Table of simple cubic graphs

Last updated

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

Contents

Connectivity

The number of connected simple cubic graphs on 4, 6, 8, 10, ... vertices is 1, 2, 5, 19, ... (sequence A002851 in the 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

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 v be the vertices of the graph and describe the Hamiltonian circle along the p vertices by the edge sequence v0v1, v1v2, ...,vp−2vp−1, vp−1v0. Halting at a vertex vi, there is one unique vertex vj at a distance di joined by a chord with vi,

The vector [d0, d1, ..., dp−1] of the p integers is a suitable, although not unique, representation of the cubic Hamiltonian graph. This is augmented by two additional rules:

  1. If a di > p/2, replace it by di − p;
  2. avoid repetition of a sequence of di 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

are preferred (if they exist), and the redundant part is then replaced by a semicolon and a dash "; –". The LCF notation [5, −9, 7, −7, 9, −5]4, for example, and would at that stage be condensed to [5, −9, 7; –]4.

Table

4 vertices

diam.girthAut.connect.LCF names picture
13244[2]4 K4 GraphY4W6EE2118918.jpg

6 vertices

diam.girthAut.connect.LCF names picture
23123[2, 3, −2]2prism graph Y3 GraphY6W21EE2507449.jpg
24724[3]6 K3, 3, utility graph GraphY6W21EE2413532.jpg

8 vertices

diam.girthAut.connect.LCF names pictures
33162[2, 2, −2, −2]2 Y8W50EE3373868.jpg
3343[4, −2, 4, 2]2 or [2, 3, −2, 3; –] Y8W46EE3097135.jpg
23123[2, 4, −2, 3, 3, 4, −3, −3] Y8W44EE3003607.jpg
34484[−3, 3]4 cube graph GraphY8W48EE2939381.jpg
24164[4]8 or [4, −3, 3, 4]2 Wagner graph GraphY8W44EE2909522.jpg

10 vertices

diam.girthAut.connect.LCF names pictures
53321Edge 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
GraphY10W111EE4260094.jpg
4342[4, 2, 3, −2, −4, −3, 2, 2, −2, −2] GraphY10W91EE3941746.jpg
3382[2, −3, −2, 2, 2; –] GraphY10W90EE4039508.jpg
33162[−2, −2, 3, 3, 3; –] Y10W90EE3890980.jpg
43162[2, 2, −2, −2, 5]2 GraphY10W93EE4069426.jpg
3323[2, 3, −2, 5, −3]2
[3, −2, 4, −3, 4, 2, −4, −2, −4, 2]
GraphY10W85EE3744960.jpg
33123[2, −4, −2, 5, 2, 4, −2, 4, 5, −4] GraphY10W81EE3677120.jpg
3323[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]
GraphY10W82EE3600347.jpg
3343[−4, 3, 3, 5, −3, −3, 4, 2, 5, −2]
[3, −4, −3, −3, 2, 3, −2, 4, −3, 3]
GraphY10W85EE3668162.jpg
3363[3, −3, 5, −3, 2, 4, −2, 5, 3, −4] Y10W84EE3625442.jpg
3343[2, 3, −2, 3, −3; –]
[−4, 4, 2, 5, −2]2
Y10W87EE3769671.jpg
3363[5, −2, 2, 4, −2, 5, 2, −4, −2, 2] GraphY10W84EE3801880.jpg
3383[2, 5, −2, 5, 5]2
[2, 4, −2, 3, 4; –]
GraphY10W83EE3701785.jpg
34483[5, −3, −3, 3, 3]2 GraphY10W85EE3583204.jpg
3484[5, −4, 4, −4, 4]2
[5, −4, −3, 3, 4, 5, −3, 4, −4, 3]
GraphY10W79EE3472233.jpg
3444[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]
GraphY10W81EE3497449.jpg
34204[5]10
[−3, 3]5
[5, 5, −3, 5, 3]2
GraphY10W85EE3540679.jpg
34204[−4, 4, −3, 5, 3]2 Pentagonal prism, G5, 2 GraphY10W85EE3547908.jpg
251204 Petersen graph Y10W75EE3421829.jpg

