  We analyze properties of semigroups generated by Schr\"odinger operators
$-\Delta+V$ or polyharmonic operators $-(-\Delta)^m$, on metric graphs both on
$L^p$-spaces and spaces of continuous functions. In the case of spatially
constant potentials, we provide a semi-explicit formula for their kernel. Under
an additional sub-exponential growth condition on the graph, we prove
analyticity, ultracontractivity, and pointwise kernel estimates for these
semigroups; we also show that their generators' spectra coincide on all
relevant function spaces and present a Kre\u{\i}n-type dimension reduction,
showing that their spectral values are determined by the spectra of generalized
discrete Laplacians acting on various spaces of functions supported on
combinatorial graphs.
