  In this article we give a geometrical description of the (in general
non-selfadjoint) in/out Laplacian $\mathcal{L}^{+/-} = (d^{+/-})^* d$ and
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 $\mathcal{L}^{+/-} = (d^{+/-})^* d$
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 $\mathcal{C}$ of a weighted digraph as coclosed forms
on the arcs, i.e. as the kernel of the discrete divergence $d^*$. 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.
