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We investigate dispersive and Strichartz estimates for the Schrödinger time evolution propagator on a star-shaped metric graph. The linear operator, , taken into consideration is the self-adjoint extension of the Laplacian, subject to a wide class of coupling conditions. The study relies on an explicit spectral representation of the solution in terms of the resolvent kernel which is further analyzed using results from oscillatory integrals. As an application, we obtain the global well-posedness for a class of semilinear Schrödinger equations. [ABSTRACT FROM AUTHOR]