2009 West Java earthquake: Difference between revisions

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| name = Holt graph
| image = [[File:Holt graph.svg|220px]]
| image_caption = In the Holt graph, all vertices are equivalent, and all edges are equivalent, but edges are not necessarily equivalent to their inverses.
| namesake = Derek F. Holt
| vertices = 27
| edges = 54
| automorphisms= 54
| girth = 5
| diameter = 3
| radius = 3
| chromatic_number = 3
| chromatic_index = 5
| properties = [[Vertex-transitive graph|Vertex-transitive]]<br>[[Edge-transitive graph|Edge-transitive]]<br>[[Half-transitive graph|Half-transitive]]<br>[[Hamiltonian graph|Hamiltonian]]<br>[[Eulerian graph|Eulerian]]<br>[[Cayley graph]]
}}
In the [[mathematics|mathematical]] field of [[graph theory]], the '''Holt graph''' or '''Doyle graph''' is the smallest [[half-transitive graph]], that is, the smallest example of a [[Vertex-transitive graph|vertex-transitive]] and [[Edge-transitive graph|edge-transitive]] graph which is not also [[symmetric graph|symmetric]].<ref>Doyle, P. "A 27-Vertex Graph That Is Vertex-Transitive and Edge-Transitive But Not L-Transitive." October 1998. [http://arxiv.org/abs/math/0703861/]</ref><ref>{{citation|title=Constructing Graphs which are ½-Transitive|first1=Brian|last1=Alspach|first2=Dragan|last2=Marušič|first3=Lewis|last3=Nowitz|journal=[[Journal of the Australian Mathematical Society]] (Series A)|volume=56|issue=3|pages=391–402|year=1994|doi=10.1017/S1446788700035564|url=http://anziamj.austms.org.au/JAMSA/V56/Part3/Alspach.html}}.</ref> Such graphs are not common.<ref>Jonathan L. Gross, Jay Yellen, ''Handbook of Graph Theory'', CRC Press, 2004, ISBN 1-58488-090-2, p. 491.</ref> It is named after Peter G. Doyle and Derek F. Holt, who discovered the same graph independently in 1976<ref>{{citation|first=P. G.|last=Doyle|title=On Transitive Graphs|series=Senior Thesis|publisher=Harvard College|year=1976}}. As cited by MathWorld.</ref> and 1981<ref name="holt">{{citation|title=A graph which is edge transitive but not arc transitive|first=Derek F.|last=Holt|journal=[[Journal of Graph Theory]]|volume=5|issue=2|pages=201–204|year=1981|doi=10.1002/jgt.3190050210}}.</ref> respectively.
 
The Holt Graph has [[graph diameter|diameter]]&nbsp;3, radius&nbsp;3 and [[girth (graph theory)|girth]]&nbsp;5, [[chromatic number]]&nbsp;3, [[chromatic index]]&nbsp;5 and is [[Hamiltonian graph|Hamiltonian]] with {{formatnum:98472}} distinct Hamiltonian cycles.<ref name="Mathworld">{{MathWorld|urlname=DoyleGraph|title=Doyle Graph}}</ref> It is also a 4-[[k-vertex-connected graph|vertex-connected]] and a 4-[[k-edge-connected graph|edge-connected]] graph.
 
It has an [[graph automorphism|automorphism group]] of order 54 automorphisms.<ref name="Mathworld"/> This is a smaller group than a symmetric graph with the same number of vertices and edges would have. The graph drawing on the right highlights this, in that it lacks reflectional symmetry.
 
The characteristic polynomial of the Holt graph is
:<math>(x^3-6x+2)^6(x+2)^4(x-1)^4(x-4).\ </math>
 
==Gallery==
<gallery>
Image:Holt graph 3COL.svg|The [[chromatic number]] of the Holt graph is&nbsp;3.
Image:Holt graph 5color edge.svg|The [[chromatic index]] of the Holt graph is&nbsp;5.
Image:Holt graph hamiltonian.svg|The Holt graph is [[Hamiltonian graph|Hamiltonian]].
</gallery>
 
== References ==
{{reflist}}
 
[[Category:Individual graphs]]
[[Category:Regular graphs]]

Latest revision as of 19:51, 26 July 2014

Handbags is one of the very most functional accessories.

Girls choose plenty of effort selecting their pocketbook to match their ensembles. Additionally they take much time to choose the pocket book together with the colour of their outfit. For folks who do believe in taking staff that is an excessive amount of with them, there are vibrant and attractive coin purses and clutch bags to choose from.

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