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 *K*_{4} 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?

**Addendum**. (*27Nov11*).
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:

independentof a specific embedding.) – Hans Stricker Jan 22 '13 at 15:37