(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, 400591, Cluj Napoca, Romaa, mradap@yahoo.com
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.
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 1002. 135
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.
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(2 3-1 +1) members. Because, there are 3-1 +1 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(2 3-1 +1) members. Sce, there are 3-1 ad 3-1 commo edges (full blac color these fgures). 137
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
COMPUTING FIRST AND SECOND ZAGREB INDEX, Therefore, by usg equatos (1) ad (2), we hae ad = V(Ca (C 6 )) =2 7 +3 +1 +1. 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,. 1 3 3 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
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 1 3 7 2 3 1 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
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