Geometric and spectral analysis on weighted digraphs

In this article we give a geometrical description of the (in general non-selfadjoint) in/out Laplacian ${\mathcal{L}}^{+\mathrm{/}-}=(d^{+\mathrm{/}-}{)}^{\ast }d$ and adjacency matrix on digraphs with arbitrary weights, where $(d^{+\mathrm{/}-}{)}^{\ast }$ is the adjoint of the evaluation map $d^{+\mathrm{/}-}$ 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 ${\mathcal{L}}^{+\mathrm{/}-}=(d^{+\mathrm{/}-}{)}^{\ast }d$ 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 $\mathcal{C}$ of a weighted digraph as coclosed forms on the arcs, i.e. as the kernel of the discrete divergence $d^{\ast }$. Moreover, $\mathcal{C}$ is perpendicular to the set of discrete gradients of functions on the vertices. We also give formulas to compute the capacity of a cut and the value of a flow in terms of ${\mathcal{L}}^{-}$ and $d$. We illustrate the results with many concrete examples.