Research Article Computing Topological Invariants of Deep Neural Networks Xiujun Zhang , 1 Nazeran Idrees , 2 Salma Kanwal , 3 Muhammad Jawwad Saif, 4 and Fatima Saeed 2 1 School of Computer Science, Chengdu University, Chengdu, China 2 Department of Mathematics, Government College University Faisalabad, Faisalabad 38000, Pakistan 3 Department of Mathematics, Lahore College for Women University, Lahore 54000, Pakistan 4 Department of Applied Chemistry, Government College University Faisalabad, Faisalabad 38000, Pakistan Correspondence should be addressed to Nazeran Idrees; nazeranidrees@gcuf.edu.pk and Salma Kanwal; salma.kanwal055@gmail.com Received 8 May 2022; Revised 27 July 2022; Accepted 12 September 2022; Published 7 October 2022 Academic Editor: Baiyuan Ding Copyright © 2022 Xiujun Zhang et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. e present study deals with the topological properties of deep neural networks. e 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. 1. Introduction Neural networks are not only studied in artificial intelligence but also have got great applications in intrusion detection systems, image processing, localization, medicine, and chemical and environmental sciences [1–3]. Neural net- works are used to model and learn complex and nonlinear relationships, which is very important in real life because many of the relationships of inputs and outputs are non- linear and complex. Artificial neural networks are the backbone of robotics, defense technology, and neural chemistry. Neural networks are not only being widely used as a tool for predictive analysis but also trained successfully to model processes including crystallization, adsorption, distillation, gasification, dry reforming, and filtration in neural chemistry [4–8]. e topological index associates a unique number to a graph or network, which provides correlation with the physiochemical properties of the network. Degree-based topological index depends upon the connectivity of the network. e first degree-based topological index, called the Randi´ c index, was formulated by Milan Randi´ c [9] while analyzing the boiling point of paraffin. Over the last three decades, hundreds of topological indices have been for- mulated by researchers, which are helpful in studying the different properties of chemical graphs like reactivity, sta- bility, boiling point, enthalpy of formation, and Kovat’s constant and inherits physical properties of materials such as stress, elasticity, strain, mechanical strength, and many others. Bollob´ as and Erdős [10] introduced the general Randi´ c index given by equation (1). e first and second Zagreb indices were introduced by Gutman and Trinajsti´ c [11] in 1972, which appeared during the analysis of π-electron energy of atoms. e multiplicative version of these Zagreb indices (the first multiplicative Zagreb index and the second multi- plicative Zagreb index) of a graph were formulated by Ghorbani and Azimi [12]. Shirdel et al. [13] introduced a new version of Zagreb indices named as the hyper-Zagreb index. e widely used atom-bond connectivity (ABC) index is Hindawi Computational Intelligence and Neuroscience Volume 2022, Article ID 9051908, 11 pages https://doi.org/10.1155/2022/9051908