A graph theoretical Gauss-Bonnet-Chern Theorem

TL;DR

This paper establishes a discrete version of the Gauss-Bonnet-Chern theorem for finite graphs, linking curvature sums to the Euler characteristic, thus bridging differential geometry and graph theory.

Contribution

It introduces a novel discrete Gauss-Bonnet-Chern theorem applicable to finite graphs, extending classical differential geometry results to combinatorial structures.

Findings

Sum of curvature over all vertices equals Euler characteristic

Provides a new geometric interpretation for graph invariants

Bridges concepts between differential geometry and graph theory

Abstract

We prove a discrete Gauss-Bonnet-Chern theorem which states where summing the curvature over all vertices of a finite graph G=(V,E) gives the Euler characteristic of G.