Lyapunov exponent for quantum graphs that are elements of a subshift of finite type

We consider the Schrödinger operator on the quantum graph whose edges connect the points of $\mathbb{Z}$. 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 $[0,\mathrm{\infty })$. The number of points $E$ of this subset in $[(\pi (j-1){)}^2,(\pi j{)}^2]$ is the same for all $j\in \mathbb{N}$.