On the Topological Properties of the Certain Neural Networks

Liu, Jia-Bao; Zhao, Jing; Wang, Shaohui; Javaid, M.; Cao, Jinde

doi:10.1515/jaiscr-2018-0016

Abstract

A topological index is a numeric quantity associated with a network or a graph that characterizes its whole structural properties. In [Javaid and Cao, Neural Computing and Applications, DOI 10.1007/s00521-017-2972-1], the various degree-based topological indices for the probabilistic neural networks are studied. We extend this study by considering the calculations of the other topological indices, and derive the analytical closed formulas for these new topological indices of the probabilistic neural network. Moreover, a comparative study using computer-based graphs has been carried out first time to clarify the nature of the computed topological descriptors for the probabilistic neural networks. Our results extend some known conclusions.

References

[1] J. Cao, R. Li, Fixed-time synchronization of delayed memristor-based recurrent neural networks, Sci. China. Inf. Sci. 60(3) (2017) 032201.10.1007/s11432-016-0555-2
Search in Google Scholar Back to article
[2] Y. Huo, J. B-Liu, J. Cao, Synchronization analysis of coupled calcium oscillators based on two regular coupling schemes, Neurocomputing 165 (2015) 126-132.10.1016/j.neucom.2015.03.001
Search in Google Scholar Back to article
[3] Z. Guo, J. Wang, Z. Yan, Attractivity analysis of memristor-based cellular neural networks with time-varying delays, IEEE Trans. Neural Netw. Learn. Syst. 25 (2013) 704-717.10.1109/TNNLS.2013.228055624807948
Search in Google Scholar Back to article
[4] J. Devillers, A. T. Balaban, Topological Indices and Related Descriptors in QSAR and QSPR, Gordon Breach, Amsterdam 1999.10.1201/9781482296945
Search in Google Scholar Back to article
[5] M. Karelson, Molecular Descriptors in QSAR/QSPR, Wiley, New York, 2000.
Search in Google Scholar Back to article
[6] M. Javaid, J. B-Liu, M. A. Rehman, S. H. Wang, On the Certain Topological Indices of Titania Nanotube TiO2[m,n], Zeitschrift f¨ur Naturforschung A 72(7) 2017 647-654.10.1515/zna-2017-0101
Search in Google Scholar Back to article
[7] M. Imran, S. Hafi, W. Gao, M. R. Farahani, On topological properties of sierpinski networks, Chaos, Solitons and Fractals 98 (2017) 199-204.10.1016/j.chaos.2017.03.036
Open DOI Search in Google Scholar Back to article
[8] S. H. Wang, B. Wei, Multiplicative Zagreb indices of k-trees, Discrete Appl. Math. 180 (2015) 168-175.10.1016/j.dam.2014.08.017
Search in Google Scholar Back to article
[9] M. Javaid, Jinde Cao, Computing topological indices of probabilistic neural network, Neural Comput. Applic. (2017). doi:10.1007/s 00521-017-2972.
Open DOI Search in Google Scholar Back to article
[10] W. Gao, M. K. Siddiqui, Molecular descriptors of nanotube, oxide, silicate, and triangulene networks, Journal of Chemistry 2017 (2017).10.1155/2017/6540754
Search in Google Scholar Back to article
[11] J. B-Liu, S. H.Wang, C.Wang, S. Haya, Further results on computation of topological indices of certain networks, IET Control Theory & Applications 11(13) (2017) 2065-2071. 10.1049/iet-cta.2016.1237
Search in Google Scholar Back to article
[12] J. A. Bondy, U. S. R. Murty, Graph Theory with Applications, Macmillan, New York, 1976.10.1007/978-1-349-03521-2
Search in Google Scholar Back to article
[13] H. Wiener, Structural determination of the paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17-20.10.1021/ja01193a005
Search in Google Scholar Back to article
