Harries graph

In the mathematical field of graph theory, the Harries graph or Harries (3-10)-cage is a 3-regular, undirected graph with 70 vertices and 105 edges.

The Harries graph has chromatic number 2, chromatic index 3, radius 6, diameter 6, girth 10 and is Hamiltonian. It is also a 3-vertex-connected and 3-edge-connected, non-planar, cubic graph. It has book thickness 3 and queue number 2.

The characteristic polynomial of the Harries graph is

$${\displaystyle (x-3)(x-1)^{4}(x+1)^{4}(x+3)(x^{2}-6)(x^{2}-2)(x^{4}-6x^{2}+2)^{5}(x^{4}-6x^{2}+3)^{4}(x^{4}-6x^{2}+6)^{5}.\,}$$

In 1972, A. T. Balaban published a (3-10)-cage graph, a cubic graph that has as few vertices as possible for girth 10. It was the first (3-10)-cage discovered but it was not unique.

The complete list of (3-10)-cage and the proof of minimality was given by O'Keefe and Wong in 1980. There exist three distinct (3-10)-cage graphs—the Balaban 10-cage, the Harries graph and the Harries–Wong graph. Moreover, the Harries–Wong graph and Harries graph are cospectral graphs.

• 
  The chromatic number of the Harries graph is 2.
• 
  The chromatic index of the Harries graph is 3.
• 
  Alternative drawing of the Harries graph.
• 
  Alternative drawing emphasizing the graph's 4 orbits.