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arxiv: 1202.4514 · v1 · pith:2CPZ5EYMnew · submitted 2012-02-21 · 🧮 math.DG · cs.DM· math.GN

On index expectation and curvature for networks

Oliver Knill This is my paper
classification 🧮 math.DG cs.DMmath.GN
keywords curvatureexpectationfinitefunctionindexsimplearbitrarycomplements
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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 the vertex x. This result complements and links Gauss-Bonnet sum K(x) = chi(G) and Poincare-Hopf sum i(x) = chi(G) which both hold for arbitrary finite simple graphs.

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  1. The energy of a simplicial complex

    math.CO 2019-07 unverdicted novelty 5.0

    The sum of entries in the inverse of the intersection matrix of a simplicial complex equals its Euler characteristic, and so does the difference between the numbers of positive and negative eigenvalues of that matrix.