Summary: This article is dedicated to the problem of the existence of a finitely- axiomatizable strongly minimal theory \({\mathcal T}\) of graphs. It is shown that this problem is equivalent to the problem of the existence of such a theory \({\mathcal T}\) with the additional condition of bounded valency and homogeneity of the components of models of \({\mathcal T}\). A description of groups that act transitively on graphs of bounded valency with finitely- axiomatizable theories is the main result of the article. It turns out that these groups modulo the stabilizer of a vertex are infinite finitely-representable groups with a finite set of nontrivial classes of conjugate elements such that each nontrivial cyclic subgroup has nontrivial intersection with one of these classes.