  We consider families of finite quantum graphs of increasing size and we are
interested in how eigenfunctions are distributed over the graph. As a measure
for the distribution of an eigenfunction on a graph we introduce the entropy,
it has the property that a large value of the entropy of an eigenfunction
implies that it cannot be localised on a small set on the graph. We then derive
lower bounds for the entropy of eigenfunctions which depend on the topology of
the graph and the boundary conditions at the vertices. The optimal bounds are
obtained for expanders with large girth, the bounds are similar to the ones
obtained by Anantharaman et.al. for eigenfunctions on manifolds of negative
curvature, and are based on the entropic uncertainty principle. For comparison
we compute as well the average behaviour of entropies on Neumann star graphs,
where the entropies are much smaller. Finally we compare our lower bounds with
numerical results for regular graphs and star graphs with different boundary
conditions.
