Exterior derivatives and Laplacians on digraphs
Australasian Journal of Combinatorics
Given a digraph G = (V, E) with the set of vertices V and the set of edges E, let d: F → Ω1 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 Δ: F → F and the label difference map d̂: F → (Ω1)* of F into the dual space (Ω1)* of Ω1 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 ďφ is equal to the value of the Hermitian product between Δφ and ψ and that dφ is a flow in G associated to Δφ.
Original Publication Date
Lee, Min Ho, "Exterior derivatives and Laplacians on digraphs" (2004). Faculty Publications. 3049.