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 \([0,\infty )\) . The number of points E of this subset in \([(\pi (j-1))^2, (\pi j)^2]\) is the same for all \(j\in {{\mathbb {N}}}\) .
