Table of simple cubic graphs

The connected 3-regular (cubic) simple graphs are listed for small vertex numbers.

Connectivity

The number of connected simple cubic graphs on 4, 6, 8, 10, ... vertices is 1, 2, 5, 19, ... Template:OEIS. A classification according to edge connectivity is made as follows: the 1-connected and 2-connected graphs are defined as usual. This leaves the other graphs in the 3-connected class because each 3-regular graph can be split by cutting all edges adjacent to any of the vertices. To refine this definition in the light of the algebra of coupling of angular momenta (see below), a subdivision of the 3-connected graphs is helpful. We shall call

Non-trivially 3-connected those that can be split by 3 edge cuts into sub-graphs with at least two vertices remaining in each part
Cyclically 4-connected—all those not 1-connected, not 2-connected, and not non-trivially 3-connected

This declares the numbers 3 and 4 in the fourth column of the tables below.

Pictures

Ball-and-stick models of the graphs in another column of the table show the vertices and edges in the style of images of molecular bonds. Comments on the individual pictures contain girth, diameter, Wiener index, Estrada index and Kirchhoff index. Aut is the order of the Automorphism group of the graph. A Hamiltonian circuit (where present) is indicated by enumerating vertices along that path from 1 upwards. (The positions of the vertices have been defined by minimizing a pair potential defined by the squared difference of the Euclidean and graph theoretic distance, placed in a Molfile, then rendered by Jmol.)

LCF notation

The LCF notation is a notation by Joshua Lederberg, Coxeter and Frucht, for the representation of cubic graphs that are Hamiltonian.

The two edges along the cycle adjacent to any of the vertices are not written down.

Let Template:Math be the vertices of the graph and describe the Hamiltonian circle along the Template:Math vertices by the edge sequence Template:Math. Halting at a vertex Template:Math, there is one unique vertex Template:Math at a distance Template:Math joined by a chord with Template:Math,

j = i + d_{i} (mod p), 2 \leq d_{i} \leq p - 2.

The vector Template:Math of the Template:Math integers is a suitable, although not unique, representation of the cubic Hamiltonian graph. This is augmented by two additional rules:

If a Template:Math, replace it by Template:Math;
avoid repetition of a sequence of Template:Math if these are periodic and replace them by an exponential notation.

Since the starting vertex of the path is of no importance, the numbers in the representation may be cyclically permuted. If a graph contains different Hamiltonian circuits, one may select one of these to accommodate the notation. The same graph may have different LCF notations, depending on precisely how the vertices are arranged.

Often the anti-palindromic representations with

d_{p - 1 - i} = - d_{i} (mod p), i = 0, 1, \dots p / 2 - 1

are preferred (if they exist), and the redundant part is then replaced by a semicolon and a dash "; –". The LCF notation Template:Math, for example, and would at that stage be condensed to Template:Math.

Table

4 vertices

diam.	girth	Aut.	connect.	LCF	names	picture
1	3	24	4	[2]⁴	K₄

6 vertices

diam.	girth	Aut.	connect.	LCF	names	picture
2	3	12	3	[2, 3, −2]²	prism graph Y₃
2	4	72	4	[3]⁶	K_{3, 3}, utility graph

8 vertices

diam.	girth	Aut.	connect.	LCF	names	pictures
3	3	16	2	[2, 2, −2, −2]²
3	3	4	3	[4, −2, 4, 2]² or [2, 3, −2, 3; –]
2	3	12	3	[2, 4, −2, 3, 3, 4, −3, −3]
3	4	48	4	[−3, 3]⁴	cubical graph
2	4	16	4	[4]⁸ or [4, −3, 3, 4]²	Wagner graph

