  We consider Schroedinger operators on compact and non-compact (finite) metric
graphs. For such operators we analyse their spectra, prove that their
resolvents can be represented as integral operators and introduce trace-class
regularisations of the resolvents. Our main result is a complete asymptotic
expansion of the trace of the (regularised) heat-semigroup generated by the
Schroedinger operator. We also determine the leading coefficients in the
expansion explicitly.
