A graph theoretical Poincare-Hopf Theorem

We introduce the index i(v) = 1− χ(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 ∈ S(v) | f(w) − f(v) < 0 } is the subgraph of the unit sphere S(v) generated by the vertices {w ∈ V | (v, w) ∈ E }. We prove that the sum ∑ v∈V i(v) of the indices always is equal to the Euler characteristic χ(G) of the graph G. This is a discrete Poincaré-Hopf theorem in a discrete Morse setting.