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 $\chi = 0$ and $K_4$ has $\chi=-2$. Is there an analog for the Gauss-Bonnet theorem for graphs, something akin to:
[total turn angle] $+$ [enclosed curvature] $= \tau + \omega = 2 \pi \chi$ ?
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: