COMPUTING FIRST AND SECOND ZAGREB INDEX, FIRST AND SECOND ZAGREB POLYNOMIAL OF CAPRA- DESIGNED PLANAR BENZENOID SERIES Ca n (C 6 )

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1 (RECOMMENDED CITATION) COMPUTING FIRST AND SECOND ZAGREB INDEX, FIRST AND SECOND ZAGREB POLYNOMIAL OF CAPRA- DESIGNED PLANAR BENZENOID SERIES Ca (C 6 ) MOHAMMAD REZA FARAHANI a, MIRANDA PETRONELLA VLAD b ABSTRACT. I graph theory, arous polyomals ad topologcal dces are ow, as arats uder graph automorphsm. I ths paper, we focus o the structure of Capra-desged plaar bezeod seres Ca (C 6 ), 0 ad compute o t seeral topologcal dces ad polyomals: frst ad secod Zagreb polyomals ad ther correspodg dces. Keywords: Capra Operato, bezeod seres, Frst Zagreb dex, secod Zagreb dex, Frst Zagreb polyomal, secod Zagreb polyomal. INTRODUCTION Let G=(V,E) be a molecular graph wth the ertex set V(G) ad the edge set E(G). V (G) =, E(G) =e are the umber of ertces ad edges. A molecular graph s a smple fte graph such that ts ertces correspod to the atoms ad the edges to the chemcal bods. The dstace d(u,) the graph G s the umber of edges a shortest path betwee two ertces u ad. The umber of ertex pars at ut dstace equals the umber of edges. A topologcal dex of a graph s a umber related to that graph ad s arat uder graph automorphsm. Weer dex W(G) s the oldest topologcal dex [1-5], whch has foud may chemcal applcatos. It s defed as: W G u V G V G d u a Departmet of Mathematcs of Ira Uersty of Scece ad Techology, (IUST), Narma, Tehra 16844, Ira. Mr_Faraha@Mathdep.ust.ac.r b Dmtre Catemr Uersty, Bucharest, Faculty of Ecoomc Sceces, No 56 Teodor Mhal Street, , Cluj Napoca, Romaa, mradap@yahoo.com

2 MOHAMMAD REZA FARAHANI, MIRANDA PETRONELLA VLAD Hyper-Weer dex s a more recetly troduced dstace-based molecular descrptor [6]: WW G d u d u W G d u 134 u V G V G u V G V G Deote by d(g,) the umber of ertex pars of G lyg at dstace to each other ad by d(g) the topologcal dameter (.e, the logest topologcal dstace G). The Weer ad hyper-weer dces of G ca be expressed as [7, 8]: d G W G d G d G W W G d G Other oldest graph arat s the Frst Zagreb dex, whch was formally troduced by Gutma ad Trajst [9, 10]. It s deoted by M 1 (G) ad s defed as the sum of squares of the ertex degrees: M G d d u d V G e u E G where d s the degree of ertex. Next, Gutma troduced the Secod Zagreb dex M 2 (G) as: M G d u d e u E G Some basc propertes of M 1 (G) ca be foud ref. [9]. For a surey o theory ad applcatos of Zagreb dces see ref. [10]. Related to the two aboe topologcal dces, we hae the frst Zagreb Polyomal M 1 (G,x) ad secod Zagreb Polyomal M 2 (G,x), respectely. They are defed as: d u d M G x x e u E G M G x x e u E G d u d There was a ast research cocerg Zagreb dces ad Weer dex wth ts modfcatos [6] ad relatos betwee Weer, hyper-weer ad Zagreb dces [9-26]. WHAT IT IS THE CAPRA OPERATION? A mappg s a ew drawg of a arbtrary plaar graph G o the plae. I graph theory, there are may dfferet mappgs (or drawg); oe of them s Capra operato. Ths method eables oe to buld a ew structure of a plaar graph G.

