Exterior derivatives and Laplacians on digraphs

Given a digraph G = (V,E) with the set of vertices V and the set of edges E, let d : F → Ω be the exterior derivative map from the space of complex-valued functions on V to the complex vector space spanned by E. We introduce the Laplacian ∆ : F → F and the label difference map d̂ : F → (Ω1)∗ of F into the dual space (Ω1)∗ of Ω and establish their connections with d. In particular, we prove that, given elements φ and ψ of F, the image of the conjugate dψ of dψ under d̂φ is equal to the value of the Hermitian product between ∆φ and ψ and that d̂φ is a flow in G associated to ∆φ.