On finite metric graphs the set of all realizations Laplace operator in edgewise defined $L^2$-spaces are studied. These by coupling boundary conditions at vertices most which define non-self-adjoint operators. In [Hussein, Krej\v{c}i\v{r}\'{\i}k, Siegl, Trans. Amer. Math. Soc., 367(4):2921--2957, 2015] a notion regularity means Cayley transform parametrizing matrices has been proposed. The mai...