THE ECCENTRIC CONNECTIVITY INDEX OF ARMCHAIR POLYHEX NANOTUBES

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1 Macedonian Journal of Chemistry and Chemical Engineering, Vol., No.,. () MJCCA ISSN Received: August, UDC:. :.]:. Acceted: January, Original scientific aer THE ECCENTRIC CONNECTIVITY INDEX OF ARMCHAIR POLYHEX NANOTUBES Mahboubeh Saheli, Ali Reza Ashrafi Institute of Nanoscience and Nanotechnology, University of Kashan, Kashan -, Iran The eccentric connectivity index ξ(g) of the grah G is defined as ξ(g) = Σ u V(G) deg(u)ε(u) where deg(u) denotes the degree of vertex u and ε(u) is the largest distance between u and any other vertex v of G. In this aer an exact exression for the eccentric connectivity index of an armchair olyhex nanotube is given. Key words: eccentric connectivity index; armchair nanotube ЕКСЦЕНТИРЧEН ИНДЕКС НА ПОВРЗУВАЊЕ НА НАНОЦЕВКИ СО ФОРМА НА ПОЛИХЕКСАГОНАЛНА ФОТЕЛЈА Ексцентричен индекс на поврзување ξ(g) на графот G е дефиниран како ξ(g) = Σ u V(G) deg(u)ε(u), каде deg(u) го означува степенот на темето u, а ε(u) е најголемото растојание помеѓу u и кое било друго теме v од G. Во овој труд е изведен егзактен израз за ексцентричен индекс на поврзување на наноцевки со форма на полихексагонална фотелја. Клучни зборови: ексцентричен индекс на поврзување; наноцевка со форма на фотелја. INTRODUCTION The grah theory has successfully rovided chemists with a variety of very useful tools, namely, the toological index. A toological index is a numeric uantity of the structural grah of a molecule. This grah has atoms as vertices and two atoms are adjacent if there is a bond between them. Suose u and v are vertices of the grah G. We define their distance d(u, v) as the length of the shortest ath connecting u and v in G. For a given vertex u of V(G) its eccentricity ε(u) is the largest distance between u and any other vertex v of G. The maximum eccentricity over all vertices of G is called the diameter of G and denoted by D(G) and the minimum eccentricity among the vertices of G is called radius of G and denoted by R(G). The set of vertices whose eccentricity is eual to the radius of G is called the center of G. It is well known that each tree has either one or two vertices in its center. The eccentric connectivity index ξ(g) of the grah G is defined as ξ ( G) = u V ( G) deg( u) ε ( u) [ ]. The mathematical roerties of this toological index are studied in some recent aers [ ]. In some research aers [ ], the authors have comuted some toological indices of armchair nanotubes. The aim of this article is to continue this roblem and comute the eccentric connectivity index of an armchair olyhex nanotube, Figure.. MAIN RESULTS In this section, the eccentric connectivity index of the molecular grah of an armchair olyhex nanotube TUVC [,] is comuted, where is the number of rows in D ercetion of TUVC [,] (Figure ), and is the number of vertical crenels.

2 M. Saheli, A. R. Ashrafi For simlicity, we denote this nanotube by T = T[,]. Notice that is even. Set E = E(T) and V = V(T). Obviously, V = and E = ( ) + / = (/ ). To comute the eccentric connectivity index of this nanotube, two cases is odd and even are considered. / +. From Figure a, one can see that for every vertex x, ε(x) =. If + then ( + )/ ε(x) + /. In this case, if x is a vertex of ( + )/ th row of T then ε(x) = ( + )/. Also, by choosing a vertex y from the i th or ( i) th row of T, one can see that ε(y) = + / i. Finally, we assume that / + <. Then ε(x) + /. From Figure b, there are ( + ) rows with eccentricity and for each i, i /, there are two rows with ε = + / i. Aly these calculations to rove the following euations: (i) / + ξ ( G) = = ( ) ( )/ Fig.. An armchair olyhex nanotube TUVC [, ] (ii) + < ξ ( G) = ( + ) + [( ( + / )) + ( ( + / )) ( ( / ( / )))] = (iii) + Fig.. An armchair olyhex lattice with = and = Theorem. Suose that is odd. Then ( ) + ( ) ξ G = < Proof. We first notice that if x and y are vertices in the same row of nanotube then ε(x) = ε(y). Consider three cases that (i) / + ; (ii) / + < ; and, (iii) +. Suose ξ ( G) = [ ( + / ( ) / )] + + ( + ) = + + This comletes our roof. In Table we consider the case that is even and the eccentricities of vertices (one vertex from each row) in some nanotubes with small arameters of and are comuted. In Table, for some nanotubes with small arameters and, the number of rows with eccentricity is given. Our calculations given in the roof of Theorem, result from these tables. Maced. J. Chem. Chem. Eng., (), ()

3 The eccentric connectivity index of armchair olyhex nanotubes T a b l e The eccentricity of T[,]( is odd) for some vlues of and The tye of tubes TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] The seuence of eccentricities,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, TUVC [,],,,,,,,,,,,,,, TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,],,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, TUVC [,],,,,,,,,,,,,,, TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,],,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, TUVC [,],,,,,,,,,,,,,, TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,],,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, TUVC [,],,,,,,,,,,,,,, T a b l e The number of rows with eccentricity, when is odd (b) (a) Fig.. The longest ath containing a vertex at the first row, when is odd and (a) / + ; (b) / + < ; (c) + (c) Theorem. Suose that is even. Then ( ) + ξ ( G) = < Proof. We notice again that if x and y are vertices in the same row of the nanotube then ε(x) = ε(y). In the same way as Theorem, we consider three searate cases that (i) / + ; (ii) / + < < + ; and, (iii) +. From Figures a-c and a similar method as Theorem, we have, Maced. J. Chem. Chem. Eng., (), ()

