Timeline for Gauss-Bonnet Theorem for Graphs?
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 22, 2017 at 11:55 | comment | added | Harry Richman | It's probably worth mentioning that $S(x) = 2-v(x)$ is the Euler characteristic of a tubular neighborhood of the vertex $x$, if you take any embedding of the graph in 3-space (i.e. a sphere for $v(x) = 0$, a pair of pants for $v(x) = 3$). By usual Gauss-Bonnet, this is equal to the total curvature of this neighborhood-surface up to a constant, and summing over all vertices gives the curvature of the tubular neighborhood of the whole graph, a closed orientable surface of genus $1-\chi(G)$. | |
| May 30, 2010 at 10:19 | vote | accept | Joseph O'Rourke | ||
| May 28, 2010 at 15:55 | comment | added | Tom Boardman | I like this answer! Intuitively a negative curvature point is one with 'more space than one might expect around it' and a positive curvature point is one with less. Ties up quite nicely I think... | |
| May 28, 2010 at 14:51 | comment | added | Joseph O'Rourke | Very nice! Thanks! Presumably, then, there is a version for, say, a simple cycle in the graph enclosing a certain amount of "scalar curvature." | |
| May 28, 2010 at 13:39 | history | answered | Benoît Kloeckner | CC BY-SA 2.5 |