Faculty Publications

Exterior Derivatives And Laplacians On Digraphs

Document Type

Article

Journal/Book/Conference Title

Australasian Journal of Combinatorics

Volume

29

First Page

85

Last Page

94

Abstract

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.

Department

Department of Mathematics

Original Publication Date

12-1-2004

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