User oliver - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T11:35:44Z http://mathoverflow.net/feeds/user/13903 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/26256/gauss-bonnet-theorem-for-graphs/59551#59551 Answer by Oliver for Gauss-Bonnet Theorem for Graphs? Oliver 2011-03-25T12:17:16Z 2011-03-25T12:17:16Z <p>There are different type of curvatures for graphs. In two dimensions its not the degree of the point which matters but the length of the circles at the point like in differential geometry. This is different from the degree if graphs with boundary are considered. The simplest curvature for two dimensional graphs (I defined the dimension for graphs in <a href="http://arxiv.org/abs/1009.2292" rel="nofollow">http://arxiv.org/abs/1009.2292</a>). For the curvature K(x) = 6-|S(x)|, any two dimensional graph G has total curvature 6 chi(G). More interesting and subtle is a second order curvature 2|S1(x)|-|S2(x)| which is differential geometrically motivated because curvature is a second order difference notion. Gauss-Bonnet is now not true in general and it seems that some smoothness condition needs to be satisfied. It is true for all two dimensional graphs non-negative curvature and for graphs which are "smooth" enough in some sense which I still struggle to define. I myself got more interested in Gauss-Bonnet-Chern for n-dimensional graphs and have a result there. There is a natural Chern-Euler form on n-dimensional graphs those total sum is the Euler characteristic for an n-dimensional graph. This form is defined also in odd dimensions but seems to be zero (it is in 3 dimensions but I still struggle to verify this in higher dimensions). Note that all this is purely graph theoretical. No additional structure on the graph is necessary. No embedding in an ambient Euclidean space necessary for example. The inductive dimension as defined in the above mentioned paper (revised here: <a href="http://www.math.harvard.edu/~knill/graphgeometry/papers/1.pdf" rel="nofollow">http://www.math.harvard.edu/~knill/graphgeometry/papers/1.pdf</a>) is a natural way to define polyhedra and polytopes as graphs which become n-dimensional graphs after some truncation or stellation procedure. The inductive dimension for graphs looks similar to inductive dimension usually considered in topology but is utterly different. With the usual inductive dimension, any graph would be one dimensional.</p>