\noindent Abstract: Together with Arkady Kurnosov and Sven Gnutzmann\\ \noindent Given a quantum hamiltonian, represented as a $N\times N$ Hermitian matrix $H$, we derive an expression for the largest $\it Lyapunov \ exponent$ of the $\it classical\ trajectories$ in the $\it phase-space$ appropriate for the dynamics induced by $H$. To this end we associate to $H$ a graph with $N$ vertices and derive a $\it quantum\ map$ on functions defined on the directed edges of the graph. Using the $\it semi-classical\ approach\ in\ the \ reverse\ direction$ we obtain the corresponding classical evolution (Liouvillian) operator. Using ergodic theory methods (Sinai, Ruelle, Bowen, Pollicot...) we obtain close expressions for the Lyapunov exponent, as well as for its variance. Applications for random matrix models will be presented. \end{document}
