The paper discusses the Nonlinear Dirac Equation with Kerr-type nonlinearity (i.e., |ψ| p−2 ψ) on noncompact metric graphs with a finite number of edges, in the case of Kirchhoff-type vertex conditions. Precisely, we prove local well-posedness for the associated Cauchy problem in the operator domain and, for infinite N -star graphs, the existence of standing waves bifurcating from the trivial solution at ω = mc 2 , for any p > 2. In the Appendix we also discuss the nonrelativistic limit of the Dirac-Kirchhoff operator.