This page is modeled after the handy wikipedia page Table of simple cubic graphs of “small” connected 3-regular graphs, where by small I mean at most 12 vertices.

These graphs are obtained using the SageMath command `graphs(n, [4]*n)`

, where n = 5,6,7,… .

**5 vertices**: Let denote the vertex set. There is (up to isomorphism) exactly one 4-regular connected graphs on 5 vertices. By Ore’s Theorem, this graph is Hamiltonian. By Euler’s Theorem, it is Eulerian.

*4reg5a*: The only such 4-regular graph is the complete graph .

We have

- diameter = 1
- girth = 3
- If G denotes the automorphism group then G has cardinality 120 and is generated by (3,4), (2,3), (1,2), (0,1). (In this case, clearly, .)
- edge set:

**6 vertices**: Let denote the vertex set. There is (up to isomorphism) exactly one 4-regular connected graphs on 6 vertices. By Ore’s Theorem, this graph is Hamiltonian. By Euler’s Theorem, it is Eulerian.

*4reg6a*: The first (and only) such 4-regular graph is the graph having edge set: .

We have

- diameter = 2
- girth = 3
- If G denotes the automorphism group then G has cardinality 48 and is generated by

**7 vertices**: Let denote the vertex set. There are (up to isomorphism) exactly 2 4-regular connected graphs on 7 vertices. By Ore’s Theorem, these graphs are Hamiltonian. By Euler’s Theorem, they are Eulerian.

*4reg7a*: The 1st such 4-regular graph is the graph having edge set: . This is an Eulerian, Hamiltonian (by Ore’s Theorem), vertex transitive (but not edge transitive) graph.

We have

- diameter = 2
- girth = 3
- If G denotes the automorphism group then G has cardinality 14 and is generated by (1,5)(2,4)(3,6), (0,1,3,2,4,6,5).

*4reg7b*: The 2nd such 4-regular graph is the graph having edge set: . This is an Eulerian, Hamiltonian graph (by Ore’s Theorem) which is neither vertex transitive nor edge transitive.

We have

- diameter = 2
- girth = 3
- If G denotes the automorphism group then G has cardinality 48 and is generated by (3,4), (2,5), (1,3)(4,6), (0,2)

**8 vertices**: Let denote the vertex set. There are (up to isomorphism) exactly six 4-regular connected graphs on 8 vertices. By Ore’s Theorem, these graphs are Hamiltonian. By Euler’s Theorem, they are Eulerian.

*4reg8a*: The 1st such 4-regular graph is the graph having edge set: . This is a vertex transitive (but not edge transitive) graph.

We have

- diameter = 2
- girth = 3
- If G denotes the automorphism group then G has cardinality 12 and is generated by (1,7)(2,3)(5,6), (0,1)(2,4)(3,5)(6,7).

*4reg8b*: The 2nd such 4-regular graph is the graph having edge set: . This is a vertex transitive (but not edge transitive) graph.

We have

- diameter = 2
- girth = 3
- If G denotes the automorphism group then G has cardinality 48 and is generated by (1,7)(2,3)(5,6), (0,1)(2,4)(3,5)(6,7).

*4reg8c*: The 3rd such 4-regular graph is the graph having edge set: . This is a strongly regular (with “trivial” parameters (8, 4, 0, 4)), vertex transitive, edge transitive graph.

We have

- diameter = 2
- girth = 4
- If G denotes the automorphism group then G has cardinality and is generated by (5,6), (4,7), (3,4), (2,5), (1,2), (0,1)(2,3)(4,5)(6,7).

*4reg8d*: The 4th such 4-regular graph is the graph having edge set: . This graph is not vertex transitive, nor edge transitive.

We have

- diameter = 2
- girth = 3
- If G denotes the automorphism group then G has cardinality 16 and is generated by (3,5), (1,4), (0,2)(1,3)(4,5)(6,7), (0,6)(2,7).

*4reg8e*: The 5th such 4-regular graph is the graph having edge set: . This graph is not vertex transitive, nor edge transitive.

We have

- diameter = 2
- girth = 3
- If G denotes the automorphism group then G has cardinality 4 and is generated by (0,1)(2,4)(3,6)(5,7), (0,2)(1,4)(3,6).

*4reg8f*: The 6th (and last) such 4-regular graph is the bipartite graph having edge set: . This graph is not vertex transitive, nor edge transitive.

We have

- diameter = 2
- girth = 3
- If G denotes the automorphism group then G has cardinality 12 and is generated by (3,4)(6,7), (1,2), (0,3)(5,6).