A graph theoretical Poincare-Hopf Theorem

Details A graph theoretical Poincare-Hopf Theorem A graph theoretical Poincare-Hopf Theorem

2012-01-05

arxiv.org/abs/1201.1162v1

Mathematics

Differential Geometry

9 figures

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

A graph theoretical Poincare-Hopf Theorem does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with A graph theoretical Poincare-Hopf Theorem, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A graph theoretical Poincare-Hopf Theorem will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-609904