| Robertson graph | |
| Vertices: | 19 |
| Edges: | 38 |
| Automorphisms: | 24 (D12) |
| Girth: | 5 |
| Diameter: | 3 |
| Radius: | 3 |
| Chromatic Number: | 3 |
| Chromatic Index: | 5 |
| Properties: | Cage Hamiltonian |
| Book Thickness: | 3 |
| Queue Number: | 2 |
In the mathematical field of graph theory, the Robertson graph or (4,5)-cage, is a 4-regular undirected graph with 19 vertices and 38 edges named after Neil Robertson.[1]
The Robertson graph is the unique (4,5)-cage graph and was discovered by Robertson in 1964.[2] As a cage graph, it is the smallest 4-regular graph with girth 5.
It has chromatic number 3, chromatic index 5, diameter 3, radius 3 and is both 4-vertex-connected and 4-edge-connected. It has book thickness 3 and queue number 2.[3]
The Robertson graph is also a Hamiltonian graph which possesses distinct directed Hamiltonian cycles.
The Robertson graph is one of the smallest graphs with cop number 4.[4]
The Robertson graph is not a vertex-transitive graph and its full automorphism group is isomorphic to the dihedral group of order 24, the group of symmetries of a regular dodecagon, including both rotations and reflections.[5]
The characteristic polynomial of the Robertson graph is
(x-4)(x-1)2(x2-3)2(x2+x-5)
(x2+x-4)2(x2+x-3)2(x2+x-1).