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This thesis is broadly concerned with two problems: obtaining the mathematical model of the specific infinite self-similar graph, and investigating the connectedness of the tree-like graph in order to show its relation to the associated hyperbolic space. Our main result concerning the former problem is that, in a variety of situations, the self-similar infinite structure obtained by using our method as the graph product of a disconnected finite graph and regular rooted tree can be connecteddoi:10.20381/ruor-452 fatcat:fxqnsazxhfe4hnga7uhlv5umui