Negative periodic orbits for graph maps

Recently, Alves, Hric and Sousa Ramos proved that for any continuous piecewise monotone map of a graph into itself, its topological entropy is equal to the maximum of two quantities. The first one is the (exponential) upper growth rate, as n → ∞, of the number of periodic points of period n at which the orientation is reversed. The second one is the logarithm of the spectral radius of the map induced in the first homology group. We extend the essential part of this theorem to arbitrary continuous maps of a graph into itself and provide a substantially shorter proof of the whole theorem. © 2006 IOP Publishing Ltd and London Mathematical Society.