This paper introduces a new class of topological structures on undirected simple graphs, called extreme graphical topological spaces, using the concepts of extreme systems and lower approximation neighborhood systems. We explore the conditions under which graphs produce either the indiscrete or discrete topology. Several fundamental properties related to both topology and graph theory are examined. Using this framework, along with the concept of extreme outer connectedness in geodesic paths, we define and study new types of connected graphs and discrete spaces, such as extreme maximally connected geodesic graphs and extreme maximally geodesic discrete spaces. Finally, we demonstrate applications of these structures by analyzing the connectedness and discrete characteristics of networks that represent X-ray structures of certain chemical compounds.