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We show that the cycle-rank r(L) of a Lyapunov graph L on a manifold M satisfies: r(L) less than or equal to g(M), where g(M) is the genus of M. This generalizes a theorem of Franks. We also show that given any integer r with 0 less than or equal to r less than or equal to g(M), r = r(L) for some Lyapunov graph L on M, dim M > 2

Cycle rank of Lyapunov graphs and the genera of manifolds