10 vertices

diam.	girth	Aut.	connect.	LCF	names	pictures
5	3	32	1		Edge list 0–1, 0–6, 0–9, 1–2, 1–5, 2–3, 2–4, 3–4, 3–5, 4–5, 6–7, 6–8, 7–8, 7–9, 8–9
4	3	4	2	[4, 2, 3, −2, −4, −3, 2, 2, −2, −2]
3	3	8	2	[2, −3, −2, 2, 2; –]
3	3	16	2	[−2, −2, 3, 3, 3; –]
4	3	16	2	[2, 2, −2, −2, 5]²
3	3	2	3	[2, 3, −2, 5, −3]² [3, −2, 4, −3, 4, 2, −4, −2, −4, 2]
3	3	12	3	[2, −4, −2, 5, 2, 4, −2, 4, 5, −4]
3	3	2	3	[5, 3, 5, −4, −3, 5, 2, 5, −2, 4] [−4, 2, 5, −2, 4, 4, 4, 5, −4, −4] [−3, 2, 4, −2, 4, 4, −4, 3, −4, −4]
3	3	4	3	[−4, 3, 3, 5, −3, −3, 4, 2, 5, −2] [3, −4, −3, −3, 2, 3, −2, 4, −3, 3]
3	3	6	3	[3, −3, 5, −3, 2, 4, −2, 5, 3, −4]
3	3	4	3	[2, 3, −2, 3, −3; –] [−4, 4, 2, 5, −2]²
3	3	6	3	[5, −2, 2, 4, −2, 5, 2, −4, −2, 2]
3	3	8	3	[2, 5, −2, 5, 5]² [2, 4, −2, 3, 4; –]
3	4	48	3	[5, −3, −3, 3, 3]²
3	4	8	4	[5, −4, 4, −4, 4]² [5, −4, −3, 3, 4, 5, −3, 4, −4, 3]
3	4	4	4	[5, −4, 4, 5, 5]² [−3, 4, −3, 3, 4; –] [4, −3, 4, 4, −4; –] [−4, 3, 5, 5, −3, 4, 4, 5, 5, −4]
3	4	20	4	[5]¹⁰ [−3, 3]⁵ [5, 5, −3, 5, 3]²
3	4	20	4	[−4, 4, −3, 5, 3]²	Pentagonal prism, G_{5, 2}
2	5	120	4		Petersen graph

