Gewirtz graph
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| Gewirtz graph | |
|---|---|
 Some embeddings with 7-fold symmetry. No 8-fold or 14-fold symmetry are possible. | |
| Vertices | 56 |
| Edges | 280 |
| Radius | 2 |
| Diameter | 2 |
| Girth | 4 |
| Chromatic number | 4 |
| Properties | Strongly regular Hamiltonian Triangle-free Vertex-transitive Edge-transitive Distance-transitive. |
| Table of graphs and parameters | |
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
- [math]\displaystyle{ (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
- ↑ Allan Gewirtz, Graphs with Maximal Even Girth, Ph.D. Dissertation in Mathematics, City University of New York, 1967.
References
- Brouwer, Andries. "Sims-Gewirtz graph". http://www.win.tue.nl/~aeb/graphs/Sims-Gewirtz.html.
- Weisstein, Eric W.. "Gewirtz graph". http://mathworld.wolfram.com/GewirtzGraph.html.
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