Harries graph
| Harries graph | |
|---|---|
|
The Harries graph | |
| Vertices | 70 |
| Edges | 105 |
| Radius | 6 |
| Diameter | 6 |
| Girth | 10 |
| Automorphisms | 120 (S5) |
| Chromatic number | 2 |
| Chromatic index | 3 |
| Properties |
Cubic Cage Triangle-free Hamiltonian |
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.[1]
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.
The characteristic polynomial of the Harries graph is
History
In 1972, A. T. Balaban published a (3-10)-cage graph, a cubic graph that has as few vertices as possible for girth 10.[2] It was the first (3-10)-cage discovered but it was not unique.[3]
The complete list of (3-10)-cage and the proof of minimality was given by O'Keefe and Wong in 1980.[4] There exist three distinct (3-10)-cage graphs—the Balaban 10-cage, the Harries graph and the Harries–Wong graph.[5] Moreover, the Harries–Wong graph and Harries graph are cospectral graphs.
Gallery
-
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.
References
- ↑ Weisstein, Eric W. "Harries Graph". MathWorld.
- ↑ A. T. Balaban, A trivalent graph of girth ten, J. Combin. Theory Ser. B 12, 1-5. 1972.
- ↑ Pisanski, T.; Boben, M.; Marušič, D.; and Orbanić, A. "The Generalized Balaban Configurations." Preprint. 2001. .
- ↑ M. O'Keefe and P.K. Wong, A smallest graph of girth 10 and valency 3, J. Combin. Theory Ser. B 29 (1980) 91-105.
- ↑ Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976.