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We present a brief overview on the existence/nonexistence of standing wavesfor the NonLinear Schr\"odinger and the NonLinear Dirac Equations (NLSE/NLDE)on metric graphs with localized nonlinearity. We first focus on the NLSE, bothin the subcritical and the critical case, and then on the NLDE, highlightingsimilarities and differences with the NLSE. Finally, we show how the twoequations are related in the nonrelativistic limit, proving the convergence ofbound states.

An overview on the standing waves of nonlinear Schr\"odinger and Dirac equations on metric graphs with localized nonlinearity