M-POLYNOMIAL METHOD FOR TOPOLOGICAL INDICES OF 3-LAYERED PROBABILISTIC NEURAL NETWORKS

Year 2019, Volume: 9 Issue: 4, 864 - 875, 01.12.2019

M. Javaid A Raheem M. Abbas J. Cao

Abstract

A molecular network can be uniquely identied by a number, polynomial or matrix. A topological index TI is a number that characterizes a molecular network completely which is used to predict the physical features of the certain changes such as bioactivities and chemical reactivities in the chemical compound. Javaid and Cao [Neural Comput. and Applic., 30 2018 , 3869-3876] studied the rst Zagreb index, second Zagreb index, general Randic index, and augmented Zagreb index for the 3-layered probabilistic neural networks PNN . In this paper, we prove the M-polynomial of the 3-layered PNN and use it as a latest developed tool to compute the certain degree based TI's. At the end, a comparison is also shown to nd the better one among all the obtained results.

Keywords

M-polynomial, Degree-based TI's, Networks, Probabilistic neural network.

References

• Amic, D., Beslo, D., Lucic, B., Nikolic, S., and Trinajstic, N.,(1998), The vertex-connectivity index revisited, J. Chem. Inf. Comput. Sci., 38, pp. 819-822.
• Araghi, L.F., Khaloozade, H., Arvan, M.R., (2009), Ship identification using probabilistic neural networks. In: Proceedings of the international multiconference of engineers and computer scientists, , pp. 18-20.
• Baca, M., Horvathova, J., Mokrisova, M., Suhanyiova, A., (2015), On topological indices of fullerenes
• Applied Mathematics and Computation, 251, pp. 154-161. Baca, M., Horvathova, J., Mokrisova, M., Andrea Semanicova-Fenovckova, Suhanyiova, A., (2015)
• On topological indices of a carbon nanotube network, Can. J. Chem. 93, pp. 1157-1160.
• Bollobas, B., Erdos, P., (1998), Graphs of extremal weights, Ars Combin., 50, pp. 225-233.
• Budak, F., Beyli, E.D.U, (2011), Detection of resistivity for antibiotics by probabilistic neural net- works, J. Med. Syst., 35, pp. 87-91.
• Bruckler, F.M., Doslic, T., Graovac, A., Gutman, I., (2011), On a class of distance-based molecular structure descriptors. Chem. Phys. Lett., 503, pp. 336–338.
• Bascil, M.S., Oztekin, H., (2012), A study on hepatitis disease diagnosis using probabilistic neural network, J. Med. Syst., 36, pp. 1603-1606.
• Devillers, J., Balaban, A.T., (1999), Topological Indices and Related Descriptors in QSAR and QSPR
• Gordon Breach, Amsterdam. Diudea M.V., (2001), QSPR/QSAR Studies by Molecular Descriptors, NOVA, New York.
• Deutsch, and Klavzar, S.,(2015), M-polynomial and degree-based topological indices. Iranian Journal of Mathematical Chemistry, 6(2), pp. 93-102.
• Furtula, B., Graovac, A., Vukicevic, D., (2010), Augmented Zagreb index, J. Math. Chem., 48, pp. 380.
• Gutman, I., Trinajsti, N., (1972), Graph theory and molecular orbitals. III. Total electron energy of alternant hydrocarbons, Chem. Phys. Lett., 17, pp. 535-538.
• Gutman, I., Polansky, O., (1986), Mathematical Concepts in Organic Chemistry, Springer-Verlag, Berlin.
• Gutman, I., (2013), Degree-based topological indices. Croat. Chem. Acta, 86, pp. 351-361.
• Gonzalez-Diaz, H., Vilar, S., Santana, L., and Uriarte, E., (2007) Medicinal Chemistry and Bioin- formatics - Current Trends in Drugs Discovery with Networks Topological Indices, Current Topics in Medicinal Chemistry, 7 (10), pp. 1015-1029.
• Gao, W., Wang, W., and Farahani, M.R., (2016), Topological indices study of molecular structure in anticancer drugs, Journal of Chemistry, Doi:10.1155/2016/3216327.
• Harary, F., (1969) Graph Theory, Addison-Wesley.
• Hall, L.H. and Kier, L.B.,(1976) Molecular Connectivity in Chemistry and Drug Research; Academic Press: Boston,239 MA, USA.
• Holmes, E., Nicholson, J.K., Tranter, G., (2001), Metabonomic characterization of genetic variations in toxicological and metabolic responses using probabilistic neural networks, Chemical Research in Toxicology, 14(2), pp. 182-191.
