Poincare Hopf for vector fields on graphs

TL;DR

This paper extends the Poincare-Hopf theorem to vector fields on finite simple graphs by defining a directed simplicial complex and an index based on Euler characteristic, linking graph theory with topological invariants.

Contribution

It introduces a novel framework for applying the Poincare-Hopf theorem to graphs using directed simplicial complexes and energy functions, bridging discrete and continuous topology.

Findings

Defined a directed simplicial complex for graphs.

Established a Poincare-Hopf type index sum equals Euler characteristic.

Extended the theorem to energy functions on complexes.

Abstract

We generalize the Poincare-Hopf theorem sum_v i(v) = X(G) to vector fields on a finite simple graph (V,E) with Whitney complex G. To do so, we define a directed simplicial complex as a finite abstract simplicial complex equipped with a bundle map F: G to V telling which vertex T(x) in x dominates the simplex x. The index i(v) of a vertex v is defined as X(F^-1(v)), where X is the Euler characteristic. We get a flow by adding a section map F: V to G. The resulting map T on G is a discrete model for a differential equation x'=F(x) on a compact manifold. Examples of directed complexes are defined by Whitney complexes defined by digraphs with no cyclic triangles or gradient fields on finite simple graphs defined by a locally injective function. The result extends to simplicial complexes equipped with an energy function H:G to Z that implements a divisor. The index sum is then the total energy.