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{{Infobox graph | |||
| name = Gewirtz graph | |||
| image = [[Image:Gewirtz graph embeddings.svg|300px]] | |||
| image_caption = Some embeddings with 7-fold symmetry. No 8-fold or 14-fold symmetry are possible. | |||
| vertices = 56 | |||
| edges = 280 | |||
| automorphisms = {{formatnum:80640}} | |||
| radius = 2 | |||
| diameter = 2 | |||
| girth = 4 | |||
| chromatic_number = 4 | |||
| properties = [[Strongly regular graph|Strongly regular]]<br>[[Hamiltonian graph|Hamiltonian]]<br>[[Triangle-free graph|Triangle-free]]<br>[[Vertex-transitive graph|Vertex-transitive]]<br>[[edge-transitive graph|Edge-transitive]]<br>[[distance-transitive graph|Distance-transitive]]. | |||
}} | |||
The '''Gewirtz graph''' is a [[strongly regular graph]] with 56 vertices and valency 10. It is named after the mathematician Allan Gewirtz, who described the graph in his dissertation.<ref>[http://genealogy.math.ndsu.nodak.edu/id.php?id=35587 Allan Gewirtz], ''Graphs with Maximal Even Girth'', Ph.D. Dissertation in Mathematics, City University of New York, 1967.</ref> | |||
==Construction== | |||
The Gewirtz graph can be constructed as follows. Consider the unique ''S''(3, 6, 22) [[Steiner system]], with 22 elements and 77 blocks. Choose a random element, and let the vertices be the 56 blocks not containing it. Two blocks are adjacent when they are disjoint. | |||
With this construction, one can embed the Gewirtz graph in the [[Higman–Sims graph]]. | |||
==Properties== | |||
The [[characteristic polynomial]] of the Gewirtz graph is | |||
: <math>(x-10)(x-2)^{35}(x+4)^{20}. \, </math> | |||
Therefore it is an [[integral graph]]. The Gewirtz graph is also determined by its spectrum. | |||
The [[independence number]] is 16. | |||
==Notes== | |||
{{Reflist}} | |||
==References== | |||
* {{cite web |url=http://www.win.tue.nl/~aeb/graphs/Sims-Gewirtz.html |title=Sims-Gewirtz graph |author=Brouwer, Andries}} | |||
* {{MathWorld|title=Gewirtz graph|urlname=GewirtzGraph}} | |||
[[Category:Individual graphs]] | |||
[[Category:Regular graphs]] | |||
[[Category:4-chromatic graphs]] | |||
Revision as of 19:51, 9 November 2013
The Gewirtz graph is a strongly regular graph with 56 vertices and valency 10. It is named after the mathematician Allan Gewirtz, who described the graph in his dissertation.[1]
Construction
The Gewirtz graph can be constructed as follows. Consider the unique S(3, 6, 22) Steiner system, with 22 elements and 77 blocks. Choose a random element, and let the vertices be the 56 blocks not containing it. Two blocks are adjacent when they are disjoint.
With this construction, one can embed the Gewirtz graph in the Higman–Sims graph.
Properties
The characteristic polynomial of the Gewirtz graph is
Therefore it is an integral graph. The Gewirtz graph is also determined by its spectrum.
The independence number is 16.
Notes
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References
- Template:Cite web
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- ↑ Allan Gewirtz, Graphs with Maximal Even Girth, Ph.D. Dissertation in Mathematics, City University of New York, 1967.