  We study a boundary value problem related to the search of standing waves for
the nonlinear Schr\"odinger equation (NLS) on graphs. Precisely we are
interested in characterizing the standing waves of NLS posed on the {\it
double-bridge graph}, in which two semi-infinite half-lines are attached at a
circle at different vertices. At the two vertices the so-called Kirchhoff
boundary conditions are imposed. The configuration of the graph is
characterized by two lengths, $L_1$ and $L_2$, and we are interested in the
existence and properties of standing waves of given frequency $\omega$. For
every $\omega>0$ only solutions supported on the circle exist (cnoidal
solutions), and only for a rational value of $L_1/L_2$; they can be extended to
every $\omega\in \mathbb{R}$. We study, for $\omega<0$, the solutions periodic
on the circle but with nontrivial components on the half-lines. The problem
turns out to be equivalent to a nonlinear boundary value problem in which the
boundary condition depends on the spectral parameter $\omega$. After
classifying the solutions with rational $L_1/L_2$, we turn to $L_1/L_2$
irrational showing that there exist standing waves only in correspondence to a
countable set of frequencies $\omega_n$. Moreover we show that the frequency
sequence $\{\omega_n\}_{n \geq 1}$ has a cluster point at $-\infty$ and it
admits at least a finite limit point, in general non-zero. Finally, any
negative real number can be a limit point of a set of admitted frequencies up
to the choice of a suitable irrational geometry $L_1/L_2$ for the graph. These
results depend on basic properties of diophantine approximation of real
numbers.
