Graph Characteristic from the Gauss-Bonnet Theorem

In this paper we explore how to characterise graphs using the Gauss-Bonnet theorem. Using the Gaussian curvatures estimated from first-order cycles we compute a global estimate of the Euler index using the Gauss-Bonnet theorem. We commence by embedding the nodes of a graph in a manifold using the heat-kernel mapping. From this mapping we are able to compute the geodesic and Euclidean distance between nodes, and these can be used to estimate the sectional curvatures of edges. Assuming that edges reside on hyper-spheres, we use Gauss's theorem to estimate the Gaussian curvature from the interior angles of geodesic triangles formed by first-order cycles in the graph. From the Gaussian curvatures we make a global estimate of the Euler index of the manifold using the Gauss-Bonnet theorem. Experiments show how the Gaussian curvatures and the Euler characteristics can be used to cluster Delaunay triangulations extracted from real world images.