# A table of small quartic graphs

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 11 vertices.

These graphs are obtained using the SageMath command graphs(n, [4]*n), where n = 5,6,7,… .

5 vertices: Let $V=\{0,1,2,3,4\}$ 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 $\Gamma = K_5$.

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, $G = S_5$.)
• edge set: $\{(0, 1), (0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)\}$

6 vertices: Let $V=\{0,1,\dots, 5\}$ 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 $\Gamma$ having edge set: $\{(0, 1), (0, 2), (0, 4), (0, 5), (1, 2), (1, 3), (1, 4), (2, 3), (2, 5), (3, 4), (3, 5), (4, 5)\}$.

We have

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

7 vertices: Let $V=\{0,1,\dots, 6\}$ 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 $\Gamma$ having edge set: $\{(0, 1), (0, 3), (0, 5), (0, 6), (1, 2), (1, 3), (1, 5), (2, 3), (2, 4), (2, 6), (3, 4), (4, 5), (4, 6), (5, 6)\}$. 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 $\Gamma$ having edge set: $\{(0, 1), (0, 3), (0, 4), (0, 6), (1, 2), (1, 5), (1, 6), (2, 3), (2, 4), (2, 6), (3, 4), (3, 5), (4, 5), (5, 6)\}$. 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 $V=\{0,1,\dots, 7\}$ 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 $\Gamma$ having edge set: $\{(0, 1), (0, 5), (0, 6), (0, 7), (1, 3), (1, 6), (1, 7), (2, 3), (2, 4), (2, 5), (2, 7), (3, 4), (3, 6), (4, 5), (4, 6), (5, 7)\}$. 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 $\Gamma$ having edge set: $\{(0, 1), (0, 5), (0, 6), (0, 7), (1, 3), (1, 6), (1, 7), (2, 3), (2, 4), (2, 5), (2, 7), (3, 4), (3, 6), (4, 5), (4, 6), (5, 7)\}$. 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 $\Gamma$ having edge set: $\{(0, 1), (0, 2), (0, 5), (0, 6), (1, 3), (1, 4), (1, 7), (2, 3), (2, 4), (2, 7), (3, 5), (3, 6), (4, 5), (4, 6), (5, 7), (6, 7)\}$. 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 $1152=2^7\cdot 3^2$ 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 $\Gamma$ having edge set: $\{(0, 1), (0, 2), (0, 4), (0, 6), (1, 3), (1, 5), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (3, 7), (4, 5), (4, 6), (5, 7), (6, 7)\}$. 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 $\Gamma$ having edge set: $\{(0, 1), (0, 2), (0, 6), (0, 7), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 7), (3, 5), (3, 7), (4, 5), (4, 6), (5, 6), (6, 7)\}$. 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 $\Gamma=K_{4,4}$ having edge set: $\{(0, 1), (0, 2), (0, 6), (0, 7), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 5), (3, 7), (4, 5), (4, 6), (5, 6), (5, 7), (6, 7)\}$. 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).

9 vertices: Let $V=\{0,1,\dots, 8\}$ denote the vertex set. There are (up to isomorphism) exactly 16 4-regular connected graphs on 9 vertices. Perhaps the most interesting of these is the strongly regular graph with parameters (9, 4, 1, 2). It has an automorphism group of cardinality 72.

Without going into details, it is possible to theoretically prove that there are no harmonic morphisms from any of these graphs to either the cycle graph $C_4$ or the complete graph $K_4$.

d4reg9-1
Gamma edges: E1 = [(0, 1), (0, 2), (0, 7), (0, 8), (1, 2), (1, 3), (1, 7), (2, 3), (2, 8), (3, 4), (3, 5), (4, 5), (4, 6), (4, 8), (5, 6), (5, 7), (6, 7), (6, 8)]
diameter:  2
girth:  3
is connected:  True
aut gp size:  12
aut gp gens:  [(1,2)(4,5)(7,8), (0,1)(3,8)(5,6), (0,4)(1,5)(2,6)(3,7)]


d4reg9-2
Gamma edges: E1 = [(0, 1), (0, 3), (0, 7), (0, 8), (1, 2), (1, 3), (1, 7), (2, 3), (2, 5), (2, 8), (3, 4), (4, 5), (4, 6), (4, 8), (5, 6), (5, 7), (6, 7), (6, 8)]
diameter:  2
girth:  3
is connected:  True
aut gp size:  2
aut gp gens:  [(0,5)(1,6)(2,8)(3,4)]


d4reg9-3
Gamma edges: E1 = [(0, 1), (0, 2), (0, 7), (0, 8), (1, 2), (1, 3), (1, 4), (2, 3), (2, 8), (3, 4), (3, 5), (4, 5), (4, 6), (5, 6), (5, 7), (6, 7), (6, 8), (7, 8)]
diameter:  2
girth:  3
is connected:  True
aut gp size:  18
aut gp gens:  [(1,7)(2,8)(3,6)(4,5), (0,1,4,6,8,2,3,5,7)]


d4reg9-4
Gamma edges: E1 = [(0, 1), (0, 5), (0, 7), (0, 8), (1, 2), (1, 4), (1, 7), (2, 3), (2, 4), (2, 5), (3, 4), (3, 6), (3, 8), (4, 5), (5, 6), (6, 7), (6, 8), (7, 8)]
diameter:  2
girth:  3
is connected:  True
aut gp size:  4
aut gp gens:  [(2,4), (0,6)(1,3)(7,8)]


