Helmholtz-Hodge Decomposition on Graphs

We propose a definition of the curl of a vector field X on a finite simple graph as the projection of X onto the orthogonal complement of circulation-free vector fields, where a vector field is circulation-free provided its line integral around every simple circuit vanishes. We justify the definition by observing that X and curl X have the same circulation and curl of the gradient and divergence of the curl vanish. This shows the gradient, curl, and divergence operators form an exact sequence, in analogy with the classical case of vector fields on Euclidean domains and yields the Helmholtz-Hodge decomposition of a vector field on a graph as the sum of a gradient, a curl, and a harmonic field. Along the way, we also prove analogues of the divergence theorem, Green's identities, and Helmholtz's theorem. A consequence of our definition is that the curl is a non-local operator, in sharp contrast to the classical case and existing notions of curl on a graph.