  Quantum graphs are commonly used as models of complex quantum systems, for
example molecules, networks of wires, and states of condensed matter. We
consider quantum statistics for indistinguishable spinless particles on a
graph, concentrating on the simplest case of abelian statistics for two
particles. In spite of the fact that graphs are locally one-dimensional, anyon
statistics emerge in a generalized form. A given graph may support a family of
independent anyon phases associated with topologically inequivalent exchange
processes. In addition, for sufficiently complex graphs, there appear new
discrete-valued phases. Our analysis is simplified by considering combinatorial
rather than metric graphs -- equivalently, a many-particle tight-binding model.
The results demonstrate that graphs provide an arena in which to study new
manifestations of quantum statistics. Possible applications include topological
quantum computing, topological insulators, the fractional quantum Hall effect,
superconductivity and molecular physics.
