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

The Berry-Keating operator H BK := −i x d dx + 1 2 [M. V. Berry and J. P. Keating, SIAM Rev. 41 (1999) 236] governing the Schrödinger dynamics is discussed in the Hilbert space L 2 (R > , dx) and on compact quantum graphs. It is proved that the spectrum of H BK defined on L 2 (R > , dx) is purely continuous and thus this quantization of H BK cannot yield the hypothetical Hilbert-Polya operator possessing as eigenvalues the nontrivial zeros of the Riemann zeta function. A complete classification of all self-adjoint extensions of H BK acting on compact quantum graphs is given together with the corresponding secular equation in form of a determinant whose zeros determine the discrete spectrum of H BK . In addition, an exact trace formula and the Weyl asymptotics of the eigenvalue counting function are derived. Furthermore, we introduce the "squared" Berry-Keating operator H 2 BK := −x 2 d 2 dx 2 − 2x d dx − 1 4 which is a special case of the Black-Scholes operator used in financial theory of option pricing. Again, all self-adjoint extensions, the corresponding secular equation, the trace formula and the Weyl asymptotics are derived for H 2 BK on compact quantum graphs. While the spectra of both H BK and H 2 BK on any compact quantum graph are discrete, their Weyl asymptotics demonstrate that neither H BK nor H 2 BK can yield as eigenvalues the nontrivial Riemann zeros. Some simple examples are worked out in detail.