Samuelson's inequality: Difference between revisions

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{{infobox graph
| name = Robertson graph
| image = [[Image:Robertson graph hamiltonian.svg|240px]]
| image_caption = The Robertson graph is Hamiltonian.
| namesake = [[Neil Robertson (mathematician)|Neil Robertson]]
| vertices = 19
| edges = 38
| automorphisms = 24 ([[dihedral group|D<sub>12</sub>]])
| girth = 5
| diameter = 3
| radius = 3
| chromatic_number = 3
| chromatic_index = 5<ref>{{MathWorld|urlname=Class2Graph|title=Class 2 Graph}}</ref>
| properties = [[Cage (graph theory)|Cage]]<br>[[Hamiltonian graph|Hamiltonian]]
}}
In the [[mathematics|mathematical]] field of [[graph theory]], the '''Robertson graph''' or '''(4,5)-cage,''' is a 4-[[regular graph|regular]] [[undirected graph]] with 19 vertices and 38 edges named after [[Neil Robertson (mathematician)|Neil Robertson]].<ref>{{MathWorld|urlname=RobertsonGraph|title=Robertson Graph}}</ref><ref>Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976.</ref>


The Robertson graph is the unique [[cage graph|(4,5)-cage graph]] and was discovered by Robertson in 1964.<ref>Robertson, N. "The Smallest Graph of Girth 5 and Valency 4." Bull. Amer. Math. Soc. 70, 824-825, 1964.</ref> As a cage graph, it is the smallest 4-regular graph with girth 5.


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It has [[chromatic number]] 3, [[chromatic index]] 5, diameter 3, radius 3 and is both  4-[[k-vertex-connected graph|vertex-connected]] and 4-[[k-edge-connected graph|edge-connected]].  
 
The Robertson graph is also a [[Hamiltonian graph]] which possesses {{formatnum:5376}} distinct directed Hamiltonian cycles.
 
==Algebraic properties==
The Robertson graph is not a [[vertex-transitive graph]] and its full automorphism group is isomorphic to the [[dihedral group]] of order 24, the group of symmetries of a regular [[dodecagon]], including both rotations and reflections.<ref name="SURV">Geoffrey Exoo & Robert Jajcay, Dynamic cage survey, Electr. J. Combin. 15, 2008.</ref>
 
The [[characteristic polynomial]] of the Robertson graph is
:<math>(x-4)(x-1)^2(x^2-3)^2(x^2+x-5)</math>
:<math>(x^2+x-4)^2(x^2+x-3)^2(x^2+x-1).\ </math>
 
==Gallery==
<gallery>
Image:Robertson graph.svg|The Robertson graph as drawn in the original publication.
Image:Robertson graph 3COL.svg|The [[chromatic number]] of the Robertson graph is&nbsp;3.
Image:Robertson graph 5color edge.svg|The [[chromatic index]] of the Robertson graph is&nbsp;5.
</gallery>
 
== References ==
{{reflist}}
 
[[Category:Individual graphs]]
[[Category:Regular graphs]]

Revision as of 06:03, 26 June 2013

Template:Infobox graph In the mathematical field of graph theory, the Robertson graph or (4,5)-cage, is a 4-regular undirected graph with 19 vertices and 38 edges named after Neil Robertson.[1][2]

The Robertson graph is the unique (4,5)-cage graph and was discovered by Robertson in 1964.[3] As a cage graph, it is the smallest 4-regular graph with girth 5.

It has chromatic number 3, chromatic index 5, diameter 3, radius 3 and is both 4-vertex-connected and 4-edge-connected.

The Robertson graph is also a Hamiltonian graph which possesses 5,376 distinct directed Hamiltonian cycles.

Algebraic properties

The Robertson graph is not a vertex-transitive graph and its full automorphism group is isomorphic to the dihedral group of order 24, the group of symmetries of a regular dodecagon, including both rotations and reflections.[4]

The characteristic polynomial of the Robertson graph is

Gallery

  • The Robertson graph as drawn in the original publication.

    The Robertson graph as drawn in the original publication.

  • The chromatic number of the Robertson graph is 3.

    The chromatic number of the Robertson graph is 3.

  • The chromatic index of the Robertson graph is 5.

    The chromatic index of the Robertson graph is 5.

References

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  1. ↑

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  2. ↑ Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 237, 1976.
  3. ↑ Robertson, N. "The Smallest Graph of Girth 5 and Valency 4." Bull. Amer. Math. Soc. 70, 824-825, 1964.
  4. ↑ Geoffrey Exoo & Robert Jajcay, Dynamic cage survey, Electr. J. Combin. 15, 2008.
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