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]
| 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 |
| Automorphisms | 80,640 |
| Chromatic number | 4 |
| Properties | Strongly regular Hamiltonian Triangle-free Vertex-transitive Edge-transitive Distance-transitive. |
| Table of graphs and parameters | |
Construction
editThe 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
editThe 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
edit- ^ Allan Gewirtz, Graphs with Maximal Even Girth, Ph.D. Dissertation in Mathematics, City University of New York, 1967.
References
edit- Brouwer, Andries. "Sims-Gewirtz graph".
- Weisstein, Eric W. "Gewirtz graph". MathWorld.