4 M. Saheli, A. R. Ashrafi (i) / + ξ ( G) = = ( ) (ii) + < < + ξ ( G) = ( + ) + [ ( + / ) (iii) + + ( + / ) ( + / ( / ))] = ξ ( G) = [ ( + / ) + ( + / ) + ( + / ) ( + / ( / ))] = This comletes our argument. In Table we consider the case that is odd and the eccentricities of vertices (one vertex from each row) in some nanotubes with small arameters of and are comuted. In Table we comute the number of rows with eccentricity, for some nanotubes with small arameters and. Our calculations given in the roof of Theorem, result from these tables. (b) (a) Fig.. The longest ath containing a vertex at the first row, when is even and (a) / + ; (b) / + < < + ; (c) + (c) T a b l e The tye of tubes TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] The Eccentricity of T[,] ( is even), for some values of and. The seuence of eccentricities,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, TUVC [,],,,,,,,,,,,,,,, TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,],,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, TUVC [,],,,,,,,,,,,,,,, TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,],,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, TUVC [,],,,,,,,,,,,,,,, TUVC [,] TUVC [,] TUVC [,] TUVC[,] TUVC [,] TUVC [,],,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, TUVC [,],,,,,,,,,,,,,,, TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,],,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, TUVC [,],,,,,,,,,,,,,,, TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,] TUVC [,],,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, TUVC [,],,,,,,,,,,,,,,, Maced. J. Chem. Chem. Eng., (), ()

5 The eccentric connectivity index of armchair olyhex nanotubes T a b l e The number of rows with eccentricity of, when is even We can combine Theorems and in the following theorem: Theorem. Suose T = T[, ] denotes an armchair olyhex nanotube with arameters and. Then the eccentric connectivity index of T is comuted as follows: ( ) + ξ ( G) = < CONCLUSIONS In this aer the eccentric connectivity index of an armchair nanotube is comuted for the first time. To the best of our knowledge it is the first aer considering the eccentric connectivity index of nanotubes into account. We resent a owerful method for calculating such indices. It is ossible to extend our method to a zig-zag olyhex nanotube. REFERENCES [] V. Sharma, R. Goswami and A. K. Madan, Eccentric connectivity index: A novel highly discriminating toological descritor for structure-roerty and structure-activity studies, J. Chem. Inf. Comut. Sci.,, (). [] H. Dureja and A. K. Madan, Sueraugmented eccentric connectivity indices: new-generation highly discriminating toological descritors for QSAR/QSPR modeling, Med. Chem. Res.,, (). [] V. Kumar, S. Sardana and A. K. Madan, Predicting anti- HIV activity of,-diary l-,-thiazolidin--ones: comutational aroaches using reformed eccentric connectivity index, J. Mol. Model,, (). [] S. Sardana and A. K. Madan, Alication of grah theory: Relationshi of molecular connectivity index, Wiener s index and eccentric connectivity index with diuretic activity, MATCH Commun. Math. Comut. Chem.,, (). [] M. Fischermann, A. Homann, D. Rautenbach, L. A. Szekely and L. Volkmann, Wiener Index versus maximum degree in trees, Discrete Al. Math.,, (). [] S. Guta, M. Singh and A. K. Madan, Alication of Grah Theory: Relationshi of Eccentric Connectivity Index and Wiener s Index with Anti-inflammatory Activity, J. Math. Anal. Al.,, (). [] A. Ilić and I. Gutman, Eccentric connectivity index of chemical trees, MATCH Commun. Math. Comut. Chem., to aear. [] B. Zhou and Z. Du, On eccentric connectivity index, MATCH Commun. Math. Comut. Chem.,, (). [] T. Doslic, M. Saheli and D. Vukičević, Eccentric connectivity index: extremal grahs and values, submitted. [] A. R. Ashrafi and A. Loghman, PI index of armchair olyhex nanotubes, ARS Combinatoria,, (). [] A. Iranmanesh and A. R. Ashrafi, Balaban index of an armchair olyhex, TUC C (R) and TUC C (S) nanotorus, J. Comut. Theor. Nanosci., (), (). [] H. Shabani and A. R. Ashrafi, Alications of the matrix ackage MATLAB in comuting the wiener olynomial of armchair olyhex nanotubes and nanotori, J. Comut. Theor. Nanosci., in ress. [] S. Yousefi and A. R. Ashrafi, Distance Matrix and Wiener Index of Armchair Polyhex Nanotubes, Studia Univ. Babes-Bolyai, Chemia, (), (). [] H. Deng, The PI Index of TUVC[; ], MATCH Commun. Math. Comut. Chem.,, (). [] M. Eliasi and B. Taeri, Distance in Armchair Polyhex Nanotubes, MATCH Commun. Math. Comut. Chem. (). [] M. V. Diudea, M. Stefu, B. Pârv and P. E. John, Armchair Polyhex Nanotubes, Croat. Chem. Acta, ( ), (). [] F. Harary, Grah Theory, Addison-Wesley, Reading MA,. Maced. J. Chem. Chem. Eng., (), ()