Graph-structured data is modeled and analyzed by graph neural networks (GNNs) in a variability of domains, such as recommendation systems, social networks, and bioinformatics. Nevertheless, their efficacy is largely dependent on their capacity to represent and deduce distinct structures that are intrinsic to graphs. This paper investigates current developments in inference and learning strategies designed to deal with discrete structures in graph data. We explore encoding strategies for graph structures into continuous representations, utilizing deep learning and probabilistic modeling approaches. We also cover methods for effective inference over discrete structures, including matching subgraphs and testing graph isomorphisms. We also study how inference and learning interact in GNNs and discuss future possibilities and difficulties in this quickly developing subject. To promote improvements in graph representation learning and graph-based reasoning problems.