Abstract
Topological indices are graph invariants determined by the distance or degree of vertices of the molecular graph. Topological indices have been used effectively in chemical graph theory in explaining the structures and predicting certain physicochemical properties of chemical compounds. In this research, we introduce the first, second, and forgotten downhill Zagreb indices and calculate those topological indices for some standard families of graphs and the join of graphs. Also, the downhill topological indices for the firefly graph, book graph, and stacked book graph are established. Finally, the downhill indices of Graphene and honeycomb network are obtained.