Graphical Homotopy Theory for Intersection Graphs of Semigroups via Path Spaces and Uniform Structures with Applications to Graphical Total Semigroups

In this article, we study the homotopy aspects of intersection graphs of topological semigroups. We begin by defining the top intersection graph TGX and investigating how the algebraic and topological properties of a topological semigroup are reflected in the global structure of this graph. In particular, we characterize when TGX is totally disconnected, bipartite, or planar in terms of the order and factorization of the underlying semigroup. We then introduce the notions of HTG-semigroups, graphical homomorphisms, and graphical homotopy relations, thereby developing a graphical homotopy framework. Within this setting, we study Gr-homotopy equivalences, Gr-contractible spaces, and retraction phenomena, including DGr-retracts and homotopy extension properties. Finally, we introduce graphical total semigroups and equip the set of Gr-path homotopy classes [Xpe] with a natural Δ-uniform topology. We show that this topology is compatible with the induced semigroup operation, yielding a topological semigroup structure. Overall, this work provides a unified algebraic, topological, and graph-theoretic perspective, and opens the door to further applications of homotopy theory in the study of intersection graphs of topological semigroups.