We consider the Schr\"odinger operator on the quantum graph whose edges connect the points of ${ \Bbb Z}$. The numbers of the edges connecting two consecutive points $n$ and $n+1$ are read along the orbits of a shift of finite type. We prove that the Lyapunov exponent is potitive for energies $E$ that do not belong to a discrete subset of $[0,\infty)$. The number of points $E$ of this subset in $[(\pi (j-1))^2, (\pi j)^2]$ is the same for all $j\in { \Bbb N}$.
Categories: math-ph math.MP