Distance property of chemical graphs

Yıl 2018, Cilt: 47 Sayı: 5, 1071 - 1093, 16.10.2018

Sakander Hayat Shahzad Ahmad Hafız Muhammad Umair Shaohui Wang

Öz

We have developed a rigorous computational technique to compute exact analytic expressions for a number of distance-based topological indices of chemical graphs. There are two main advantages of our technique over existing techniques of similar nature: first, our technique is significantly diverse as it also covers the Wiener index and eccentricity-based topological indices besides Szeged-like indices, and secondly we have considerably reduced the algorithmic and computational complexity in comparison to previous techniques. Our proposed technique generates certain vertex and edge partitions of a graph which are essential in computing the exact analytical formulas of distance-based and eccentricity-based indices. To ensure the applicability of our technique,we have computed various distance-based and eccentricity-based topological indices for certain infinite families of polyomino chain system. Moreover, we find analytical exact expressions of certain degree-based topological indices for these polyomino chains. These topological indices can be obtained as a by-product of our technique.

Anahtar Kelimeler

Combinatorial algorithms, Distance-based topological indices, Eccentricity-based topological indices, Degree-based topological indices, Polyomino chain system

Kaynakça

• Ashrafi, A.R. Doslic, T. and Saheli, M. The eccentric connectivity index of $TUC_4C_8(R)$ nanotubes, MATCH Commun. Math. Comput. Chem. 65, 221-230, 2011.
• Ashrafi, A.R. Ghorbani, M. and Jalali, M. The $PI$ and edge Szeged polynomials of an infinitefamily of fullerenes, Fullerenes, Nanotubes and Carbon Nanostructures 18 (3), 107-116,2010.
• Aouchiche, M. and Hansen, P. On a conjecture about the Szeged index, European J. Combin. 31, 1662-1666, 2010.
• Arockiaraj M. Kavithah, S.R.J. and Balasubramanian, K. Vertex-cut methods for distance- based topological indices and its application to inorganic networks, J. Math. Chem. 54, 1728-1747, 2016.
• Baca, M. Horváthová, J. Mokrisová, M. and Suhányiová, A. On topological indices of fullerenes, Appl. Math. Comput. 251, 154-161, 2015.
• Imran, M. and Hayat, S. On counting polynomials of certain polyomino chains, Bulg. Chem. Commun. 48, 332-337, 2016.
• Diudea, M.V. Nanomolecules and nanostructures: polynomials and indices, University of Kragujevac, Kragujevac, 2010.
• Diudea, M.V. Ursu O. and Nagy, Cs.L. TOPOCLUJ, Babes-Bolyai University, Cluj, 2002.
• Dureja, H. and Madan, A.K. Superaugmented eccentric connectivity indices: new-generationhighly discriminating topological descriptors for $QSAR/QSPR$ modeling, Med. Chem. Res.16, 331-341, 2007.
• Eliasi, M. and Taeri, B. Szeged index of armchair polyhex nanotubes, MATCH Commun. Math. Comput. Chem., 59 437-450, 2008.
• Estrada, E. Torres, L. Rodríguez, L. and Gutman, I. An atom-bond connectivity index: Modelling the enthalpy of formation of alkanes, Indian J. Chem., 37A, 849-855, 1998.
• The GAP Team, GAP, Groups, Algorithms and Programming, Lehrstuhl De fur Mathematik, RWTH, Aachen, 1992.
• Golomb, S.W. Polyominoes, Princeton University Press, Princeton, New Jersey, 1994.
• Graovac A. Ghorbani, M. and Hosseinzadeh, M.A. Computing fifth geometric-arithmetic index for nanostar dendrimers, J. Math. Nanosci. 1, 33-42, 2011.
• Graovac, A. and Hosseinzadeh, M.A. Computing $ABC_4$ index of nanostar dendrimers, Optoelectron.Adv. Mater. Rapid Commun. 4, 1419-1422, 2010.
• Graovac, A. Ori, O. Faghani, M. and Ashrafi, A.R. Distance property of fullerenes, Iranian J. Math. Chem. 1, 5-15, 2010.
