In this paper, we extend the classical first and second Zagreb indices to the setting of graphons. We introduce their rigorous integral definitions, \( M_1(W) \) and \( M_2(W) \), and establish their asymptotic properties, which provide a bridge between these graphon-based indices and the traditional Zagreb indices of finite graphs. Furthermore, we develop a general framework for extending arbitrary degree-based graph indices to graphons, enabling the analysis of large-scale networks. We investigate extremal problems for these indices and explore their relationship with network assortativity.

Overall, our results provide a powerful set of tools to analyze the topological properties of large real-world networks. We demonstrate their practical utility by applying the graphon framework to model and analyze complex systems in various disciplines, including chemistry. These applications highlight how our graphon-based indices can provide insights into key structural features, such as network heterogeneity and inter-group interactions.