Heawood graph

In mathematics , the Heawood graph is a graph with 14 nodes and 21 edges, which is important as an incidence graph of the Fano level , among other things . It is named after Percy Heawood .

Combinatorial properties

The Heawood graph is bipartite and can therefore be colored with 2 colors.

The Heawood graph is a cubic graph ; H. 3-regular.

It is bipartite and therefore has the chromatic number 2.

It has a diameter of 3 and is the only cubic graph that does not contain a cycle of length 5.

Symmetries

The automorphism group of the Heawood graph is isomorphic to the projective linear group and therefore has 336 elements. ${\ displaystyle PGL (2, F_ {7})}$

The Heawood graph is transitive in distance ; H. for every two pairs of points with there is an automorphism that maps up and down . ${\ displaystyle (x, y), (u, v)}$ ${\ displaystyle d (x, y) = d (u, v)}$ ${\ displaystyle x}$ ${\ displaystyle u}$ ${\ displaystyle y}$ ${\ displaystyle v}$

The Heawood graph in geometry and topology

This embedding of the Heawood graph splits the torus (which is created by identifying opposite square edges) into 7 regions.

The Heawood graph is the incidence graph of the Fano level .

It can be embedded in the torus without crossing , which it divides into 7 regions that touch each other in pairs. In particular, it is dual to the complete graph . ${\ displaystyle K_ {7}}$

Web links

Wolfram MathWorld: Heawood Graph