Topological groups, automorphisms of infinite graphs and a theorem of Trofimov

We give a short proof of a theorem of Trofimov, using the theory of topological groups. An automorphism g of a graph is bounded if there is a number M such that the distance between a vertex v and gv is less than M for all vertices v in the graph. Trofimovʹs theorem is a characterization of those locally finite infinite graphs that admit a transitive group of bounded automorphisims.