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We study a boundary value problem related to the search of standing waves forthe nonlinear Schr\"odinger equation (NLS) on graphs. Precisely we areinterested in characterizing the standing waves of NLS posed on the {\itdouble-bridge graph}, in which two semi-infinite half-lines are attached at acircle at different vertices. At the two vertices the so-called Kirchhoffboundary conditions are imposed. The configuration of the graph ischaracterized by two lengths, $L_1$ and $L_2$, and we are interested in theexistence and properties of standing waves of given frequency $\omega$. Forevery $\omega>0$ only solutions supported on the circle exist (cnoidalsolutions), and only for a rational value of $L_1/L_2$; they can be extended toevery $\omega\in \mathbb{R}$. We study, for $\omega<0$, the solutions periodicon the circle but with nontrivial components on the half-lines. The problemturns out to be equivalent to a nonlinear boundary value problem in which theboundary condition depends on the spectral parameter $\omega$. Afterclassifying 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 acountable set of frequencies $\omega_n$. Moreover we show that the frequencysequence $\{\omega_n\}_{n \geq 1}$ has a cluster point at $-\infty$ and itadmits at least a finite limit point, in general non-zero. Finally, anynegative real number can be a limit point of a set of admitted frequencies upto the choice of a suitable irrational geometry $L_1/L_2$ for the graph. Theseresults depend on basic properties of diophantine approximation of realnumbers.

Standing waves for the NLS on the double-bridge graph and a rational-irrational dichotomy