Interplanetary scintillation: Difference between revisions

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{{infobox graph
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  | name = Harries&ndash;Wong graph
| image = [[Image:Harries-wong graph.svg|220px]]
| image_caption = The Harries&ndash;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&ndash;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&ndash;Wong Graph}}</ref>
 
The Harries&ndash;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&ndash;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&ndash;105.</ref> There exist three distinct (3-10)-cage graphs—the [[Balaban 10-cage]], the [[Harries graph]] and the Harries&ndash;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&ndash;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&ndash;Wong graph is&nbsp;2.
Image:Harries-wong graph 3color edge.svg|The chromatic index of the Harries&ndash;Wong graph is&nbsp;3.
Image:harries-wong_graph_alternative_drawing.svg|Alternative drawing of the Harries&ndash;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 16:19, 7 February 2014

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