### Abstract

We consider a quantum graph with spectrum $k_0^2 \le k_1^2 \le \dots$. The vacuum energy may be defined as the constant term in the asymptotic series expansion of $\frac{1}{2} \sum_{j=1}^{\infty} k_j \exp (-t k_j)$ as $t$ goes to zero. If the spectrum was instead that of the electromagnetic field between two parallel plates this would correspond to a renormalized energy of the vacuum whose derivative with respect to the plate separation is the attractive Casimir force measurable by experiment.

Using the trace formula for the density of states of a quantum graph a simple expression for the vacuum energy as a sum over periodic orbits has been derived. In the talk I will explain some calculations used to deduce properties of the vacuum energy. Of particular physical interest is the effect of changes in the lengths of the edges on the vacuum energy. To address this we show that the periodic orbit sum is convergent and differentiable with respect to the edge lengths. This work is part of a joint project with G. Berkolaiko, S. Fulling, M. Pivarski, J. H. Wilson and B. Winn.