The theorems of Green-Stokes,Gauss-Bonnet and Poincare-Hopf in Graph Theory

By proving graph theoretical versions of Green-Stokes, Gauss-Bonnet and Poincare-Hopf, core ideas of undergraduate mathematics can be illustrated in a simple graph theoretical setting. In this pedagogical exposition we present the main proofs on a single page and add illustrations. While discrete Stokes is is old, the other two results for graphs were found only recently.

💡 Research Summary

The paper presents discrete analogues of three cornerstone theorems of undergraduate mathematics—Green‑Stokes, Gauss‑Bonnet, and Poincaré‑Hopf—within the framework of finite simple graphs. After fixing a graph (G=(V,E)), the author introduces a chain complex where 0‑forms are functions on vertices and 1‑forms are functions on edges. The exterior derivative (d) is defined as a simple difference: for a vertex function (f) and an edge (e={u,v}), (df(e)=f(v)-f(u)). This definition automatically satisfies (d\circ d=0) because the graph has no higher‑dimensional cells, mirroring the nilpotency of the continuous exterior derivative.

The discrete Green‑Stokes theorem is then stated for any vertex subset (A\subset V) with boundary (\partial A). A divergence operator (\operatorname{div}) on edge‑forms is introduced using an orientation sign (\sigma(v,e)) that records whether a vertex is the head or tail of an edge. The theorem reads \