Graph theory and topology are two important branches of mathematics concerned with relationships, structure, continuity, and connectivity. Although graph theory primarily investigates discrete structures consisting of vertices and edges, topology studies spaces and properties that remain invariant under continuous transformations. The interaction between these fields provides useful mathematical techniques for representing and investigating connectivity in both discrete and continuous settings. This paper examines the application of graph-theoretic concepts to the study of topological spaces, with particular emphasis on connectivity, path connectivity, components, adjacency structures, open covers, and graph representations associated with topological objects. The study adopts a theoretical and analytical approach based on established concepts from graph theory and general topology. It explains how graphs can serve as discrete models of topological relationships and how topological realizations of graphs provide a bridge in the opposite direction. Special attention is given to connected graphs, paths, components, cut vertices, spanning trees, intersection graphs, and the nerve of a cover. The paper further discusses applications in digital topology, network analysis, computational topology, and topological data analysis. It concludes that graph theory provides an effective discrete framework for investigating many connectivity-related aspects of topological structures, while emphasizing that graph connectivity and topological connectivity must be related through a clearly defined construction rather than treated as universally equivalent.