Harries graph
Regular graph with 70 nodes and 105 edges
title: "Harries graph" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["individual-graphs", "regular-graphs"] description: "Regular graph with 70 nodes and 105 edges" topic_path: "general/individual-graphs" source: "https://en.wikipedia.org/wiki/Harries_graph" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Regular graph with 70 nodes and 105 edges ::
::data[format=table title="Infobox graph"]
| Field | Value |
|---|---|
| name | Harries graph |
| image | Harries graph.svg |
| image_caption | The Harries graph |
| namesake | W. Harries |
| vertices | 70 |
| edges | 105 |
| automorphisms | 120 (S) |
| genus | 9 |
| girth | 10 |
| diameter | 6 |
| radius | 6 |
| chromatic_number | 2 |
| chromatic_index | 3 |
| properties | Cubic |
| Cage | |
| Triangle-free | |
| Hamiltonian | |
| book thickness | 3 |
| queue number | 2 |
| :: |
|name = Harries graph |image = Harries graph.svg |image_caption = The Harries graph |namesake = W. Harries |vertices = 70 |edges = 105 |automorphisms = 120 (S) |genus = 9 |girth = 10 |diameter = 6 |radius = 6 |chromatic_number = 2 |chromatic_index = 3 |properties = Cubic Cage Triangle-free Hamiltonian |book thickness = 3 |queue number = 2
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
: (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. 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.
Gallery
File:Harries graph 2COL.svg|The chromatic number of the Harries graph is 2. File:Harries graph 3color edge.svg|The chromatic index of the Harries graph is 3. File:harries_graph_alternative_drawing.svg|Alternative drawing of the Harries graph. File:Harries graph petersen drawing.jpg|Alternative drawing emphasizing the graph's 4 orbits.
References
References
- "Harries Graph".
- Jessica Wolz, ''Engineering Linear Layouts with SAT''. Master Thesis, University of Tübingen, 2018
- 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. [http://citeseer.ist.psu.edu/448980.html].
- 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.
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