A divergence-like characterization of admissible functions on digraphs (Q1885481)

A smooth function on a manifold with a codimension-one-foliation is called admissible if there is a Riemannian metric such that the function is the mean curvature function of the leaves. In a previous paper [Tohoku Math. J., II. Ser. 51, No. 2, 227--236 (1999; Zbl 0934.57034)] the author has established a \(1-1\)-correspondence between codimension-one-foliations of manifolds and directed graphs. The main theorem of the present paper gives a similar correspondence between smooth admissible functions and certain labelings (defined in terms of a coboundary operator) of the associated digraph. In some sense the labeling on the combinatorial side corresponds to taking the divergence of the unit normal vector of the foliation.