• Gutman, I. A formula for the Wiener number of trees and its extension to graphs containing cycles, Graph Theory Notes New York, 27, 9-15, 1994.
• Gutman, I. and Dobrynin, A.A. The Szeged index - a success story, Graph Theory Notes New York 34, 37-44, 1998.
• Hayat, S. and Imran, M. Computation of topological indices of certain networks, Appl. Math. Comput. 240, 213-228, 2014.
• HyperChem package Release 7.5 for Windows, Hypercube Inc., 1115 NW 4th Street, Gainesville, Florida 32601, USA, 2002.
• Imran, M. Hayat, S. and Malik, M.Y.H. On topological indices of certain interconnection networks, Appl. Math. Comput. 244, 936-951, 2014.
• Karelson, M. Molecular descriptors in $QSAR/QSPR$, Wiley, New York, 2000.
• Khadikar, P.V. Karmarkar, S. and Agrawal, V.K. A novel $PI$ index and its applications to $QSPR/QSAR$ studies, J. Chem. Inf. Comput. Sci., 41, 934-949, 2001.
• Klarner, D.A. Polyominoes, In: J. E. Goodman, J. O'Rourke, (eds.), Handbook of Discrete and Computational Geometry, CRC Press, Boca Raton, (1997), 225-242, Chapter 12.
• Liu, J.-B. and Cao, J. Applications of Laplacian spectra for n-prism networks, Neurocomputing, 198, 69-73, 2016.
• Liu, J.-B. Pan, X.-F. Yu, L. and Li, D. Complete characterization of bicyclic graphs withminimal Kirchhoff index, Discrete Appl. Math. 200, 95-107, 2016.
• Liu, J.-B. and Pan, X.-F. Minimizing Kirchhoff index among graphs with a given vertex bipartiteness, Appl. Math. Comput. 291, 84-88, 2016.
• Liu, J.-B. Wang, W.R. Zhang, Y.M. and Pan, X.-F. On degree resistance distance of cacti, Discrete Appl. Math. 203, 217-225, 2016.
• Liu, J.-B. Wang, C. Wang, S. and Wei, B. Zagreb indices and multiplicative Zagreb indices of Eulerian graphs, Bull. Malays. Math. Sci. Soc. DOI: 10.1007/s40840-017-0463-2.
• MATLAB and Statistics Toolbox Release 2012b, The MathWorks, Inc., Natick, Massachusetts, United States.
• Mehranian, Z. Mottaghi, A. and Ashrafi, A.R. The topological study of $IPR$ fullerenes by studying their Szeged and revised Szeged indices, J. Theor. Comput. Chem. 11 (3), 547-559,2012.
• Mottaghi, A. and Ashrafi, A.R. Topological edge properties of $C_{12n+60} fullerenes, BeilsteinJ. Nanotechnol., 4, 400-405, 2013.
• Pisanski, T. and Randi¢, M. Use of the Szeged index and the revised Szeged index for meauring network bipartivity, Discrete Appl. Math. 158, 1936-1944, 2010.
• Randi¢, M. On characterization of molecular branching, J. Amer. Chem. Soc. 97 (23), 6609-6615, 1975.
• Randi¢, M. On generalization of Wiener index for cyclic structures, Acta Chim. Slov. 49, 483-496, 2002.
• Sharma, V. Goswami, R. and Madan, A.K. Eccentric connectivity index: a novel highly discriminating topological descriptor for structure-property and structure-activity studies, J. Chem. Inf. Comput. Sci. 37, 273-282, 1997.
• Siddiqui, M.K. Imran M. and Ahmad A., On Zagreb indices, Zagreb polynomials of some nanostar dendrimers, Appl. Math. Comput. 280, 132-139, 2016.
• Vukisevi¢, D. and Furtula, B. Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem. 46, 1369-1376, 2009.
• Wang, C. Wang, S. and Wei, B. Cacti with extremal PI index, Trans. Comb. 5, 1-8, 2016.
• Wang, S. and Wei, B. Multiplicative Zagreb indices of k-trees, Discrete Appl. Math. 180, 168-175, 2015.
• Wiener, H. Structural determination of the paraffin boiling points, J. Am. Chem. Soc. 69,17-20, 1947.
• Xing, R. and Zhou, B. On the revised Szeged index, Discrete Appl. Math. 159, 69-78, 2011.
• Yarahmadia, Z. Ashrafi A.R. and Moradic, S. Extremal polyomino chains with respect to Zagreb indices, App. Math. Lett. 25, 166-171, 2012.
• Zhou, B. and Du, Z. On eccentric connectivity index, MATCH Commun. Math. Comput. Chem. 63, 181-198, 2010.