One can define the Euler characteristic χ for a graph as the number of vertices minus the number of edges. Thus an n-cycle has χ = 0 and K4 has χ = –2.
Is there an analog for the Gauss-Bonnet theorem for graphs, something akin to:
[total turn angle] + [enclosed curvature] = τ + ω = 2 π χ ?
Certainly if one embeds the graph on a manifold, then an interpretation is possible via Gauss-Bonnet on the manifold. But is there a more purely combinatorial interpretation?
A new paper on this topic just appeared on the arXiv:
Oliver Knill (who answered below back in March),
"A graph theoretical Gauss-Bonnet-Chern Theorem."
arXiv:1111.5395v1. Here is Knill's first figure: