgs with dgr i graph o aotub o p, ] thror th scod rormulatd Zagrb idx is, E dg dg p 8p Stp II. Now w cosidr 8p-pairs o dgs with dgr ad dgr 4 i graph o aotub o T C C p, ]. Thror th scod rormulatd Zagrb idx is E dg dg 4 8p 96 p Stp III. Sic thr ar 8p-4p pairs o dgs with dgr 4 i a graph o aotub o T C C p, ]. Thror th scod rormulatd Zagrb idx is, E dg dg 4 Thror, rom th stps I, II, ad III, w gt 4 p p 64 p p

9 O th Rormulatd Zagrb Idics o Crtai Naostructurs 85 E dg dg 8p 96 p 64 p p p[64 5] Thorm.7. Lt b a graph o aotorus o T C [ 4C8 p, ], th th irst rormulatd Zagrb idx is, 96p Proo. Lt b a graph o aotorus o T C C p, ] with 6p umbr o dgs. Th dgr o all th dgs i aotorus o T C C p, ] is 4. Th irst rormulatd Zagrb idx is, 4 6 p 96 p dg E Thorm.8. Lt b a graph o aotorus o T C [ 4C8 p, ], th th scod rormulatd Zagrb idx is, 9p Proo. Lt b a graph o aotorus o T C C p, ] with 6p umbr o dgs. Th dgr o all th dgs i aotorus o T C C p, ] is 4. Sic ach pair o dgs is rpatd twic i T C C p, ]. Th scod rormulatd Zagrb idx is, E 4 6 p 9 p dg dg 4. CONCLUSION I mathmatical chmistry, umbrs codig crtai structural aturs o orgaic molculs ad drivd rom th corrspodig molcular graph, ar calld molcular topologyor topological idics, it has widly dmostratd its high prormac i th discovry ad dsig o w drugs. I this papr w studid ad computd w

10 86 Krthi. Mirajar ad Priyaa Y.B rsults o rormulatd Zagrb idics or aostructurs such as aostar ddrimr D[] ad D-lattic, aotub, aotorus o T C [ 4C8 p, ] ad which ar hav may chmical applicatios. Furthr this study ca b xtdd to comput w rsults o topological idics o various amilis o chmical structurs. 5. ACKNOWLEDENT Th authors ar thaul to Uivrsity rats CommissioUC, ovt. o Idia or th iacial support through UC-RNF-04-5-SC-KAR-75098/SAIII/ REFERENCES [] Bollobás B., Erdos, P., 998, raph o xtrmal wight, Ars Combi, 50, pp. 5-. [] Das, K. C., ad utma, I., 004, Som proprtis o th scod Zagrb idx, MATCH Commu. Math. Comput. Chm., 5, pp. 0-. [] utma, I., ad Das. K.C., 004, Th irst Zagrb idx 0 yars atr, MATCH Commu. Math. Comput. Chm., 50, pp [4] utma, I., ad Triajastić, N., 97, raph thory ad molcular orbitals. Total π-lctro rgy o altrat hydrocarbos, Chm. Phys. Ltt., 7, pp [5] Harary, F., 969, raph thory, Addiso-Wsly, Radig. [6] Hosamai, S. M., ad Zaar, S., O topological proprtis o th li graphs o subdivisio graphs o crtai aostructurs-ii, pr-prit. [7] Hydari, A., ad Tari, B., 007, Szgd idx o aotubs, MATCH Commu. Math. Comput. Chm., 57, pp [8] Jažić, D., Miličvič, A., Niolič, S., Triajstić, N., ad Vuičvić, D., 007, Zagrb idics: xtsio to wightd graphs rprstig molculs cotiig htroatoms, Croat Chm Acta, 80, pp [9] Li, Y., Ya, L., Farahi, M. R., Alai, A. N., Pajsh Kaa, M. R., ad Pradp Kumar, R., 06, Th dg cctric coctivity idx o aostar Ddrimr D[], Itratioal Joural o Biology, Pharmacy ad Allid Scics, 57, pp [0] Miličvić, A., Niolić, S., ad Triajastić, N., 004, O rormulatd Zagrb idics, Mol. Divrs., 8, pp [] Nadm, M. F., Zaar, S., ad Zahid, Z., 06, O topological proprtis o th li graph o crtai aostructurs, Appl. Math. Comput., 7, pp [] Nwom,. R., Moorild, C.N., ad Vogtl, F., 00, Ddrimr ad Ddros cocpts, sythsis, Applicatios, Wily-VCH vrlag mbh ad Co. Kgaa. [] Niolić, S., Kovačvić,., Miličvič, A., ad Triajstić, N., 00, Th Zagrb idics 0 yars atr, Croat. Chm. Acta., 76, pp. -4.

11 O th Rormulatd Zagrb Idics o Crtai Naostructurs 87 [4] Zhou, B., 004, Zagrb idics, MATCH Commu. Math. Comput. Chm., 5, pp. -8. [5] Zhou, B., utma, I., 005, Furthr proprtis o Zagrb idics, MATCH Commu. Math. Comput. Chm., 54, pp.-9.

12 88 Krthi. Mirajar ad Priyaa Y.B