12 vertices

diam.	girth	Aut.	connect.	LCF	names	picture
6	3	16	1		Edge list 0–1, 0–2, 0–11, 1–2, 1–6, 2–3, 3–4, 3–5, 4–5, 4–6, 5–6, 7–8, 7–9, 7–11, 8–9, 8–10, 9–10, 10–11
5	3	16	1		Edge list 0–1, 0–6, 0–11, 1–2, 1–3, 2–3, 2–5, 3–4, 4–5, 4–6, 5–6, 7–8, 7–9, 7–11, 8–9, 8–10, 9–10, 10–11
6	3	8	1		Edge list 0–1, 0–3, 0–11, 1–2, 1–6, 2–3, 2–5, 3–4, 4–5, 4–6, 5–6, 7–8, 7–9, 7–11, 8–9, 8–10, 9–10, 10–11
5	3	32	1		Edge list 0–1, 0–6, 0–11, 1–2, 1–4, 2–3, 2–5, 3–4, 3–6, 4–5, 5–6, 7–8, 7–9, 7–11, 8–9, 8–10, 9–10, 10–11
5	3	4	2	[3, −2, −4, −3, 4, 2]² [4, 2, 3, −2, −4, −3; –]
4	3	8	2	[3, −2, −4, −3, 3, 3, 3, −3, −3, −3, 4, 2]
4	3	4	2	[4, 2, 3, −2, −4, −3, 2, 3, −2, 2, −3, −2]
4	4	64	2	[3, 3, 3, −3, −3, −3]²
4	3	16	2	[2, −3, −2, 3, 3, 3; –]
4	3	16	2	[2, 3, −2, 2, −3, −2]²
4	3	2	2	[−2, 3, 6, 3, −3, 2, −3, −2, 6, 2, 2, −2] [4, 2, −4, −2, −4, 6, 2, 2, −2, −2, 4, 6]
4	3	8	2	[6, 3, 3, 4, −3, −3, 6, −4, 2, 2, −2, −2]
5	3	4	2	[4, 2, 3, −2, −4, −3, 5, 2, 2, −2, −2, −5]
4	3	16	2	[−3, −3, −3, 5, 2, 2; –]
4	3	8	2	[2, −3, −2, 5, 2, 2; –]
4	3	4	2	[2, 4, −2, 3, −5, −4, −3, 2, 2, −2, −2, 5] [5, 2, −4, −2, −5, −5, 2, 2, −2, −2, 4, 5]
4	3	4	2	[−2, −2, 4, 4, 4, 4; –] [3, −4, −4, −3, 2, 2; –] [5, 3, 4, 4, −3, −5, −4, −4, 2, 2, −2, −2]
4	3	2	2	[4, −2, 4, 2, −4, −2, −4, 2, 2, −2, −2, 2] [5, −2, 2, 3, −2, −5, −3, 2, 2, −2, −2, 2]
5	3	16	2	[2, 2, −2, −2, −5, 5]²
4	3	8	2	[−2, −2, 4, 5, 3, 4; –]
4	3	4	2	[5, 2, −3, −2, 6, −5, 2, 2, −2, −2, 6, 3]
4	3	8	2	[4, −2, 3, 3, −4, −3, −3, 2, 2, −2, −2, 2]
4	3	8	2	[−2, −2, 5, 3, 5, 3; –] [−2, −2, 3, 5, 3, −3; –]
5	3	32	2	[2, 2, −2, −2, 6, 6]²
4	3	8	2	[−3, 2, −3, −2, 2, 2; –]
4	3	8	2	[−2, −2, 5, 2, 5, −2; –]
4	3	8	2	[6, −2, 2, 2, −2, −2, 6, 2, 2, −2, −2, 2]
4	3	48	2	[−2, −2, 2, 2]³
4	3	4	3	[2, 3, −2, 3, −3, 3; –] [−4, 6, 4, 2, 6, −2]²
4	3	4	3	[−4, 6, 3, 3, 6, −3, −3, 6, 4, 2, 6, −2] [−2, 3, −3, 4, −3, 3, 3, −4, −3, −3, 2, 3]
4	3	1	3	[−5, 2, −3, −2, 6, 4, 2, 5, −2, −4, 6, 3] [−2, 3, −3, 4, −3, 4, 2, −4, −2, −4, 2, 3] [3, −2, 3, −3, 5, −3, 2, 3, −2, −5, −3, 2]
3	3	4	3	[−5, −5, 4, 2, 6, −2, −4, 5, 5, 2, 6, −2] [4, −2, 3, 4, −4, −3, 3, −4, 2, −3, −2, 2]
3	3	8	3	[−5, −5, 3, 3, 6, −3, −3, 5, 5, 2, 6, −2] [2, 4, −2, 3, 5, −4, −3, 3, 3, −5, −3, −3]
4	3	2	3	[2, 4, −2, 3, 6, −4, −3, 2, 3, −2, 6, −3] [2, 4, −2, 3, 5, −4, −3, 4, 2, −5, −2, −4] [−5, 2, −3, −2, 5, 5, 2, 5, −2, −5, −5, 3]
4	3	2	3	[−5, 2, −3, −2, 6, 3, 3, 5, −3, −3, 6, 3] [4, −2, −4, 4, −4, 3, 3, −4, −3, −3, 4, 2] [−3, 3, 3, 4, −3, −3, 5, −4, 2, 3, −2, −5]
4	3	2	3	[2, 3, −2, 4, −3, 6, 3, −4, 2, −3, −2, 6] [−4, 5, −4, 2, 3, −2, −5, −3, 4, 2, 4, −2]
4	3	1	3	[6, 3, −4, −4, −3, 3, 6, 2, −3, −2, 4, 4] [−5, −4, 4, 2, 6, −2, −4, 5, 3, 4, 6, −3] [3, 4, 4, −3, 4, −4, −4, 3, −4, 2, −3, −2] [4, 5, −4, −4, −4, 3, −5, 2, −3, −2, 4, 4] [4, 5, −3, −5, −4, 3, −5, 2, −3, −2, 5, 3]
