Abstract
The paper [11] raised the question of describing the cardinality and types of approximations for natural families of theories. In the present paper, a partial answer to this question is given, and the study of approximation and topological properties of natural classes of theories is also continued. We consider a cycle graph consisting of one cycle or, in other words, a certain number of vertices (at least 3 if the graph is simple) connected into a closed chain. It is shown that an infinite cycle graph is approximated by finite cycle graphs. Approximations of regular graphs by finite regular graphs are considered. On the other hand, approximations of acyclic regular graphs by finite regular graphs are considered. It is proved that any infinite regular graph is pseudofinite. And also, for any k, any k-regular graph is homogeneous and pseudofinite.Examples of pseudofinite 3-regular and 4-regular graphs are given.
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