# Harries–Wong graph

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Harries–Wong graph
The Harries–Wong graph
Vertices 70
Edges 105
Diameter 6
Girth 10
Automorphisms 24 (S4)
Chromatic number 2
Chromatic index 3
Properties Cubic
Cage
Triangle-free
Hamiltonian

In the mathematical field of graph theory, the Harries–Wong graph is a 3-regular undirected graph with 70 vertices and 105 edges.[1]

The Harries–Wong 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–Wong graph is

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

## 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)-cages 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.

## References

1. ^
2. ^ A. T. Balaban, A trivalent graph of girth ten, J. Combin. Theory Ser. B 12, 1–5. 1972.
3. ^ Pisanski, T.; Boben, M.; Marušič, D.; and Orbanić, A. "The Generalized Balaban Configurations." Preprint. 2001. [1].
4. ^ 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.
5. ^ Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976.