A graph theoretical Poincare-Hopf Theorem

TL;DR

This paper establishes a discrete Poincare-Hopf theorem for graphs, linking critical points of a function to the Euler characteristic, enabling efficient computation of topological invariants in large graphs.

Contribution

It introduces a novel index for critical points in graphs and proves a discrete Poincare-Hopf theorem connecting this index to the Euler characteristic.

Findings

The sum of the index over all vertices equals the graph's Euler characteristic.

The method simplifies computation of topological invariants in large graphs.

Provides a new tool for discrete Morse theory on graphs.

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.