Geometric and spectral analysis on weighted digraphs

In this article we give a geometrical description of the (in general non-selfadjoint) in/out Laplacian L+/-=(d+/-)*dand adjacency matrix on digraphs with arbitrary weights, where (d+/-)* is the adjoint of the evaluation map d+/- on the terminal/initial vertex of each arc and d =d++d- denotes the discrete gradient. We prove that the multiplicity of the zero eigenvalue of L+/-=(d+/-)*d coincides with the number of sources/sinks of the digraph. We also show that for an acyclic digraph with combinatorial weights the spectrum is contained in the set of non-zero integers. The geometrical perspective allows to interpret the set of circulations Cof a weighted digraph as coclosed forms on the arcs, i.e. as the kernel of the discrete divergence d*. Moreover, Cis perpendicular to the set of discrete gradients of functions on the vertices. We also give formulas to compute the capacity to compute the capacity of a cut and the value of a flow in terms of L and d. We illustrate the results with many concrete examples.

directed graphs; spectral graph theory; discrete laplacian; sinks and sources; circulations and flows; value and capacity

Geometric and spectral analysis on weighted digraphs Articles