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.
A graph theoretical Poincare-Hopf Theorem
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abstract
We introduce the index i(v) = 1 - X(S(v)) for critical points of a locally injective function f on the vertex set V of a simple graph G=(V,E). Here S(v) = {w in E | (v,w) in E, f(w)-f(v)<0} is the subgraph of the unit sphere at v in G. It is the exit set of the gradient vector field. We prove that the sum of i(v) over V is always is equal to the Euler characteristic X(G) of the graph G. This is a discrete Poincare-Hopf theorem in a discrete Morse setting. It allows to compute X(G) for large graphs for which other methods become impractical.
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2019 1verdicts
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The energy of a simplicial complex
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.