Classification of harmonic structures on graphs

Graphs, viewed as one-dimensional simplicial complexes, can be given harmonic structures satisfying the Brelot axioms. In this paper, we describe all possible harmonic structures on graphs. We determine those harmonic structures which induce discrete harmonic structures when restricted to the set of vertices. Conversely, given a discrete harmonic structure on the set of vertices and an arbitrarily prescribed harmonic structure on each edge, we determine when these structures yield a harmonic structure on the graph. In addition, we provide a variety of interesting examples.