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We consider compact metric graphs with an arbitrary self adjoint realisationof the differential Laplacian. After discussing spectral properties ofLaplacians, we prove several versions of trace formulae, relating Laplacespectra to sums over periodic orbits on the graph. This includes trace formulaewith, respectively, absolutely and conditionally convergent periodic orbitsums; the convergence depending on properties of the test functions used. Wealso prove a trace formula for the heat kernel and provide small-$t$asymptotics for the trace of the heat kernel.

The trace formula for quantum graphs with general self adjoint boundary conditions