  The aim of this paper is to analyse algorithms for constructing presentations
of graph braid groups from the point of view of anyonic quantum statistics on
graphs. In the first part of this paper, we provide a comprehensive review of
an algorithm for constructing so-called minimal Morse presentations of graph
braid groups that relies on discrete Morse theory. Next, we introduce the
notion of a physical presentation of a graph braid group as a presentation
whose generators have a direct interpretation as particle exchanges. We show
how to derive a physical presentation of a graph braid group from its minimal
Morse presentation. In the second part of the paper, we study unitary
representations of graph braid groups that are constructed from their
presentations. We point out that algebraic objects called moduli spaces of flat
bundles encode all unitary representations of graph braid groups. For
$2$-connected graphs, we conclude the stabilisation of moduli spaces of flat
bundles over graph configuration spaces for large numbers of particles.
Moreover, we set out a framework for studying locally abelian anyons on graphs
whose non-abelian properties are only encoded in non-abelian topological phases
assigned to cycles of the considered graph.
