<p>A neural network is a part of machine learning models inspired by the structure of the human brain that has obtained significant applications in various areas due to its capacity to model complex patterns and relationships in data. Neural networks have a wide range of applications in pattern recognition, natural language processing and image recognition. Recently, a graph-theoretic approach to neural networks to study their structural properties and classification has been started. Using different graph-theoretic properties has applications in studying the topological properties of neural networks. In this article, for the first time, we study some NP-hard problems by combining graph-theoretic properties like vertex cover, maximum independence number, minimum domination number and perfect matching with cover pebbling number on several novel neural networks. Here we consider convolutional neural networks (CVNNs), modular neural networks (MNNs), generalised regression neural networks (GRNNs) and Hopfield neural networks (HNNs). To study their topological properties, we compute their structures of vertex cover pebbling number, maximum independence cover pebbling number, perfect matching cover pebbling number, and minimum domination cover pebbling number. We have also given the algorithm for each graph property. A pebbling move is the deletion of two pebbles from a vertex and placing a pebble on a neighboring vertex. The least number of pebbles needed to move a pebble each on every vertex simultaneously is known as the cover pebbling number. We compute the vertex cover pebbling number, maximum independence cover pebbling number, perfect matching cover pebbling number, and minimum domination cover pebbling number on CVNNs, MNNs, GRNNs and HNNs. These results provide insight into how different neural network architectures influence the computational complexity and values of various cover pebbling numbers, and also provide insights into the structural robustness or information flow within the networks.</p>