### Abstract

The vacuum (Casimir) energy of a quantized scalar field in a given geometrical situation is a certain moment of the eigenvalue density of an associated self-adjoint differential operator. For various classes of quantum graphs it has been calculated by several methods: (1) Direct calculation from the explicitly known spectrum is feasible only in simple cases. (2) Analysis of the secular equation determining the spectrum, as in the Kottos-Smilansky derivation of the trace formula, yields a sum over periodic orbits in the graph. (3) Construction of an associated integral kernel by the method of images yields a sum over closed (not necessarily periodic) orbits. We show that for the Kirchhoff boundary condition the sum over nonperiodic orbits in fact makes no contribution to the total energy, whereas for more general vertex scattering matrices (complex or frequency-dependent) it can make a nonvanishing contribution, which, however, is localized near vertices and hence can be "indexed" a posteriori by truly periodic orbits. Physically, the question of greatest interest is the sign of the Casimir effect in a given situation; for graphs it is being studied by both analytical and numerical methods. For example, in a star graph with 4 or more bonds ("pistons") of equal length and the standard Kirchhoff-Neumann boundary conditions, the vacuum force is repulsive (expanding), whereas the electromagnetic Casimir forces of laboratory interest are usually attractive. This work is done in collaboration with Justin H. Wilson, with contributions from Lev Kaplan, Gregory Berkolaiko, Jonathan Harrison, Melanie Pivarski, and Brian Winn and support from National Science Foundation Grant No. PHY-0554849.