PI Polynomial of V-Phenylenic Nanotubes and Nanotori

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1 It J Mol Sci 008, 9, 9-34 Full Research Paper Iteratioal Joural of Molecular Scieces ISSN by MDPI PI Polyomial of V-Pheyleic Naotubes ad Naotori Vahid Alamia, Amir Bahrami,* ad Behrooz Edalatzadeh 3 The Orgaizatio for Educatioal Research ad Plaig (OERP), Ira Departmet of Mathematics, Islamic Azad Uiversity, Garmsar Brach, Garmsar, Ira 3 Departmet of Mathematics ad statistics, Shahid Beheshti Uiversity, Tehra, Ira; b_edalatzadeh@sbuacir * Author to whom correspodece should be addressed s: bahrami@khayamutacir, amirbahr@gmailcom Received: October 007; i revised form: 6 November 007 / Accepted: 4 December 007 / Published: 8 February 008 Abstract: The PI polyomial of a molecular graph is defied to be the sum E(G) N(e) + V(G) ( V(G) +)/ E(G) over all edges of G, where N(e) is the umber of edges parallel to e I this paper, the PI polyomial of the pheyleic aotubes ad aotori are computed Several ope questios are also icluded Keywords: PI polyomial, molecular graph, pheyleic aotube ad aotorus Itroductio Let G be a simple molecular graph without loops, directed ad multiple edges The vertex ad edge sets of G are represeted by V(G) ad E(G), respectively A topological idex is a umeric quatity derived from the structural graph of a molecule Usage of topological idices i chemistry bega i 947, whe Harold Wieer developed the most widely kow topological descriptor, the Wieer idex, ad used it to determie physical properties of the type of alkaes kow as paraffis [] The Hosoya polyomial of a graph G is defied to be W(G;) = Σ uv V(G) d(u,v), where d(u,v) deotes the legth of a miimum path betwee u ad v I [], Hosoya used the ame Wieer polyomial while some authors later used the ame Hosoya polyomial Let G be a coected molecular graph ad e=uv a edge of G, eu (e G) deotes the umber of edges lyig closer to the vertex u tha the vertex v, ad ev (e G) is the umber of edges lyig closer to

2 It J Mol Sci 008, 9 30 the vertex v tha the vertex u The Padmakar-Iva (PI) idex of a graph G is defied to be PI (G) = Σ e E(G) [ eu (e G) + ev (e G)], see [3] ad [4] I a series of papers [5, 6] Ashrafi et al defied a ew polyomial which they amed the Padmakar-Iva polyomial They abbreviated this ew polyomial as PI(G,), for a molecular graph G We defie PI(G;) = Σ uv V(G) N(u,v), where for a edge e = uv, N(u,v) = eu (e G) + ev (e G) ad zero otherwisethis polyomial is very importat i computig the PI idex This ewly proposed polyomial, PI(G,), does ot coicide with the Wieer polyomial (W (G,)) for acyclic molecules I a series of papers [7, 8] Diudea et al ivestigated the structure ad computed the Hosoya polyomial of some aotubes ad aotori Gutma et al [9] also computed the Hosoya polyomials of some bezeoid graphs I [0] Shouju et al ivestigated the Hosoya polyomials of armchair ope-eded aotubes Also, i [5] ad [6] the authors computed the PI ad Wieer Polyomial of some aotubes ad aotori I this paper we cotiue this program to compute the PI polyomial of V-pheyleic aotubes ad aotori, usig the molecular graphs i Figures ad Throughout this paper, the otatio is the same as i [] ad [] Figure A V-Pheyleic Naotube i= i= i=4 Figure A V-Pheyleic Naotorus

3 It J Mol Sci 008, 9 3 Results ad Discussio The ovel pheyleic ad aphthyleic lattices proposed ca be cotructed from a square et embedded o the toroidal surface I this sectio, the PI polyomial of a V-Pheyleic aotube ad aotorus are computed Followig Diudea [3] we deote a V-Pheyleic aotube by T=VPH[4,m] We also deote a V-Pheyleic aotorus by H=VPHY[4,m] Let G be a arbitrary graph For every edge e, we defie N (e) = E(G) - ( eu (e G) + ev (e G)) By Theorem i [6] we have: PI(G, ) = e E(G) E(G) V(G) + + E(G) So it is eough to compute N(e), for every edge e E(G) From above the argumet ad Figures ad, it is easy to see that E(T) =36m, E(H) =36m ad V(T) =4m, V(H) =4m I the followig theorem we compute the PI polyomial of the molecular grapht i Figure Theorem PI(T,)=( (36m-6) ) (8m) +( (36m-4) ) (4m-)+ ( (36m--8m) ) (8m) 36m 6 (6m), if m 4m 36m 6 i+ 36m 8m + {( )} + ( m)(4) if m > ad m < i= 36m 6 i+ 36m 0+ {( )} + ( m )(4), m i= +(4m+)(m+)-36m+ Proof: To compute the PI polyomial of T, it is eough to calculate N(e) To do this, we cosider three cases: that e is vertical, horizotal or oblique If e is horizotal a similar proof as Lemma i [4] shows that N(e)=8m Also, if e is a vertical edge i oe hexago or octago the N(e) = 4,, respectively We cosider the set A(T) of oblique edges i T For every e i A(T), we have two cases: Case : m A similar argumet as Lemma i [4] gives that N(e)=4 Case : m > We deote the i th row of oblique edges i A(T) by A i (see Figure ) It is easy to see that by graph symmetry each elemet of A i has the same umber of parallels If e A i ad i (m- -m ), by computatios, we have N(e)=4+i-, also if (m- -m )+ i m, the N(e)=8- If m>, the N(e)=8m For >m because of symmetry computatios are similar to upper part of graph So we have:

