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{{infobox graph | |||
| name = Hall–Janko graph | |||
| image = [[Image:Hall janko graph.svg|300px]] | |||
| image_caption = HJ as [[Foster graph]] (90 outer vertices) plus [[Steiner system]] S(3,4,10) (10 inner vertices). | |||
| namesake = [[Zvonimir Janko]]<br>[[Marshall Hall (mathematician)|Marshall Hall]] | |||
| vertices = 100 | |||
| edges = 1800 | |||
| automorphisms = | |||
| diameter = 2 | |||
| girth = 3 | |||
| radius = 2 | |||
| chromatic_number = 10 | |||
| chromatic_index = | |||
| properties = [[Strongly regular graph|Strongly regular]]<br>[[Vertex-transitive graph|Vertex-transitive]]<br>[[Cayley graph]]<br>[[Eulerian graph|Eulerian]]<br>[[Hamiltonian graph|Hamiltonian]]<br>[[integral graph|Integral]] | |||
}} | |||
In the [[mathematics|mathematical]] field of [[graph theory]], the '''Hall–Janko graph''', also known as the '''Hall-Janko-Wales graph''', is a 36-[[regular graph|regular]] [[undirected graph]] with 100 vertices and 1800 edges.<ref>{{MathWorld | urlname=Hall-JankoGraph | title=Hall-Janko graph}}</ref> | |||
It is a [[Rank 3 permutation group|rank 3]] [[strongly regular graph]] with parameters (100,36,14,12) and a maximum [[coclique]] of size 10. This parameter set is not unique, it is however uniquely determined by its parameters as a rank 3 graph. The Hall–Janko graph was originally constructed by D. Wales to establish the existence of the [[Hall-Janko group]] as an [[Index of a subgroup|index]] 2 subgroup of its [[Graph automorphism|automorphism group]]. | |||
The Hall–Janko graph can be constructed out of objects in U<sub>3</sub>(3), the simple group of order 6048:<ref>Andries E. Brouwer, "[http://www.win.tue.nl/~aeb/drg/graphs/HallJanko.html Hall-Janko graph]".</ref><ref>Andries E. Brouwer, "[http://www.win.tue.nl/~aeb/drg/graphs/U3_3.html U<sub>3</sub>(3) graph]".</ref> | |||
* In U<sub>3</sub>(3) there are 36 simple maximal subgroups of order 168. These are the vertices of a subgraph, the U<sub>3</sub>(3) graph. A 168-subgroup has 14 maximal subgroups of order 24, isomorphic to S<sub>4</sub>. Two 168-subgroups are called adjacent when they intersect in a 24-subgroup. The U<sub>3</sub>(3) graph is strongly regular, with parameters (36,14,4,6) | |||
* There are 63 involutions (elements of order 2). A 168-subgroup contains 21 involutions, which are defined to be neighbors. | |||
* Outside U<sub>3</sub>(3) let there be a 100th vertex '''C''', whose neighbors are the 36 168-subgroups. A 168-subgroup then has 14 common neighbors with C and in all 1+14+21 neighbors. | |||
* An involution is found in 12 of the 168-subgroups. C and an involution are non-adjacent, with 12 common neighbors. | |||
* Two involutions are defined as adjacent when they generate a dihedral subgroup of order 8.<ref>Robert A. Wilson, 'The Finite Simple Groups', Springer-Verlag (2009), p. 224.</ref> An involution has 24 involutions as neighbors. | |||
The characteristic polynomial of the Hall–Janko graph is <math>(x-36) (x-6)^{36} (x+4)^{63}</math>. Therefore the Hall–Janko graph is an [[integral graph]]: its [[Spectral graph theory|spectrum]] consists entirely of integers. | |||
==References== | |||
{{reflist}} | |||
{{DEFAULTSORT:Hall-Janko Graph}} | |||
[[Category:Group theory]] | |||
[[Category:Individual graphs]] | |||
[[Category:Regular graphs]] | |||
Revision as of 20:45, 4 June 2013
In the mathematical field of graph theory, the Hall–Janko graph, also known as the Hall-Janko-Wales graph, is a 36-regular undirected graph with 100 vertices and 1800 edges.[1]
It is a rank 3 strongly regular graph with parameters (100,36,14,12) and a maximum coclique of size 10. This parameter set is not unique, it is however uniquely determined by its parameters as a rank 3 graph. The Hall–Janko graph was originally constructed by D. Wales to establish the existence of the Hall-Janko group as an index 2 subgroup of its automorphism group.
The Hall–Janko graph can be constructed out of objects in U3(3), the simple group of order 6048:[2][3]
- In U3(3) there are 36 simple maximal subgroups of order 168. These are the vertices of a subgraph, the U3(3) graph. A 168-subgroup has 14 maximal subgroups of order 24, isomorphic to S4. Two 168-subgroups are called adjacent when they intersect in a 24-subgroup. The U3(3) graph is strongly regular, with parameters (36,14,4,6)
- There are 63 involutions (elements of order 2). A 168-subgroup contains 21 involutions, which are defined to be neighbors.
- Outside U3(3) let there be a 100th vertex C, whose neighbors are the 36 168-subgroups. A 168-subgroup then has 14 common neighbors with C and in all 1+14+21 neighbors.
- An involution is found in 12 of the 168-subgroups. C and an involution are non-adjacent, with 12 common neighbors.
- Two involutions are defined as adjacent when they generate a dihedral subgroup of order 8.[4] An involution has 24 involutions as neighbors.
The characteristic polynomial of the Hall–Janko graph is . Therefore the Hall–Janko graph is an integral graph: its spectrum consists entirely of integers.
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. - ↑ Andries E. Brouwer, "Hall-Janko graph".
- ↑ Andries E. Brouwer, "U3(3) graph".
- ↑ Robert A. Wilson, 'The Finite Simple Groups', Springer-Verlag (2009), p. 224.