In this paper, we present the $\mathrm{\Psi }$-intersection graph ${\mathrm{\Psi }}_{\mathcal{S}}(\mathcal{B})$ associated with an $S$-act over topological semigroups. We investigate how the internal structure of $S$-acts, particularly the presence and distribution of simple subacts, determines fundamental graph-theoretic properties of ${\mathrm{\Psi }}_{\mathcal{S}}(\mathcal{B})$. Characterizations of connectivity, bipartiteness, girth, and diameter are established in terms of decomposition properties and chain conditions on subacts. We further analyze the clique structure of ${\mathrm{\Psi }}_{\mathcal{S}}(\mathcal{B})$, showing that every clique contains at most one simple subact and describing maximal cliques through families of non-simple subacts intersecting via simple components. Domination-type parameters are also studied by introducing indecomposable controlling sets and deriving bounds for the control number of ${\mathrm{\Psi }}_{\mathcal{S}}(\mathcal{B})$. Finally, by using monophonic convexity and the $m$-topology on graphs, we examine induced path structures and their interaction with algebraic paths in $S$-acts, revealing connections between geodesic behavior and semigroup actions. These results provide a unified algebraic, topological, and graph-theoretic framework and open new combinatorial and metric approaches to the study of topological path semigroups.

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.