In this paper, a random number technique for computing the value of a linkage characteristic polynomial is shown to be an effective method for identifying isomorphic graphs. The technique has been applied to the topological synthesis of one- degree-of- freedom, epicyclic gear trains with up to six links. All the permissible graphs of epicyclic gear trains were generated by a systematic procedure, and the isomorphic graphs were identified by comparing the values of their corresponding linkage characteristic polynomials. It is shown that there are 26 nonisomorphic rotation graphs and 80 displacement nonisomorphic graphs from which all the six-link, one-degree-of- freedom, epicyclic gear trains can be derived.