Hidden symmetries in non-self-adjoint graphs

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 main point presented here is that not only existence this essential for basic spectral properties, but also its poles and asymptotic behaviour. It shown these asymptotics can be characterized using quasi-Weierstrass normal form exposes some "hidden" symmetries system. Thereby, one analyse theory mostly Laplacians, well-posedness time-dependent heat-, wave- Schr\"odinger equations on as initial-boundary value problems. particular, generators $C_0$- analytic semigroups $C_0$-cosine functions characterized. star-shaped characterization bounded $C_0$-groups thus operators similar to self-adjoint ones obtained.