3 COMPUTING FIRST AND SECOND ZAGREB INDEX, Let G be a cyclc plaar graph. Capra map operato s acheed as follows: () sert two ertces o eery edge of G; () () add pedat ertces to the aboe serted oes ad coect the pedat ertces order (-1,+3) aroud the boudary of a face of G. By rug these steps for eery face/cycle of G, oe obtas the Capra-trasform of G Ca(G), see Fgure 1. Fgure 1. Examples of Capra operato o the square face (top row) ad mappg Capra of plaar hexago (bottom row). By teratg the Capra-operato o the hexago (.e. bezee graph C 6 ) ad ts Ca-trasforms, a bezeod seres (Fgures 2 ad 3) ca be desged. We wll use the Capra-desged bezee seres to calculate some coectty dces (see below). Ths method was troduced by M.V. Dudea ad used may papers [27-36]. Sce Capra of plaar bezeod seres has a ery remarable structure, we loze t. We deote Capra operato by Ca, ths paper, as orgally Dudea dd. Thus, Capra operato of arbtrary graph G s Ca(G), terato of Capra wll be deoted by CaCa(G) (or we deote Ca 2 (G)) (Fgures 2 ad 3). The bezee molecule s a usual molecule chemstry, physcs ad ao sceces. Ths molecule s ery useful to sythesze aromatc compouds. We use the Capra operato to geerate ew structures of molecular graph bezee seres. Theorem 1. Let Ca(C 6 ) be the frst member of Capra of bezeod seres. The, Hosoya polyomal of Ca(C 6 ) s equal to: H(Ca(C 6 ),x)=24+30x 1 +48x 2 +57x 3 +x54x 4 +45x 5 +30x 6 +12x 7 ad the Weer dex of Ca(C 6 ) s equal to

4 MOHAMMAD REZA FARAHANI, MIRANDA PETRONELLA VLAD Hosoya polyomal H(G) s equal to x d u It s easy to u V G V G see that Weer dex s obtaed from Hosoya polyomal as the frst derate, x=1. Fgure 2. The frst two graphs Ca(C 6 ) ad Ca 2 (C 6 ) from the Capra of plaar bezeod seres, together wth the molecular graph of bezee (deoted here Ca 0 (C 6 )) 136 Fgure 3. Graph Ca 3 (C 6 ) s the thrd member of Capra plaar bezeod seres.

5 COMPUTING FIRST AND SECOND ZAGREB INDEX, By these termologes, we hae the followg theorem: Theorem 2. Cosder the graph G=Ca (C 6 ) as the terate Capra of plaar bezeod seres. The: Frst Zagreb polyomal of G s equal to M 1 (Ca (C 6 ),x)=(3(7 )-2(3 )-3)x 6 +4(3 )x 5 +(3 +3)x 4 ad the Frst Zagreb dex s M 1 (Ca (C 6 ))=18(7 )+12(3 )-6. Secod Zagreb polyomal of G s equal to M 2 Ca (C 6 ),x)=(3(7 )-2(3 )-3)x 9 4(3 )x 6 (3 +3)x 4 ad the Secod Zagreb dex of G s M 2 (Ca (C 6 ))=27(7 )+10(3 )-15. RESULTS AND DISCUSSION Capra trasforms of a plaar bezeod seres s a famly of molecular graphs whch are geeralzatos of bezee molecule C 6. I other words, we cosder the base member of ths famly s the plaar bezee, deoted here Ca 0 (C 6 )=C 6 =bezee. It s easy to see that Ca (C 6 )=Ca(Ca -1 (C 6 )) (Fgures 2 ad 3) [27-36]. I addto, we eed the followg defto. Defto 3. [21] Let G be a molecular graph ad d s the degree of ertex V G We dde ertex set V(G) ad edge set E(G) of graph G to seeral parttos, as follow: V V G d ad E e u E G d du Obously, d such that M d V G ad Max d V G Now, we start to proof of the aboe theorem. Proof of Theorem 2. Let G=Ca (C 6 ) ( 0) be the Capra plaar bezeod seres. By costructo, the structure Ca (C 6 ) collects see tmes of structure Ca -1 (C 6 ) (we call "flower" the substructure Ca -1 (C 6 ) the graph Ca (C 6 )). Therefore, by smple ducto o, the ertex set of Ca (C 6 ) wll hae 7 V(Ca (C 6 )) -6( ) members. Because, there are ad 3-1 commo ertces betwee see flowers Ca -1 (C 6 ) Ca (C 6 ), mared by full blac color the aboe fgures. Smlarly, the edge set E(Ca (C 6 )) hae 7 E(Ca (C 6 )) -6( ) members. Sce, there are 3-1 ad 3-1 commo edges (full blac color these fgures). 137

6 MOHAMMAD REZA FARAHANI, MIRANDA PETRONELLA VLAD Now, we sole the recurse sequeces V(Ca (C 6 )) ad E(Ca (C 6 )). Frst, suppose = V(Ca (C 6 )) ad e = E(Ca (C 6 )) so ad e e Thus, we hae j j j (1) where 0 =6 s the umber of ertces bezee C 6 (Fgure 2) ad s equal to O the other had, sce Hece (2) 138

