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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?

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:

2 Addendum pointing to new paper on the topic.

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?

Addendum. (27Nov11). A new paper on this topic just appeared on the arXiv: Oliver Knill, "A graph theoretical Gauss-Bonnet-Chern Theorem." arXiv:1111.5395v1. Here is Knill's first figure:

1

# Gauss-Bonnet Theorem for Graphs?

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?