Assume Ω is a connected and simple graph with edge set E(Ω) and vertex set V (Ω). In chemical graph theory, the Geometric-Arithmetic index, Sombor index, and first Zagreb index of graph are three well-defined and studied topological indices. In the present study, we introduce the new graph invariant, we call it as GSM(Ω) index, this new graph invariant is the product of Geometric-Arithmetic index, Sombor index, and first Zagreb index. Furthermore we discuss the effect on GSM(Ω) of inserting and deleting an edge into a graph Ω. In addition, we investigate the connections between the GSM(G) index and various well-studied topological indices.We discussed nano JD*-Homeomorphisms in nano topological spaces in this study. We address the fundamental characteristics and noteworthy characterizations of nano JD*-Homeomorphisms in nano topological spaces. Our analysis demonstrates how they relate to other ideas already known, such as nano semihomeomorphisms, nano pre-homeomorphisms, and nano g-homeomorphisms