4 It J Mol Sci 008, 9 3 e is vertical E = ( (36m-6) ) (8m) +( (36m-4) ) (4m-) ad e is horizotal E =( (36m--8m) ) (8m) Also: e is oblique 36m 6 (6m), if m 4m 36m 6 i+ 36m 8m = {( )} + ( m)(4) if m > ad m < i= 36m 6 i+ 36m 0+ {( )} + ( m )(4), m i= Thus: E(T) V(T) + PI(T, ) = + E(T) e E(T) E = E + E + + ( V(T) +) ( V(T) +)/ - E(T) e is horizotal e is vertical e is oblique =( (36m-6) ) (8m) +( (36m-4) ) (4m-)+ ( (36m--8m) ) (8m)+ 4m i= i= {( {( 36m 6 36m 6 i+ 36m 6 i+ (6m) )} + ( m)(4) )} + ( m )(4) 36m 8m 36m 0+ +(4m+)(m+)-36m+, if m if m > ad m <, m Which completes the proof I our ext theorem we cosider a V-Pheyleic aotorus H ad calculate its Padmakar-Iva polyomial, PI(H,), Figure E(T) V(H) + Theorem PI(H, ) = + E(H) =( (36m-8) ) (8m) +( (36m-) ) (4m) + e E(H) ( (36m-8m) 36m 6z+ ) (8m)+ (6m) (4m+)(m+)-36m, where z=mi{m,}

5 It J Mol Sci 008, 9 33 Proof: To prove the theorem, we apply a similar method as i Theorem It is easily see that N(e)=8 for each vertical edge i hexagos, that is two times more twice the tube case by horizotal symmetry A vertical edge i a octago has parallels, as i Theorem Also N(e)=8m for each horizotal edge Let z = mi{ m, }, for each oblique edge e we have N( e ) = 6z So: Thus: PI(H, ) = e E(H) E(H) N ( e is vertical e is horizotal = ( (36m-8) ) (8m) +( (36m-) ) (4m) e is oblique e) V(H) + + (8m)+ (6m) =( (36m-8m) ) (8m) 36m 6z+ = (6m) E(H) 36m 6z+ =( (36m-8) ) (8m) +( (36m-) ) (4m) + ( (36m-8m) ) +(4m+)(m+)-36m ad this completes the proof We coclude our paper with the followig ope questios: Questio : Let F(x)= k k ( ) x be a polyomial of degree Is there a V-pheyleic aotube or k = 0 aotorus T such that PI (T,x) = F(x)? Questio : Is it true that for every polyomial F(x) with positive coefficiets ad of degree, there exists a V-pheyleic aotube or aotorus T, such that PI (T,x) = F(x)? Questio 3: What is the relatio betwee the Hosoya polyomial ad PI polyomial of a V- pheyleic aotube or aotorus? Refereces ad Notes Wieer, H Structural determiatio of the paraffie boilig poits J Am Chem Soc 947, 69, 7-0 Hosoya, H O some coutig polyomials i chemistry Disc Appl Math 988, 9, Khadikar, P V O a ovel structural descriptor PI idex Nat Acad Sci Lett 000, 3, Khadikar, P V; Karmarkar, S; Agrawal, V K PI idex of polyacees ad its use i developig QSPR Nat Acad Sci Lett 000, 3, Ashrafi, A R; Maoochehria, B; Yousefi-Azari, H O the PI polyomial of a graph Util Math 006, 7, Maoochehria, B; Yousefi-Azari, H; Ashrafi, A R PI polyomial of some bezeoid graphs MATCH Commu Math Comput Chem 007, 57,

6 It J Mol Sci 008, Diudea, M V Hosoya Polyomial i Tori MATCH Commu Math Comput Chem 00, 45, 09-8 Kostatiova, E V; Diudea, M V The Wieer polyomial derivatives ad other topological idexes i chemical research Croat Chem Acta 000, 73, Gutma, I; Klavzar, S; Petkovsek, M E; Zigert, P O Hosoya Polyomials of Bezeoid Graphs MATCH Commu Math Comput Chem 00, 43, Shouju, ; Hepig, Z Hosoya polyomials of armchair ope-eded aotubes It J Quatum Chem 007, 07, Camero, PJ Combiatorics: Topics, Techiques, Algorithms; Cambridge Uiversity Press: Cambridge, 994; pp -50 Triajstic, N Chemical graph theory, d ed; CRC Press: Boca Rato, FL, 99; pp Diudea, MV Pheyleic ad aphthyleic tori Fuller Naotub Carbo Naostruct 00, 0, Ashrafi, A R; Loghma, A Padmakar-Iva Idex of TUC 4 C 8 (S) Naotubes J Comput Theor Naosci 006, 3, by MDPI ( Reproductio is permitted for ocommercial purposes