Molecular structure of DNA via Zagreb connection descriptors

Topological indices quantify the connectivity and structural properties of chemical compounds. We use the topological indices for predicting and evaluating the numerous properties of molecules, such as boiling temperatures, toxicity, and biological activity. Zagreb connection indices are a useful tool for studying the structural characteristics of the DNA backbone network. These indices provide important information on the arrangement and connections between nucleotide bases inside the DNA molecule. These indices show compactness, complexity, and topological properties in order to predict DNA bending propensity, DNA-protein interaction, and DNA stability. DNA folding patterns and the impact of mutations on DNA networks are areas of further research for these topological indices. In this study, we calculate Zagreb connection indices and modified Zagreb connection indices for backbone DNA network and subdivided backbone DNA network. Furthermore, we compute the hyper-Zagreb connection index, the inverse sum connection index, and the harmonic connection index.

Computing Topological Invariants of Deep Neural Networks

A deep neural network has multiple layers to learn more complex patterns and is built to simulate the activity of the human brain. Currently, it provides the best solutions to many problems in image recognition, speech recognition, and natural language processing. The present study deals with the topological properties of deep neural networks. The topological index is a numeric quantity associated to the connectivity of the network and is correlated to the efficiency and accuracy of the output of the network. Different degree-related topological indices such as Zagreb index, Randic index, atom-bond connectivity index, geometric-arithmetic index, forgotten index, multiple Zagreb indices, and hyper-Zagreb index of deep neural network with a finite number of hidden layers are computed in this study.

A Novel Fast Feedforward Neural Networks Training Algorithm

In this paper1 a new neural networks training algorithm is presented. The algorithm originates from the Recursive Least Squares (RLS) method commonly used in adaptive filtering. It uses the QR decomposition in conjunction with the Givens rotations for solving a normal equation - resulting from minimization of the loss function. An important parameter in neural networks is training time. Many commonly used algorithms require a big number of iterations in order to achieve a satisfactory outcome while other algorithms are effective only for small neural networks. The proposed solution is characterized by a very short convergence time compared to the well-known backpropagation method and its variants. The paper contains a complete mathematical derivation of the proposed algorithm. There are presented extensive simulation results using various benchmarks including function approximation, classification, encoder, and parity problems. Obtained results show the advantages of the featured algorithm which outperforms commonly used recent state-of-the-art neural networks training algorithms, including the Adam optimizer and the Nesterov’s accelerated gradient.

Study of Generalized Hourglass Section in Carbon Nanocone via Connection Number

In 1972, Gutman and Trinajstic showed that total $\pi$ -energy of a molecule depends upon a numeric quantity which is often called as Zagreb index. In the same report, they also discussed another numeric quantity depending on the number of atoms at a distance two from a particular atom and proved influencing results on $\pi$ -energy of a molecule. In modern literature, this quantity is named as connection number. In this article, we will describe some Zagreb connection numbers for hourglass section in carbon nanocone network with different lengths of cycle in the central core.

Topological Properties of Four-Layered Neural Networks

A topological property or index of a network is a numeric number which characterises the whole structure of the underlying network. It is used to predict the certain changes in the bio, chemical and physical activities of the networks. The 4-layered probabilistic neural networks are more general than the 3-layered probabilistic neural networks. Javaid and Cao [Neural Comput. and Applic., DOI 10.1007/s00521-017-2972-1] and Liu et al. [Journal of Artificial Intelligence and Soft Computing Research, 8(2018), 225-266] studied the certain degree and distance based topological indices (TI’s) of the 3-layered probabilistic neural networks. In this paper, we extend this study to the 4-layered probabilistic neural networks and compute the certain degree-based TI’s. In the end, a comparison between all the computed indices is included and it is also proved that the TI’s of the 4-layered probabilistic neural networks are better being strictly greater than the 3-layered probabilistic neural networks.

Torus Pairwise Disjoint-Path Routing

Modern supercomputers include hundreds of thousands of processors and they are thus massively parallel systems. The interconnection network of a system is in charge of mutually connecting these processors. Recently, the torus has become a very popular interconnection network topology. For example, the Fujitsu K, IBM Blue Gene/L, IBM Blue Gene/P, and Cray Titan supercomputers all rely on this topology. The pairwise disjoint-path routing problem in a torus network is addressed in this paper. This fundamental problem consists of the selection of mutually vertex disjoint paths between given vertex pairs. Proposing a solution to this problem has critical implications, such as increased system dependability and more efficient data transfers, and provides concrete implementation of green and sustainable computing as well as security, privacy, and trust, for instance, for the Internet of Things (IoT). Then, the correctness and complexities of the proposed routing algorithm are formally established. Precisely, in an n-dimensional k-ary torus ( n < k , k ≥ 5 ), the proposed algorithm connects c ( c ≤ n ) vertex pairs with mutually vertex-disjoint paths of lengths at most 2 k ( c - 1 ) + n ⌊ k / 2 ⌋ , and the worst-case time complexity of the algorithm is O ( n c 4 ) . Finally, empirical evaluation of the proposed algorithm is conducted in order to inspect its practical behavior.