7 COMPUTING FIRST AND SECOND ZAGREB INDEX, Therefore, by usg equatos (1) ad (2), we hae ad = V(Ca (C 6 )) = By usg a smlar argumet ad (1), we ca see that e e e e e It s easy to see that, the frst member of recurse sequece e s e 0 =6, (Fgure 2). Now, by usg (2), we hae e ad the sze of edge set E(Ca (C 6 )) s equal to: e = E(Ca (C 6 )) =3(7 +3 ), Also, accordg to Fgures 2 ad 3, we see that the umber of ertces of degree two the graph Ca (C 6 ) (we deote by ) s equal to. The sx remoed ertces are the commo oes betwee the sx flowers "Ca -1 (C 6 )" wth degree three. By usg a smlar argumet ad smple ducto, we hae the umbers of edges of graph Ca (C 6 ), whch are the set E or E (deoted by Now, we sole the recurse sequece coclude Thus, e ). ad we It s obous that, accordg to the structure of bezee, Also, e E E 3 3 ad accordg to the aboe defto, t s obous that, for Capra of plaar bezeod seres G=Ca (C 6 ) we hae two parttos: V V Ca C d ad V V Ca C d wth the sze ad respectely. 139

8 MOHAMMAD REZA FARAHANI, MIRANDA PETRONELLA VLAD O the other had, accordg to the structure of Capra plaar bezeod seres Ca (C 6 ), there are edges, such that the frst pot of them s a ertex wth degree two. Amog these edges, there exst edges, of whch the frst ad ed pot of them hae degree 2 (the members of E or E ). Thus, e E E e. So, the sze of edge set E ad E s equal to e Now, t s obous that: e E E e e Now, we ow the sze of all sets V V E E E E E ad E So, we ca calculate the Frst ad Secod Zagreb Polyomal of Capra plaar bezeod seres G=Ca (C 6 ), as follow: Frst Zagreb Polyomal of G=Ca (C 6 ): M G x x e E G d u d x x x e E e E e E E x E x E x x x x Secod Zagreb Polyomal of G Ca C : d u d M G x x x x x e E G e E e E e E x x x hae: M G Also, accordg to defto of Frst ad Secod Zagreb dex, we M G x x x 140

9 COMPUTING FIRST AND SECOND ZAGREB INDEX, ad M G M G x x x Of course, by usg V 2 ad V 3, we hae M G Thus, we completed the proof of the theorem 3. ACKNOWLEDGMENTS The frst author s thaful to Dr. M. Alaeya ad Dr. A. Aghaja of Departmet of Mathematcs, Ira Uersty of Scece ad Techology (IUST) for ther precous support ad suggestos. REFERENCES 1. H. Weer. J. Am. Chem. Soc. 1947, 69, A.A. Dobry, R. Etrger ad I. Gutma. Acta Appl. Math. 2001, 66, D.E. Needham, I.C. We ad P.G. Seybold. J. Am. Chem. Soc. 1988, 110, G. Rucer ad C. Rucer. J. Chem. If. Comput. Sc. 1999, 39, B. Zhou ad I. Gutma, Chemcal Physcs Letters. 2004, 394, I. Gutma, Ida J. Chem, 1997, 36, D.J. Kle, I. Luots ad I. Gutma. J. Chem. If. Comput. Sc. 1995, 35, B. Zhou ad I. Gutma, Chemcal Physcs Letters. 2004, 394, I. Gutma ad N. Trajst, Chem. Phys. Lett. 1972, 17, I. Gutma, B. Ruscc, N. Trajst ad C.F. Wlcox, J. Chem. Phys. 1975, 62, I. Gutma ad K.C. Das, MATCH Commu. Math. Comput. Chem. 2004, 50, S. Nolc, G. Koacec, A. Mlcec ad N. Trajst, Croat. Chem. Acta. 2003, 76, A. Behtoe, M. Jaesar ad B. Taer, Appl. Math. Lett. 2009, 22, K.C. Das, I. Gutma ad B. Zhou, Joural of Mathematcal Chemstry. 2009, 46, M.H. Khalfeh, H. Yousef-Azar ad A.R. Ashraf, Comput. Appl. Math. 2008, 56, S. L ad H. Zhou, Appl. Math. Lett. 2010, 22, M. Lu ad B. Lu, MATCH Commu. Math. Comput. Chem. 2010, 63, M. Ghorba ad M. Ghaz, Dgest. J. Naomater. Bos. 2010, 5(4), M. Ghorba ad M. Ghaz, Dgest. J. Naomater. Bos. 2010, 5(4),

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