3	4	4	3	[4, 6, −4, −4, −4, 3, 3, 6, −3, −3, 4, 4] [−5, −4, 3, 3, 6, −3, −3, 5, 3, 4, 6, −3] [4, −3, 5, −4, −4, 3, 3, −5, −3, −3, 3, 4]
3	4	16	3	[3, 3, 4, −3, −3, 4; –] [3, 6, −3, −3, 6, 3]²
4	3	1	3	[4, −2, 5, 2, −4, −2, 3, −5, 2, −3, −2, 2] [5, −2, 2, 4, −2, −5, 3, −4, 2, −3, −2, 2] [2, −5, −2, −4, 2, 5, −2, 2, 5, −2, −5, 4]	Frucht graph
4	3	4	3	[−2, 6, 2, −4, −2, 3, 3, 6, −3, −3, 2, 4] [−2, 2, 5, −2, −5, 3, 3, −5, −3, −3, 2, 5]
4	3	2	3	[2, 4, −2, 6, 2, −4, −2, 4, 2, 6, −2, −4] [2, 5, −2, 2, 6, −2, −5, 2, 3, −2, 6, −3]
4	3	2	3	[6, 3, −3, −5, −3, 3, 6, 2, −3, −2, 5, 3] [3, 5, 3, −3, 4, −3, −5, 3, −4, 2, −3, −2] [−5, −3, 4, 2, 5, −2, −4, 5, 3, −5, 3, −3]
4	4	12	3	[3, −3, 5, −3, −5, 3, 3, −5, −3, −3, 3, 5]
4	3	2	3	[4, 2, 4, −2, −4, 4; –] [3, 5, 2, −3, −2, 5; –] [6, 2, −3, −2, 6, 3]²
4	3	2	3	[3, 6, 4, −3, 6, 3, −4, 6, −3, 2, 6, −2] [4, −4, 5, 3, −4, 6, −3, −5, 2, 4, −2, 6] [−5, 5, 3, −5, 4, −3, −5, 5, −4, 2, 5, −2]
3	3	1	3	[6, −5, 2, 6, −2, 6, 6, 3, 5, 6, −3, 6] [6, 2, −5, −2, 4, 6, 6, 3, −4, 5, −3, 6] [5, 5, 6, 4, 6, −5, −5, −4, 6, 2, 6, −2] [−4, 4, −3, 3, 6, −4, −3, 2, 4, −2, 6, 3] [6, 2, −4, −2, 4, 4, 6, 4, −4, −4, 4, −4] [−3, 2, 5, −2, −5, 3, 4, −5, −3, 3, −4, 5] [−5, 2, −4, −2, 4, 4, 5, 5, −4, −4, 4, −5]
3	3	2	3	[2, 6, −2, 5, 6, 4, 5, 6, −5, −4, 6, −5] [5, 6, −4, −4, 5, −5, 2, 6, −2, −5, 4, 4] [2, 4, −2, −5, 4, −4, 3, 4, −4, −3, 5, −4] [2, −5, −2, 4, −5, 4, 4, −4, 5, −4, −4, 5]
4	3	4	3	[2, 4, −2, −5, 5]² [−5, 2, 4, −2, 6, 3, −4, 5, −3, 2, 6, −2]
4	3	2	3	[−4, −4, 4, 2, 6, −2, −4, 4, 4, 4, 6, −4] [−4, −3, 4, 2, 5, −2, −4, 4, 4, −5, 3, −4] [−3, 5, 3, 4, −5, −3, −5, −4, 2, 3, −2, 5]
3	3	2	3	[2, 5, −2, 4, 4, 5; –] [2, 4, −2, 4, 4, −4; –] [−5, 5, 6, 2, 6, −2]² [5, −2, 4, 6, 3, −5, −4, −3, 2, 6, −2, 2]
3	3	2	3	[3, 6, −4, −3, 5, 6, 2, 6, −2, −5, 4, 6] [2, −5, −2, 4, 5, 6, 4, −4, 5, −5, −4, 6] [5, −4, 4, −4, 3, −5, −4, −3, 2, 4, −2, 4]
4	3	2	3	[6, −5, 2, 4, −2, 5, 6, −4, 5, 2, −5, −2] [−2, 4, 5, 6, −5, −4, 2, −5, −2, 6, 2, 5] [5, −2, 4, −5, 4, −5, −4, 2, −4, −2, 5, 2]
4	3	1	3	[2, −5, −2, 6, 3, 6, 4, −3, 5, 6, −4, 6] [6, 3, −3, 4, −3, 4, 6, −4, 2, −4, −2, 3] [5, −4, 6, −4, 2, −5, −2, 3, 6, 4, −3, 4] [5, −3, 5, 6, 2, −5, −2, −5, 3, 6, 3, −3] [−5, 2, −5, −2, 6, 3, 5, 5, −3, 5, 6, −5] [−3, 4, 5, −5, −5, −4, 2, −5, −2, 3, 5, 5] [5, 5, 5, −5, 4, −5, −5, −5, −4, 2, 5, −2]
3	3	2	3	[5, −3, 6, 3, −5, −5, −3, 2, 6, −2, 3, 5] [2, 6, −2, −5, 5, 3, 5, 6, −3, −5, 5, −5] [5, 5, 5, 6, −5, −5, −5, −5, 2, 6, −2, 5] [4, −3, 5, 2, −4, −2, 3, −5, 3, −3, 3, −3] [5, 5, −3, −5, 4, −5, −5, 2, −4, −2, 5, 3]
4	3	4	3	[2, 4, −2, 5, 3, −4; –] [5, −3, 2, 5, −2, −5; –] [3, 6, 3, −3, 6, −3, 2, 6, −2, 2, 6, −2]
