Graph theory and topology provide complementary mathematical frameworks for studying complex systems. Graphs describe systems through vertices and edges, while topology investigates global structural properties such as connectivity, holes, cycles, continuity, and higher-dimensional relationships. In many modern applications, however, pairwise relationships are not sufficient to represent the full structure of a system. Simplicial complexes extend graphs by incorporating higher-order interactions among three or more elements, thereby enabling a richer topological description. This paper examines graph-theoretic methods for studying topological networks and simplicial complexes. It focuses on graph connectivity, paths, cycles, components, cliques, adjacency structures, incidence relations, clique complexes, nerves, boundary operators, Betti numbers, and higher-order connectivity. The study adopts a theoretical and analytical methodology and shows how graphs serve as the one-dimensional foundation of simplicial complexes, while simplicial complexes generalize graph-based representations to higher dimensions. The paper also discusses applications in social networks, communication systems, biological networks, sensor networks, image analysis, and topological data analysis. Particular attention is given to the transition from graphs to simplicial complexes through clique expansion and filtration-based constructions. The analysis indicates that graph-theoretic techniques are useful for identifying local and global structural properties, whereas simplicial methods are necessary when higher-order relationships and topological holes must be represented. The paper concludes that the integration of graph theory and simplicial topology provides a powerful mathematical framework for studying complex networked systems.