  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.

Differential Geometry;Computational Geometry;Discrete Mathematics;General Topology