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math.DG2012★ 21 cited
An index formula for simple graphs
Oliver Knill
Gauss-Bonnet for simple graphs G assures that the sum of curvatures K(x) over the vertex set V of G is the Euler characteristic X(G). Poincare-Hopf tells that for any injective fun…
math.DG2012★ 36 cited
On index expectation and curvature for networks
Oliver Knill
We prove that the expectation value of the index function i(x) over a probability space of injective function f on any finite simple graph G=(V,E) is equal to the curvature K(x) at…