The theoretical investigations of the previous chapters are verified, underlined and extended by various numerical experiments. This chapter includes numerical experiments on the convergence of the finite element semidiscretization and the Crank-Nicolson multigrid method. Further discussions on the computation of equilateral and non-equilateral quantum graph spectra are included as well as numerical results in the context of spectral solutions such as on the decay of coefficients and simulations of heat, fractional diffusion and reaction-diffusion equations. Finally, we compare the finite element and spectral Galerkin methods in terms of their achieved accuracy in relation to their complexity.