Questions about the project
Question | Answer |
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In page 3 you showed that the Euler characteristic of the complete graph K_n is 1 and each point has curvature 1/n. In page 18, example 3.2 you claimed the K_4 has Euler characteristic 0 and the curvature is constant 0. How are this two thing compatible? Also, K_4 is a 3 dimensional graph with your definition but you also mentioned that all 3 dimensional graphs have constant curvature 0. | Yes, the 4-vertex clique K4 is 3 dimensional. But the the unit sphere of each point is K3 which has Euler characteristic 1, not 2. In the definition of geometric graphs it is assumed that the Euler characteristic of spheres is either 0 for odd dim spheres or 2 for even dimensional spheres. While K4 is three dimensional, it is in a nongeometric a In smaller dimensions, the triangle K3 is 2 dimensional but its unit spheres are not circles but K2 graphs, which are one dimensional but not 1 dimensional in the geometric sense. Every geometric 1 dim graph has unit spheres of Euler characteristic 2 and so consist of two discrete points. Every 1 dim graph a finite union of circles. When looking at dimension only, then there are many more graphs which are one dimensional, like any tree. |
Is the theorem 1 valid for all finite graphs or only for those "geometric" ones? My understanding is that Theorem 1 is valid for all finite graphs as long as we use the definition of curvature as in equation 1 and valid only for the "geometric" ones if we use definition 2. Is this correct? | A general version is mentioned in the introduction. It is valid for all finite graphs without self loops and multiple connections. The main theme in the paper however is a version for geometric graphs which resemble manifolds. In that case one can say more. For example that positive sectional curvature implies that the graph is finite or that in dimension is 3, the curvature form is constant 0. |
At the end of page 2 you defined V_{k}(p) as the number of k-1 dimensional simplices in the unit sphere. Then what is the meaning of V_{0}(p)? Shouldn't the sum in equation 1 start at k=2? |
V_{k}(p) is the number of k-dimensional simplices (=(k+1) vertex cliques)
in S(p) like v_{k} is the number of k-dimensional simplices in G. The notation
V_{-1}=1 means that the empty graph is embedded once. It
validates the transfer equations also for k=0, where it is
sum_p V_{-1}(p)= v_{0} and holds because v_{0} is the number of vertices.
The assumption that the empty graph has dimension -1 is the seed
of inductive dimension: dim(G) = 1+ (sum_{p in V} dim(S(p)))/|V| |
How come that the results are graph theoretical but concrete graphs in the example section are built in concrete space? | Polyhedra are often realized in R^{n} first to build the graph. Once the graph is built, one can forget about the construction. |