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{{infobox graph | |||
| name = Harries–Wong graph | |||
| image = [[Image:Harries-wong graph.svg|220px]] | |||
| image_caption = The Harries–Wong graph | |||
| namesake = | |||
| vertices = 70 | |||
| edges = 105 | |||
| automorphisms = 24 ([[Symmetric group|''S''<sub>4</sub>]]) | |||
| girth = 10 | |||
| diameter = 6 | |||
| radius = 6 | |||
| chromatic_number = 2 | |||
| chromatic_index = 3 | |||
| properties = [[Cubic graph|Cubic]]<br>[[Cage (graph theory)|Cage]]<br>[[Triangle-free graph|Triangle-free]]<br>[[Hamiltonian graph|Hamiltonian]] | |||
}} | |||
In the [[mathematics|mathematical]] field of [[graph theory]], the '''Harries–Wong graph'''<!-- | |||
, named after ?????? Haries and ?????? Wong, | |||
--> is a 3-[[regular graph|regular]] [[undirected graph]] with 70 vertices and 105 edges.<ref>{{MathWorld|urlname=Harries-WongGraph|title= Harries–Wong Graph}}</ref> | |||
The Harries–Wong graph has [[chromatic number]] 2, [[chromatic index]] 3, radius 6, diameter 6, girth 10 and is [[Hamiltonian graph|Hamiltonian]]. It is also a 3-[[k-vertex-connected graph|vertex-connected]] and 3-[[k-edge-connected graph|edge-connected]] [[planar graph|non-planar]] [[cubic graph]]. | |||
The characteristic polynomial of the Harries–Wong graph is | |||
: <math>(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. \, </math> | |||
==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.<ref>A. T. Balaban, A trivalent graph of girth ten, J. Combin. Theory Ser. B 12, 1–5. 1972.</ref> It was the first (3-10)-cage discovered but it was not unique.<ref>Pisanski, T.; Boben, M.; Marušič, D.; and Orbanić, A. "The Generalized Balaban Configurations." Preprint. 2001. [http://citeseer.ist.psu.edu/448980.html].</ref> | |||
The complete list of (3-10)-cages and the proof of minimality was given by O'Keefe and Wong in 1980.<ref>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.</ref> There exist three distinct (3-10)-cage graphs—the [[Balaban 10-cage]], the [[Harries graph]] and the Harries–Wong graph.<ref>[[John Adrian Bondy|Bondy, J. A.]] and [[U. S. R. Murty|Murty, U. S. R.]] Graph Theory with Applications. New York: North Holland, p. 237, 1976.</ref> Moreover, the Harries–Wong graph and Harries graph are [[Spectral graph theory|cospectral graphs]]. | |||
==Gallery== | |||
<gallery> | |||
Image:Harries-wong graph 2COL.svg|The chromatic number of the Harries–Wong graph is 2. | |||
Image:Harries-wong graph 3color edge.svg|The chromatic index of the Harries–Wong graph is 3. | |||
Image:harries-wong_graph_alternative_drawing.svg|Alternative drawing of the Harries–Wong graph. | |||
Image:Harries-Wong graph - the 8 orbits.jpg|The 8 orbits of the Harries–Wong graph. | |||
</gallery> | |||
== References == | |||
{{reflist}} | |||
{{DEFAULTSORT:Harries-Wong graph}} | |||
[[Category:Individual graphs]] | |||
[[Category:Regular graphs]] | |||
Revision as of 00:42, 2 October 2013
Template:Infobox graph 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
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.
Gallery
-
The chromatic number of the Harries–Wong graph is 2. -
The chromatic index of the Harries–Wong graph is 3. -
Alternative drawing of the Harries–Wong graph. -
The 8 orbits of the Harries–Wong graph.
References
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Hostgator tutorials on how to install Wordpress need not be complicated, especially when you will be dealing with a web hosting service that is friendly for novice webmasters and a blogging platform that is as intuitive as riding a bike. After that you can get Hostgator to host your domain and use the wordpress to do the blogging. Once you start site flipping, trust me you will not be able to stop. I cut my webmaster teeth on Control Panel many years ago, but since had left for other hosting companies with more commercial (cough, cough) interfaces. If you don't like it, you can chalk it up to experience and go on. First, find a good starter template design. When I signed up, I did a search for current "HostGator codes" on the web, which enabled me to receive a one-word entry for a discount. Your posts, comments, and pictures will all be imported into your new WordPress blog. - ↑ 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. [1].
- ↑ 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.