[14] O. Ivanciuc, T. S. Balaban, A. T. Balaban, Reciprocal distance matrix, related local vertex invariants and topological indices, J. Math. Chem. 12 (1993) 309-318.10.1007/BF01164642
Open DOI Search in Google Scholar Back to article
[15] R. Todeschini, V. Consonni, Molecular descriptors for chemoinformatics, vol I, vol II. Wiley-VCH, Weinheim (2009) 934-938.10.1002/9783527628766
Search in Google Scholar Back to article
[16] K. Xu, K. C. Das, N. Trinajstic, The Harary Index of a Graph, Springer Briefs in Mathematical Methods, DOI:10.1007/97836624584335.10.1007/97836624584335
Open DOI Search in Google Scholar Back to article
[17] I. Gutman, Selected properties of the Schultz molecular topological index, J. Chem. Inf. Comput. Sci. 34 (1994) 1087-1089.10.1021/ci00021a009
Open DOI Search in Google Scholar Back to article
[18] H. Narumi, M. Hatayama, Simple topological index. a newly devised index charaterizing the topological nature of structural isomers of saturated hydrocarbons, Mem. Fac. Eng. Hokkaido Univ. 16 (1984) 209-214.
Search in Google Scholar Back to article
[19] D. J. Klein, V. R. Rosenfeld, The Narumi-Katayama degree-product index and the degreeproduct polynomial, in: I. Gutman, B. Furtula (Eds.), Novel Molecular Structure Descriptors- Theory and Applications II, Univ. Kragujevac, Kragujevac (2010) 79-90.
Search in Google Scholar Back to article
[20] I. Gutman, Some Properties of the Wiener Polynomials, Graph Theory Notes New York 25 (1993) 13-18.
Search in Google Scholar Back to article
[21] C. Wang, J.-B.Liu, S. Wang, Sharp upper bounds for multiplicative Zagreb indices of bipartite graphs with given diameter, Discrete Appl. Math. 227 (2017) 156-165.10.1016/j.dam.2017.04.037
Search in Google Scholar Back to article
[22] I. Gutman, N. Trinajstić, Graph theory and molecular orbits. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535-538.10.1016/0009-2614(72)85099-1
Open DOI Search in Google Scholar Back to article
[23] B. Furtula, I. Gutman, Z. Kovijani´c Vuki´cevi´c, G. Lekishvili, G. Popivoda, On an old/new degreebased topological index, Sciences mathematiques 40 (2015) 19-31.
Search in Google Scholar Back to article
[24] S. Mukwembi, A note on diameter and the degree sequence of a graph, Appl. Math. Lett. 25 (2012) 175-178.10.1016/j.aml.2011.08.010
Open DOI Search in Google Scholar Back to article
[25] R. Todeschini, D. Ballabio, V. Consonni, Novel molecular descriptors based on functions of new vertex degrees, in: I. Gutman, B. Furtula (Eds.), Novel Molecular Structure Descriptors Theory and Applications I, Univ. Kragujevac, Kragujevac (2010) 72-100.
Search in Google Scholar Back to article
[26] R. Todeschini, V. Consonni, New local vertex invariants and molecular descriptors based on functions of the vertex degrees, MATCH Commun. Math. Comput. Chem. 64 (2010) 359-372.
Search in Google Scholar Back to article
[27] S. Wang, B. Wei, Multiplicative Zagreb indices of k-trees, Discrete Appl. Math. 180 (2015) 168-175.10.1016/j.dam.2014.08.017
Search in Google Scholar Back to article
[28] O. Ivanciuc, QSAR comparative study of Wiener descriptors for weighted molecular graphs, J. Chem. Inf. Comput. Sci. 40 (2000) 1412-1422.10.1021/ci000068y11128100
Open DOI Search in Google Scholar Back to article
[29] O. Ivanciuc, T. Ivanciuc, AT. Balaban, The complementary distance matrix, a new molecular graph metric, ACH Models Chem. 137 (2000) 57-82.
Search in Google Scholar Back to article

On the Topological Properties of the Certain Neural Networks

Abstract

Paradigm

My account