Gewirtz graph
In today's article we are going to talk about Gewirtz graph. This is a topic that has been of interest to many people throughout history and that continues to generate debate today. From its origins to its implications in today's society, Gewirtz graph has been the object of study and reflection by experts in different fields. Throughout this article, we will explore the different aspects related to Gewirtz graph, from its impacts on everyday life to its influence on popular culture. Without a doubt, Gewirtz graph is a fascinating topic that deserves our attention and reflection.
| 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 | |
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
- ^ Allan Gewirtz, Graphs with Maximal Even Girth, Ph.D. Dissertation in Mathematics, City University of New York, 1967.
References
- Brouwer, Andries. "Sims-Gewirtz graph".
- Weisstein, Eric W. "Gewirtz graph". MathWorld.