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We treat the stationary (cubic) nonlinear Schr\"odinger equation (NSLE) onsimplest graphs. Formulation of the problem and exact analytical solutions ofNLSE are presented for star graphs consisting of three bonds. It is shown thatthe method can be extended for the case of arbitrary number of bonds of stargraphs and for other simplest topologies such as tree and loop graphs. The caseof repulsive and attractive nonlinearities are treated separately.

Stationary Nonlinear Schr\"odinger Equation on Simplest Graphs: Boundary conditions and exact solutions