The folded map on an identification graph and its application

In this article, we define a new operation on a graph, namely the u-graph and find a necessary and sufficient condition for the folded map to be a homomorphism. Also, we introduce the limit folded map on the u-graph. We deduce the relation between a special family of graphs and their folded maps. We construct a finite sequence of folded maps on a finite number of connected u-graphs. Under certain conditions, we derive the relationship between a u-graph and its subgraphs. Furthermore, we present some applications for new graph operations.