Harries graph explained

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.^[1]

The characteristic polynomial of the Harries graph is

$$(x-3)\; (x-1)^4\; (x+1)^4\; (x+3)\; (x^2-6)\; (x^2-2)\; (x^4-6x^2+2)^5\; (x^4-6x^2+3)^4\; (x^4-6x^2+6)^5.\; \backslash ,$$

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.

Notes and References

1. Jessica Wolz, Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018
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. http://citeseer.ist.psu.edu/448980.html.
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.