  We establish several properties of the integrated density of states for
random quantum graphs: Under appropriate ergodicity and amenability
assumptions, the integrated density of states can be defined using an
exhaustion procedure by compact subgraphs. A trace per unit volume formula
holds, similarly as in the Euclidean case. Our setting includes periodic
graphs. For a model where the edge length are random and vary independently in
a smooth way we prove a Wegner estimate and related regularity results for the
integrated density of states. These results are illustrated for an example
based on the Kagome lattice. In the periodic case we characterise all compactly
supported eigenfunctions and calculate the position and size of discontinuities
of the integrated density of states.
