Abstract: | 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 (i.e. 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. |