d4reg9-5
Gamma edges: E1 = [(0, 1), (0, 3), (0, 5), (0, 8), (1, 2), (1, 4), (1, 7), (2, 3), (2, 5), (2, 7), (3, 4), (3, 8), (4, 5), (4, 6), (5, 6), (6, 7), (6, 8), (7, 8)]
diameter:  2
girth:  3
is connected:  True
aut gp size:  12
aut gp gens:  [(1,5)(2,4)(6,7), (0,1)(2,3)(4,5)(7,8)]


d4reg9-6
Gamma edges: E1 = [(0, 1), (0, 3), (0, 7), (0, 8), (1, 2), (1, 5), (1, 6), (2, 3), (2, 5), (2, 6), (3, 4), (3, 8), (4, 5), (4, 7), (4, 8), (5, 6), (6, 7), (7, 8)]
diameter:  2
girth:  3
is connected:  True
aut gp size:  8
aut gp gens:  [(2,6)(3,7), (0,3)(1,2)(4,7)(5,6)]


d4reg9-7
Gamma edges: E1 = [(0, 1), (0, 3), (0, 4), (0, 8), (1, 2), (1, 3), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (4, 5), (4, 8), (5, 6), (5, 7), (6, 7), (6, 8), (7, 8)]
diameter:  2
girth:  3
is connected:  True
aut gp size:  2
aut gp gens:  [(0,3)(1,4)(2,8)(5,6)]


d4reg9-8
Gamma edges: E1 = [(0, 1), (0, 3), (0, 7), (0, 8), (1, 2), (1, 3), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (4, 5), (4, 6), (4, 8), (5, 6), (5, 8), (6, 7), (7, 8)]
diameter:  2
girth:  3
is connected:  True
aut gp size:  2
aut gp gens:  [(0,8)(1,5)(2,6)(3,4)]


d4reg9-9
Gamma edges: E1 = [(0, 1), (0, 3), (0, 6), (0, 8), (1, 2), (1, 3), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (4, 5), (4, 7), (4, 8), (5, 6), (5, 8), (6, 7), (7, 8)]
diameter:  2
girth:  3
is connected:  True
aut gp size:  4
aut gp gens:  [(5,7), (0,3)(2,6)(4,8)]


d4reg9-10
Gamma edges: E1 = [(0, 1), (0, 3), (0, 5), (0, 8), (1, 2), (1, 4), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (3, 7), (4, 5), (4, 8), (5, 6), (6, 7), (6, 8), (7, 8)]
diameter:  2
girth:  3
is connected:  True
aut gp size:  16
aut gp gens:  [(2,6)(3,8), (1,5), (0,1)(2,3)(4,5)(6,8)]


d4reg9-11
Gamma edges: E1 = [(0, 1), (0, 3), (0, 7), (0, 8), (1, 2), (1, 4), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (3, 5), (4, 5), (4, 8), (5, 6), (6, 7), (6, 8), (7, 8)]
diameter:  2
girth:  3
is connected:  True
aut gp size:  8
aut gp gens:  [(2,4)(7,8), (0,2)(3,7)(4,6)(5,8)]


d4reg9-12
Gamma edges: E1 = [(0, 1), (0, 3), (0, 6), (0, 8), (1, 2), (1, 4), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (3, 7), (4, 5), (4, 8), (5, 6), (5, 8), (6, 7), (7, 8)]
diameter:  2
girth:  3
is connected:  True
aut gp size:  18
aut gp gens:  [(1,6)(2,5)(3,8)(4,7), (0,1,6)(2,7,3)(4,5,8), (0,2)(1,3)(5,8(6,7)]


d4reg9-13
Gamma edges: E1 = [(0, 1), (0, 3), (0, 4), (0, 8), (1, 2), (1, 5), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (3, 7), (4, 5), (4, 8), (5, 6), (6, 7), (6, 8), (7, 8)]
diameter:  2
girth:  3
is connected:  True
aut gp size:  8
aut gp gens:  [(2,6)(3,8), (0,1)(2,3)(4,5)(6,8), (0,4)(1,5)]


d4reg9-14
Gamma edges: E1 = [(0, 1), (0, 3), (0, 4), (0, 8), (1, 2), (1, 5), (1, 8), (2, 3), (2, 5), (2, 7), (3, 4), (3, 7), (4, 5), (4, 6), (5, 6), (6, 7), (6, 8), (7, 8)]
diameter:  2
girth:  3
is connected:  True
aut gp size:  72
aut gp gens:  [(2,5)(3,4)(6,7), (1,3)(4,8)(5,7), (0,1)(2,3)(4,5)]


d4reg9-15
Gamma edges: E1 = [(0, 1), (0, 4), (0, 6), (0, 8), (1, 2), (1, 3), (1, 5), (2, 3), (2, 4), (2, 7), (3, 4), (3, 7), (4, 5), (5, 6), (5, 8), (6, 7), (6, 8), (7, 8)]
diameter:  2
girth:  3
is connected:  True
aut gp size:  32
aut gp gens:  [(6,8), (2,3), (1,4), (0,1)(2,6)(3,8)(4,5)]


d4reg9-16
Gamma edges: E1 = [(0, 1), (0, 3), (0, 7), (0, 8), (1, 2), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 7), (3, 8), (4, 5), (4, 6), (5, 6), (6, 7), (6, 8), (7, 8)]
diameter:  2
girth:  3
is connected:  True
aut gp size:  16
aut gp gens:  [(7,8), (4,5), (0,1)(2,3)(4,7)(5,8), (0,2)(1,3)(4,7)(5,8)]


10 vertices: Let $V=\{0,1,\dots, 9\}$ denote the vertex set. There are (up to isomorphism) exactly 59 4-regular connected graphs on 10 vertices. One of these actually has an automorphism group of cardinality 1.

11 vertices: There are (up to isomorphism) exactly 265 4-regular connected graphs on 11 vertices.