A discrete Gauss-Bonnet type theorem

@article{Knill2010ADG,
  title={A discrete Gauss-Bonnet type theorem},
  author={Oliver Knill},
  journal={arXiv: General Topology},
  year={2010}
}
  • O. Knill
  • Published 2010
  • Mathematics
  • arXiv: General Topology
We prove a prototype curvature theorem for subgraphs G of the flat triangular tesselation which play the analogue of "domains" in two dimensional Euclidean space: The Pusieux curvature K(p) = 2|S1(p)| - |S2(p)| is equal to 12 times the Euler characteristic when summed over the boundary of G. Here |S1(p)| is the arc length of the unit sphere of p and |S2(p)| is the arc length of the sphere of radius 2. This curvature 12 formula is discrete Gauss-Bonnet formula or Hopf Umlaufsatz. The curvature… Expand
An index formula for simple graphs
  • O. Knill
  • Mathematics, Computer Science
  • ArXiv
  • 2012
TLDR
It is proved that any odd dimensional geometric graph G has zero curvature, and the same integral geometric index formula is valid if f is a Morse function and if S(x) is a sufficiently small geodesic sphere around x and B(f,x) intersected with the level surface. Expand
The Jordan-Brouwer theorem for graphs
  • O. Knill
  • Mathematics, Computer Science
  • ArXiv
  • 2015
We prove a discrete Jordan-Brouwer-Schoenflies separation theorem telling that a (d-1)-sphere H embedded in a d-sphere G defines two different connected graphs A,B in G such a way that theExpand
BARYCENTRIC CHARACTERISTIC NUMBERS
If G is the category of finite simple graphs G = (V,E), the linear space V of valuations on G has a basis given by the f-numbers vk(G) counting complete subgraphs Kk+1 in G. The barycentricExpand
Gauss-Bonnet for multi-linear valuations
  • O. Knill
  • Computer Science, Mathematics
  • ArXiv
  • 2016
We prove Gauss-Bonnet and Poincare-Hopf formulas for multi-linear valuations on finite simple graphs G=(V,E) and answer affirmatively a conjecture of Gruenbaum from 1970 by constructing higher orderExpand
On the Dimension and Euler characteristic of random graphs
  • O. Knill
  • Mathematics, Computer Science
  • ArXiv
  • 2011
TLDR
It is shown here that the average dimension E[dim] is a computable polynomial of degree n(n-1)/2 in p and the explicit formulas for the signature polynomials f and g allow experimentally to explore limiting laws for the dimension of large graphs. Expand
Graphs with Eulerian unit spheres
  • O. Knill
  • Mathematics, Computer Science
  • ArXiv
  • 2015
TLDR
It is proved here that G is an Eulerian sphere if and only if the degrees of all the (d-2)-dimensional sub-simplices in G are even. Expand
The strong ring of simplicial complexes
  • O. Knill
  • Mathematics, Computer Science
  • ArXiv
  • 2017
TLDR
The isomorphism of R with a subring of the strong Sabidussi ring shows that the multiplicative primes in R are the simplicial complexes and that every connected element in the strong ring has a unique prime factorization. Expand
The graph spectrum of barycentric refinements
  • O. Knill
  • Computer Science, Mathematics
  • ArXiv
  • 2015
TLDR
It is proved that for any finite simple graph G, the spectral functions F(G(m) of successive refinements converge for m to infinity uniformly on compact subsets of (0,1) and exponentially fast to a universal limiting eigenvalue distribution function F which only depends on the clique number respectively the dimension d of the largest complete subgraph of G and not on the starting graph G. Expand
Crystallization in Two Dimensions and a Discrete Gauss–Bonnet Theorem
TLDR
It is shown that the emerging field of discrete differential geometry can be usefully brought to bear on crystallization problems by appealing to a discrete Gauss–Bonnet theorem which relates the sum/integral of the curvature to topological invariants. Expand
A Sard theorem for graph theory
  • O. Knill
  • Computer Science, Mathematics
  • ArXiv
  • 2015
TLDR
Eigenfunctions of geometric graphs and especially the second eigenvector of 3-spheres, which by Courant-Fiedler has exactly two nodal regions, are illustrated. Expand
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