• Javaid, M., Rehman, M.U., Cao, J., (2017), Topological indices of rhombus type silicate and oxide networks, Can. J. Chem. 95(2), pp. 134-143.
• Javaid, M. Cao, J., (2017), Computing topological indices of probabilistic neural network, Neural Comput. and Applic., 30(2018), 3869-3876.
• Kowalski, P.A., Kulczycki, P., Interval probabilistic neural network, Neural Comput. Applic. DOI 1007/s00521 − 015 − 2109 − 3.
• Klavzar, S., Gutman, I., (1996), A Comparison of the Schultz molecular topological index with the Wiener index, J. Chem. Inf. Comput. Sci., 36, pp. 1001–1003.
• Kim, D., Kim, D.H., Chang, S., (2008), Application of probabilistic neural network to design break- water armor blocks, Ocean Engineering, 35, pp. 294-300.
• Kulli, V., Stone, B., Wang, S., Wei, B., (2017) Generalized multiplicative indices of polycyclic aromatic hydrocarbons and benzenoid systems, Zeitschrift f¨ur Naturforschung A, 72(6)a, pp. 573–576.
• Labanowski,J.K., Motoc I., and Dammkoehler, R.A., (1991), The physical meaning of topological indices, Computers Chem., 1(15), pp. 47-53.
• Lee, J.-J., Yun, C.-B., (2007), Damage localization for bridges using probabilistic neural networks
• KSCE Journal of Civil Engineering 11(2), pp. 111-120. Matamala A. R., and Estrada, E., (2005), Generalised topological indices: Optimisation methodology and physico-chemical interpretation, Chemical Physics Letters, 410, pp. 343-347.
• Meshoul, S., and Batouche, M., (2010), A novel approach for online signature verification using fisher based probabilistic neural network, In: Proceedings of IEEE symposium on computers and communications, pp. 314-319.
• Munir, M., Nazeer, W., Shahzadi, Z., Kang, S.M., (2016), M-polynomial and degree-based topological indices of polyhex nanotubes, Symmetry ,8, pp. 149-159.
• Polya, G., Kombinatorische Anzahlbestimmungen fur Gruppen, (1936), Graphen und chemische Verbindungen, Acta Math., 68, pp. 145-253.
• Rajan, B., William, A., Grigorious, C., and Stephen, S., (2012) On certain topological indices of silicate, honeycomb and hexagonal networks, J. Comp. Math. Sci., 5, pp. 530-535.
• Randic, M., (1975), On characterization of molecular branching, J. Am. Chem. Soc., 97, pp. 6609-6615.
• Rucker, G., Rucker, C., (1999), On topological indices, boiling points, and cycloalkanes. J. Chem. Inf. Comput. Sci., 39, pp. 788-802.
• Shafiei, F., (2015), Relationship between topological indices and thermodynamic properties and of the monocarboxylic acids applications in QSPR, Iranian Journal of Mathematical Chemistry, 1(6), pp. 28.
• Standal, I.B., Rainuzzo, J., Axelson, D.E., Valdersnes, S., Julshamn, K., Aursand, M., (2012), Clas- sification of geographical origin by PNN analysis of fatty acid data and level of contaminants in oils from Peruvian anchovy, J. Am. Oil Chem. Soc., 89(7), pp. 1173-1182.
• Specht, D.F., (1990), Probabilistic neural networks, Neural Netw., 3, pp. 109-118.
• Tran, T., Nguyen, T., Tsai, P., Kong, X., (2011) BSPNN: boosted subspace probabilistic neural network for email security. Artif. Intell. Rev., 35, pp. 369-382.
• Tran, T.P., Cao, L., Tran, D., Nguyen, C.D., Novel intrusion detection using probabilistic neural network and adaptive boosting, Int. J. Comput. Sci. Inf. Secur, 6, pp. 83-91.
• Wiener, H.J., (1947), Structural determination of paraffin boiling points, J. Amer. Chem. Soc., 69, pp. 17-20.
• Wang, Y., Adali, T., Kung, S. Y., Szabo, Z., (1998) Quantification and segmentation of brain tissues from MR images: a probabilistic neural network approach, Ieee Transactions on Image Processing, (8), 1165-1181.
• West, D.B, 1996, Introduction to Graph Theory, USA Printce Hall.
• Yan, F., Shang, Q., Xia, S., Wang, Q., and Ma, P., (2015), Application of topological index in predicting ionic liquids densities by the quantitative structure property relationship method, J. Chem. Eng. Data, 60, pp. 734-739.
• Wang, C., Liu, J., Wang, S., (2017) Sharp upper bounds for multiplicative Zagreb indices of bipartite graphs with given diameter, Discrete Applied Mathematics, 227, pp. 156-165