On the Topological Properties of the Certain Neural Networks

Open access


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.

[1] J. Cao, R. Li, Fixed-time synchronization of delayed memristor-based recurrent neural networks, Sci. China. Inf. Sci. 60(3) (2017) 032201.

[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.

[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.

[4] J. Devillers, A. T. Balaban, Topological Indices and Related Descriptors in QSAR and QSPR, Gordon Breach, Amsterdam 1999.

[5] M. Karelson, Molecular Descriptors in QSAR/QSPR, Wiley, New York, 2000.

[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.

[7] M. Imran, S. Hafi, W. Gao, M. R. Farahani, On topological properties of sierpinski networks, Chaos, Solitons and Fractals 98 (2017) 199-204.

[8] S. H. Wang, B. Wei, Multiplicative Zagreb indices of k-trees, Discrete Appl. Math. 180 (2015) 168-175.

[9] M. Javaid, Jinde Cao, Computing topological indices of probabilistic neural network, Neural Comput. Applic. (2017). doi:

[10] W. Gao, M. K. Siddiqui, Molecular descriptors of nanotube, oxide, silicate, and triangulene networks, Journal of Chemistry 2017 (2017).

[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.

[12] J. A. Bondy, U. S. R. Murty, Graph Theory with Applications, Macmillan, New York, 1976.

[13] H. Wiener, Structural determination of the paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17-20.

[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.

[15] R. Todeschini, V. Consonni, Molecular descriptors for chemoinformatics, vol I, vol II. Wiley-VCH, Weinheim (2009) 934-938.

[16] K. Xu, K. C. Das, N. Trinajstic, The Harary Index of a Graph, Springer Briefs in Mathematical Methods, DOI:10.1007/97836624584335.

[17] I. Gutman, Selected properties of the Schultz molecular topological index, J. Chem. Inf. Comput. Sci. 34 (1994) 1087-1089.

[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.

[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.

[20] I. Gutman, Some Properties of the Wiener Polynomials, Graph Theory Notes New York 25 (1993) 13-18.

[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.

[22] I. Gutman, N. Trinajstić, Graph theory and molecular orbits. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535-538.

[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.

[24] S. Mukwembi, A note on diameter and the degree sequence of a graph, Appl. Math. Lett. 25 (2012) 175-178.

[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.

[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.

[27] S. Wang, B. Wei, Multiplicative Zagreb indices of k-trees, Discrete Appl. Math. 180 (2015) 168-175.

[28] O. Ivanciuc, QSAR comparative study of Wiener descriptors for weighted molecular graphs, J. Chem. Inf. Comput. Sci. 40 (2000) 1412-1422.

[29] O. Ivanciuc, T. Ivanciuc, AT. Balaban, The complementary distance matrix, a new molecular graph metric, ACH Models Chem. 137 (2000) 57-82.

Journal of Artificial Intelligence and Soft Computing Research

The Journal of Polish Neural Network Society, the University of Social Sciences in Lodz & Czestochowa University of Technology

Journal Information

CiteScore 2017: 5.00

SCImago Journal Rank (SJR) 2017: 0.492
Source Normalized Impact per Paper (SNIP) 2017: 2.813

Cited By


All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 414 414 37
PDF Downloads 210 210 8