The databases contained on this page seek to provide an encompassing list of important properties for exhaustive sets of cubic graphs. The objective is to allow cross-referencing of such properties to identify special cubic graphs of interest to researchers. Each set contains, for all cubic graphs of a given size, a *property signature* that describes the properties of that graph. The properties currently listed are:

ID number as output by GENREG [1] with the command: genreg k 3 3

Hamiltonicity (0 for non-Hamiltonian, 1 for Hamiltonian)

Number of undirected Hamiltonian cycles (0 for non-Hamiltonian graphs)

Hyperhamiltonicity (0 for non-Hyperhamiltonian, 1 for Hyperhamiltonian)

Hypohamiltonicity (0 for non-Hypohamiltonian, 1 for Hypohamiltonian)

Edge-connectivity

Cyclic edge-connectivity

Girth

Planarity (0 for non-planar, 1 for planar)

Bipartiteness (0 for non-bipartite, 1 for bipartite)

Snark (0 for non-Snark, 1 for Snark)

Filar the graph is located in (number of triangles)

Gene (0 for descendant, 1 for gene)

Mutant (0 for non-mutant, 1 for mutant)

Number of ancestor genes (1 if graph is a gene)

Size of largest ancestor gene (the size of the graph for a gene)

Second-largest eigenvalue

Diameter

Currently the datbases are available for all cubic graphs up to 20 vertices. They can be downloaded individually as text files, or as a combined Excel file.

- 4-vertex cubic graphs
- 6-vertex cubic graphs
- 8-vertex cubic graphs
- 10-vertex cubic graphs
- 12-vertex cubic graphs
- 14-vertex cubic graphs
- 16-vertex cubic graphs
- 18-vertex cubic graphs
- 20-vertex cubic graphs
- Combined Excel file

A 10-vertex graph with the following property signature:

**4 1 4 0 0 2 2 3 1 0 0 5 0 0 3 4 2.41421 3**

- Has ID
**#4**as output by GENREG with the command: genreg 10 3 3 - Is
**Hamiltonian** - Contains
**4**undirected Hamiltonian cycles - Is
**non-Hyperhamiltonian** - Is
**non-Hypohamiltonian** - Is
**2**-edge-connected - Is
**2**-cyclic-edge-connected - Has girth
**3** - Is
**planar** - Is
**non-bipartite** - Is
**not**a Snark - Is in the
**5th**filar - Is
**not**a Gene - is
**not**a mutant - Contains
**3**ancestor genes - Has a largest ancestor gene of size
**4** - Has second-largest eigenvalue of
**2.41421** - Has diameter
**3**

If you believe there are additional properties we should be including in the property signature, please contact Michael Haythorpe.

[1] **M. Meringer**: Fast Generation of Regular Graphs and Construction of Cages.*Journal of Graph Theory 30*, 137-146, 1999.