4	3	2	3	[6, 2, −4, −2, −5, 3, 6, 2, −3, −2, 4, 5] [2, 3, −2, 4, −3, 4, 5, −4, 2, −4, −2, −5] [−5, 2, −4, −2, −5, 4, 2, 5, −2, −4, 4, 5]
3	3	2	3	[5, 2, 5, −2, 5, −5; –] [6, 2, −4, −2, 4, 6]² [2, −5, −2, 6, 2, 6, −2, 3, 5, 6, −3, 6] [−5, −2, 6, 6, 2, 5, −2, 5, 6, 6, −5, 2]
3	3	12	3	[−5, 3, 3, 5, −3, −3, 4, 5, −5, 2, −4, −2]
3	3	2	3	[6, −4, 3, 4, −5, −3, 6, −4, 2, 4, −2, 5] [−4, 6, −4, 2, 5, −2, 5, 6, 4, −5, 4, −5] [5, −5, 4, −5, 3, −5, −4, −3, 5, 2, 5, −2]
4	3	12	3	[−4, 5, 2, −4, −2, 5; –]	Dürer graph
3	3	4	3	[2, 5, −2, 5, 3, 5; –] [6, −2, 6, 6, 6, 2]² [5, −2, 6, 6, 2, −5, −2, 3, 6, 6, −3, 2]
3	3	4	3	[6, −2, 6, 4, 6, 4, 6, −4, 6, −4, 6, 2] [5, 6, −3, 3, 5, −5, −3, 6, 2, −5, −2, 3]
3	3	4	3	[4, −2, 4, 6, −4, 2, −4, −2, 2, 6, −2, 2] [5, −2, 5, 6, 2, −5, −2, −5, 2, 6, −2, 2]
3	3	24	3	[6, −2, 2]⁴	Truncated tetrahedron
3	3	12	3		Tietze's Graph
3	3	36	3	[2, 6, −2, 6]³
4	4	24	4	[−3, 3]⁶ [3, −5, 5, −3, −5, 5]²	G_{6, 2}, Y₆
3	4	4	4	[6, −3, 6, 6, 3, 6]² [6, 6, −5, 5, 6, 6]² [3, −3, 4, −3, 3, 4; –] [5, −3, 6, 6, 3, −5]² [5, −3, −5, 4, 4, −5; –] [6, 6, −3, −5, 4, 4, 6, 6, −4, −4, 5, 3]
3	4	8	4	[−4, 4, 4, 6, 6, −4]² [6, −5, 5, −5, 5, 6]² [4, −3, 3, 5, −4, −3; –] [−4, −4, 4, 4, −5, 5]²
3	4	2	4	[−4, 6, 3, 6, 6, −3, 5, 6, 4, 6, 6, −5] [−5, 4, 6, 6, 6, −4, 5, 5, 6, 6, 6, −5] [5, −3, 4, 6, 3, −5, −4, −3, 3, 6, 3, −3] [4, −4, 6, 4, −4, 5, 5, −4, 6, 4, −5, −5] [4, −5, −3, 4, −4, 5, 3, −4, 5, −3, −5, 3]
3	4	2	4	[3, 4, 5, −3, 5, −4; –] [3, 6, −4, −3, 4, 6]² [−4, 5, 5, −4, 5, 5; –] [3, 6, −4, −3, 4, 4, 5, 6, −4, −4, 4, −5] [4, −5, 5, 6, −4, 5, 5, −5, 5, 6, −5, −5] [4, −4, 5, −4, −4, 3, 4, −5, −3, 4, −4, 4]
3	4	8	4	[4, −4, 6]⁴ [3, 6, 3, −3, 6, −3]² [−3, 6, 4, −4, 6, 3, −4, 6, −3, 3, 6, 4]	Bidiakis cube
3	4	16	4	[6, −5, 5]⁴ [3, 4, −4, −3, 4, −4]²
3	4	2	4	[−3, 5, −3, 4, 4, 5; –] [4, −5, 5, 6, −4, 6]² [−3, 4, −3, 4, 4, −4; –] [5, 6, −3, −5, 4, −5, 3, 6, −4, −3, 5, 3] [5, 6, 4, −5, 5, −5, −4, 6, 3, −5, 5, −3]
3	4	4	4	[4, −3, 4, 5, −4, 4; –] [4, 5, −5, 5, −4, 5; –] [−5, −3, 4, 5, −5, 4; –]
3	4	2	4	[6, −4, 6, −4, 3, 5, 6, −3, 6, 4, −5, 4] [6, −4, 3, −4, 4, −3, 6, 3, −4, 4, −3, 4] [5, 6, −4, 3, 5, −5, −3, 6, 3, −5, 4, −3] [5, −5, 4, 6, −5, −5, −4, 3, 5, 6, −3, 5] [5, 5, −4, 4, 5, −5, −5, −4, 3, −5, 4, −3]
3	4	4	4	[6, −3, 5, 6, −5, 3, 6, −5, −3, 6, 3, 5] [3, −4, 5, −3, 4, 6, 4, −5, −4, 4, −4, 6]
3	4	8	4	[5, 6, 6, −4, 5, −5, 4, 6, 6, −5, −4, 4]
3	5	16	4	[4, −5, 4, −5, −4, 4; –]
3	4	4	4	[6, 4, 6, 6, 6, −4]² [−3, 4, −3, 5, 3, −4; –] [−5, 3, 6, 6, −3, 5, 5, 5, 6, 6, −5, −5] [−3, 3, 6, 4, −3, 5, 5, −4, 6, 3, −5, −5]
4	4	8	4	[3, 5, 5, −3, 5, 5; –] [−3, 5, −3, 5, 3, 5; –] [5, −3, 5, 5, 5, −5; –]
3	4	48	4	[5, −5, −3, 3]³ [−5, 5]⁶	Franklin graph
3	4	24	4	[6]¹² [6, 6, −3, −5, 5, 3]²
3	5	18	4	[6, −5, −4, 4, −5, 4, 6, −4, 5, −4, 4, 5]

