  A theoretical framework for sesquilinear forms defined on the direct sum of
Hilbert spaces is developed in the first part. Conditions for the boundedness,
ellipticity and coercivity of the sesquilinear form are proved. A criterion of
E.-M. Ouhabaz is used in order to prove qualitative properties of the abstract
Cauchy problem having as generator the operator associated with the
sesquilinear form. In the second part we analyze quantum graphs as a special
case of forms on subspaces of the direct sum of Hilbert spaces. First, we set
up a framework for handling quantum graphs in the case of infinite networks.
Then, the operator associated with such systems is identified and investigated.
Finally, we turn our attention to symmetry properties of the associated
parabolic problem and we investigate the connection with the physical concept
of a gauge symmetry.
