Topological entropy, homological growth and zeta functions on graphs
  In connection with the Entropy Conjecture it is known that the topological entropy of a continuous graph map is bounded from below by the spectral radius of the induced map on the first homology group. We show that in the case of a piecewise monotone graph map, its topological entropy is equal precisely to the maximum of the mentioned spectral radius and the exponential growth rate of the number of periodic points of negative type. This nontrivially extends a result of Milnor and Thurston on piecewise monotone interval maps. For this purpose we generalize the concept of Milnor-Thurston zeta function incorporating in the Lefschetz zeta function. The methods developed in the paper can be used also in a more general setting. 
Jo\~ao F. Alves, Roman Hric and Jos\'e Sousa Ramos