The paper discusses the Nonlinear Dirac Equation with Kerr-type nonlinearity (i.e., $\psi^{p-2}\psi$) on noncompact metric graphs a finite number of edges, in case Kirchhoff-type vertex conditions. Precisely, we prove local well-posedness for associated Cauchy problem operator domain and, infinite $N$-star graphs, existence standing waves bifurcating from trivial solution at $\omega=mc^2$, any ...