  We present an integral formulation of the vacuum energy of Schr\"odinger
operators on finite metric graphs. Local vertex matching conditions on the
graph are classified according to the general scheme of Kostrykin and Schrader.
While the vacuum energy of the graph can contain finite ambiguities the Casimir
force on a bond with compactly supported potential is well defined. The vacuum
energy is determined from the zeta function of the graph Schr\"odinger operator
which is derived from an appropriate secular equation via the argument
principle. A quantum graph has an associated probabilistic classical dynamics
which is generically both ergodic and mixing. The results therefore present an
analytic formulation of the vacuum energy of this quasi-one-dimensional quantum
system which is classically chaotic.
