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 connected

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... . have the hyperbolic metric space associated to it). This addresses a question about the existence of the optimal depth for the breadth-first search algorithm and also has possible applications to the recent research topics in Psychological and Brain Sciences. We approach the connectedness problem by showing the similarity of obtained geometric structures to well known algebraic structures such as groupoid and pseudogroup. One of our main results is that, under the assumption that the emerged geometric self-similar structure is connected, it is naturally associated to the hyperbolic metric space. Thus, the variety of well known methods can be applied in further study. We also show that the connectedness of our structure can be reached in the finite number of steps or can not be reached at all. This gives the grounds for the optimal application of the breadth-first search algorithm.