  Simple cycles, also known as self-avoiding polygons, are cycles on graphs
which are not allowed to visit any vertex more than once. We present an exact
formula for enumerating the simple cycles of any length on any directed graph
involving a sum over its induced subgraphs. This result stems from an Hopf
algebra, which we construct explicitly, and which provides further means of
counting simple cycles. Finally, we obtain a more general theorem asserting
that any Lie idempotent can be used to enumerate simple cycles.
