WOLFRAM|DEMONSTRATIONS PROJECT

Gauss-Bonnet and Poincaré-Hopf for Graphs

choose graph type

Cycle

index for graph type

1

0

0

1

=χ=

V

∑

x=1

i(x)

1

3

1

3

1

3

1

=χ=

V

∑

x=1

K[x]

Two representations of a selected graph are shown. On the left, the vertices are labeled with the indices, which add up to the Euler characteristic of the graph. On the right, the graph is shown with curvatures, which also sum to the Euler characteristic. The left graphic illustrates a Gauss–Bonnet theorem for graphs. The right graphic illustrates a Poincaré–Hopf theorem for graphs.