# Dyck graph

Dyck graph
The Dyck graph
Named after W. Dyck
Vertices 32
Edges 48
Diameter 5
Girth 6
Automorphisms 192
Chromatic number 2
Chromatic index 3
Properties Symmetric
Cubic
Hamiltonian
Bipartite
Cayley graph

In the mathematical field of graph theory, the Dyck graph is a 3-regular graph with 32 vertices and 48 edges, named after Walther von Dyck.[1][2]

It is Hamiltonian with 120 distinct Hamiltonian cycles. It has chromatic number 2, chromatic index 3, radius 5, diameter 5 and girth 6. It is also a 3-vertex-connected and a 3-edge-connected graph.

The Dyck graph is a toroidal graph, and the dual of its symmetric toroidal embedding is the Shrikhande graph, a strongly regular graph both symmetric and hamiltonian.

## Algebraic properties

The automorphism group of the Dyck graph is a group of order 192.[3] It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore, the Dyck graph is a symmetric graph. It has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the Foster census, the Dyck graph, referenced as F32A, is the only cubic symmetric graph on 32 vertices.[4]

The characteristic polynomial of the Dyck graph is equal to ${\displaystyle (x-3)(x-1)^{9}(x+1)^{9}(x+3)(x^{2}-5)^{6}}$.

## Dyck map

The Dyck graph is the skeleton of a symmetric tessellation of a surface of genus three by twelve octagons, known as the Dyck map or Dyck tiling. The dual graph for this tiling is the complete tripartite graph K4,4,4.[5][6]

## References

1. ^ Dyck, W. (1881), "Über Aufstellung und Untersuchung von Gruppe und Irrationalität regulärer Riemann'scher Flächen", Math. Ann., 17: 473, doi:10.1007/bf01446929.
2. ^
3. ^ Royle, G. F032A data
4. ^ Conder, M.; Dobcsányi, P. (2002), "Trivalent symmetric graphs up to 768 vertices", J. Combin. Math. Combin. Comput., 40: 41–63.
5. ^ Dyck, W. (1880), "Notiz über eine reguläre Riemannsche Fläche vom Geschlecht 3 und die zugehörige Normalkurve 4. Ordnung", Math. Ann., 17: 510–516, doi:10.1007/bf01446930.
6. ^ Ceulemans, A. (2004), "The tetrakisoctahedral group of the Dyck graph and its molecular realization.", Molecular physics, 102 (11): 1149–1163, doi:10.1080/00268970410001728780.