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

Abstract

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

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

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

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.

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

Details

Primary Language English
Journal Section Research Article
Authors

M. JAVAİD This is me
Department of Mathematics, School of Science, University of Management and Technology (UMT), Lahore, Pakistan


A RAHEEM This is me
Department of Mathematics, National University of Singapore, Singapore.


M. ABBAS This is me
Department of Mathematics, GC University, Lahore, Pakistan.


J. CAO This is me
School of Mathematics, and Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing, 210 096, China

Publication Date December 1, 2019
Published in Issue Year 2019, Volume 9, Issue 4

Cite

Bibtex @ { twmsjaem760959, journal = {TWMS Journal of Applied and Engineering Mathematics}, issn = {2146-1147}, eissn = {2587-1013}, address = {Işık University ŞİLE KAMPÜSÜ Meşrutiyet Mahallesi, Üniversite Sokak No:2 Şile / İstanbul}, publisher = {Turkic World Mathematical Society}, year = {2019}, volume = {9}, number = {4}, pages = {864 - 875}, title = {M-POLYNOMIAL METHOD FOR TOPOLOGICAL INDICES OF 3-LAYERED PROBABILISTIC NEURAL NETWORKS}, key = {cite}, author = {Javaid, M. and Raheem, A and Abbas, M. and Cao, J.} }