  We develop a full characterization of abelian quantum statistics on graphs.
We explain how the number of anyon phases is related to connectivity. For
2-connected graphs the independence of quantum statistics with respect to the
number of particles is proven. For non-planar 3-connected graphs we identify
bosons and fermions as the only possible statistics, whereas for planar
3-connected graphs we show that one anyon phase exists. Our approach also
yields an alternative proof of the structure theorem for the first homology
group of n-particle graph configuration spaces. Finally, we determine the
topological gauge potentials for 2-connected graphs.
