Abstract :
The link of a vertex υ of a graph G is the subgraph induced by the set of vertices of G adjacent to υ. If all the links of G are isomorphic to a given graph H, then G is called locally H, or locally homogeneous. We deal with the problem of characterization of all connected locally graphs for H fixed. It turns out that the problem is closely related to the theory of covering spaces. This fundamental observation is generalized and developed in Section 1 of the paper. Theoretical results from Section 1 are used for deriving of some new results on locally homogeneous graphs and reproving of some old ones. For instance, Theorem 2.7 establishes that if a graph H has a limited number of edges then either there is no locally H graph, or there are infinitely many finite connected locally H graphs.