The LCF entries are absent above if the graph has no Hamiltonian cycle, which is rare (see Tait's conjecture). In this case a list of edges between pairs of vertices labeled 0 to n−1 in the third column serves as an identifier.

Vector coupling coefficients

Each 4-connected (in the above sense) simple cubic graph on Template:Math vertices defines a class of quantum mechanical Template:Math Template:Hyphenj symbols. Roughly speaking, each vertex represents a 3-jm symbol, the graph is converted to a digraph by assigning signs to the angular momentum quantum numbers Template:Math, the vertices are labelled with a handedness representing the order of the three Template:Math (of the three edges) in the 3Template:Hyphenjm symbol, and the graph represents a sum over the product of all these numbers assigned to the vertices.

There are 1 ([[6-j symbol|6Template:Hyphenj]]), 1 ([[9-j symbol|9Template:Hyphenj]]), 2 (12Template:Hyphenj), 5 (15Template:Hyphenj), 18 (18Template:Hyphenj), 84 (21Template:Hyphenj), 607 (24Template:Hyphenj), 6100 (27Template:Hyphenj), 78824 (30Template:Hyphenj), 1195280 (33Template:Hyphenj), 20297600 (36Template:Hyphenj), 376940415 (39Template:Hyphenj) etc. of these Template:OEIS.

If they are equivalent to certain vertex-induced binary trees (cutting one edge and finding a cut that splits the remaining graph into two trees), they are representations of recoupling coefficients, and are then also known as Yutsis graphs Template:OEIS.

References

Table of simple cubic graphs

Contents

Connectivity

Pictures

LCF notation