  On finite metric graphs the set of all realizations of the Laplace operator
in the edgewise defined $L^2$-spaces are studied. These are defined by coupling
boundary conditions at the vertices most of 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 of regularity of boundary conditions by means
of the Cayley transform of the parametrizing matrices has been proposed. The
main point presented here is that not only the existence of this Cayley
transform is essential for basic spectral properties, but also its poles and
its asymptotic behaviour. It is shown that these poles and asymptotics can be
characterized using the quasi-Weierstrass normal form which exposes some
"hidden" symmetries of the system. Thereby, one can analyse not only the
spectral theory of these mostly non-self-adjoint Laplacians, but also the
well-posedness of the time-dependent heat-, wave- and Schr\"odinger equations
on finite metric graphs as initial-boundary value problems. In particular, the
generators of $C_0$- and analytic semigroups and $C_0$-cosine operator
functions can be characterized. On star-shaped graphs a characterization of
generators of bounded $C_0$-groups and thus of operators similar to
self-adjoint ones is obtained.