12 vertices

diam.girthAut.connect.LCF names picture
63161Edge 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
GraphY12W184EE4984524.jpg
53161Edge 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
GraphY12W172EE4845339.jpg
6381Edge 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
GraphY12W178EE4778916.jpg
53321Edge 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
GraphY12W172EE4710611.jpg
5342[3, −2, −4, −3, 4, 2]2
[4, 2, 3, −2, −4, −3; –]
GraphY12W150EE4512486.jpg
4382[3, −2, −4, −3, 3, 3, 3, −3, −3, −3, 4, 2] GraphY12W149EE4463116.jpg
4342[4, 2, 3, −2, −4, −3, 2, 3, −2, 2, −3, −2] GraphY12W149EE4612066.jpg
44642[3, 3, 3, −3, −3, −3]2 GraphY12W152EE4414446.jpg
43162[2, −3, −2, 3, 3, 3; –] GraphY12W152EE4563732.jpg
43162[2, 3, −2, 2, −3, −2]2 GraphY12W152EE4713249.jpg
4322[−2, 3, 6, 3, −3, 2, −3, −2, 6, 2, 2, −2]
[4, 2, −4, −2, −4, 6, 2, 2, −2, −2, 4, 6]
GraphY12W149EE4589062.jpg
4382[6, 3, 3, 4, −3, −3, 6, −4, 2, 2, −2, −2] GraphY12W146EE4494265.jpg
5342[4, 2, 3, −2, −4, −3, 5, 2, 2, −2, −2, −5] GraphY12W154EE4630261.jpg
43162[−3, −3, −3, 5, 2, 2; –] GraphY12W153EE4576519.jpg
4382[2, −3, −2, 5, 2, 2; –] GraphY12W153EE4722986.jpg
4342[2, 4, −2, 3, −5, −4, −3, 2, 2, −2, −2, 5]
[5, 2, −4, −2, −5, −5, 2, 2, −2, −2, 4, 5]
GraphY12W143EE4558501.jpg
4342[−2, −2, 4, 4, 4, 4; –]
[3, −4, −4, −3, 2, 2; –]
[5, 3, 4, 4, −3, −5, −4, −4, 2, 2, −2, −2]
GraphY12W145EE4490052.jpg
4322[4, −2, 4, 2, −4, −2, −4, 2, 2, −2, −2, 2]
[5, −2, 2, 3, −2, −5, −3, 2, 2, −2, −2, 2]
GraphY12W148EE4695537.jpg
53162[2, 2, −2, −2, −5, 5]2 GraphY12W160EE4772073.jpg
4382[−2, −2, 4, 5, 3, 4; –] GraphY12W141EE4463910.jpg
4342[5, 2, −3, −2, 6, −5, 2, 2, −2, −2, 6, 3] GraphY12W146EE4563214.jpg
4382[4, −2, 3, 3, −4, −3, −3, 2, 2, −2, −2, 2] GraphY12W150EE4628096.jpg
4382[−2, −2, 5, 3, 5, 3; –]
[−2, −2, 3, 5, 3, −3; –]
GraphY12W147EE4505416.jpg
53322[2, 2, −2, −2, 6, 6]2 GraphY12W158EE4735563.jpg
4382[−3, 2, −3, −2, 2, 2; –] GraphY12W152EE4739504.jpg
4382[−2, −2, 5, 2, 5, −2; –] GraphY12W143EE4651523.jpg
4382[6, −2, 2, 2, −2, −2, 6, 2, 2, −2, −2, 2] GraphY12W153EE4840271.jpg
43482[−2, −2, 2, 2]3 GraphY12W162EE5042874.jpg
4343[2, 3, −2, 3, −3, 3; –]
[−4, 6, 4, 2, 6, −2]2
GraphY12W144EE4466589.jpg
4343[−4, 6, 3, 3, 6, −3, −3, 6, 4, 2, 6, −2]
[−2, 3, −3, 4, −3, 3, 3, −4, −3, −3, 2, 3]
GraphY12W140EE4361888.jpg
4313[−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]
GraphY12W142EE4432053.jpg
3343[−5, −5, 4, 2, 6, −2, −4, 5, 5, 2, 6, −2]
[4, −2, 3, 4, −4, −3, 3, −4, 2, −3, −2, 2]
GraphY12W136EE4401162.jpg
3383[−5, −5, 3, 3, 6, −3, −3, 5, 5, 2, 6, −2]
[2, 4, −2, 3, 5, −4, −3, 3, 3, −5, −3, −3]
GraphY12W136EE4311500.jpg
4323[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]
GraphY12W138EE4387324.jpg
4323[−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]
GraphY12W139EE4330141.jpg
4323[2, 3, −2, 4, −3, 6, 3, −4, 2, −3, −2, 6]
[−4, 5, −4, 2, 3, −2, −5, −3, 4, 2, 4, −2]
GraphY12W139EE4405952.jpg
4313[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]
GraphY12W136EE4291096.jpg
3443[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]
GraphY12W135EE4208576.jpg
34163[3, 3, 4, −3, −3, 4; –]
[3, 6, −3, −3, 6, 3]2
GraphY12W136EE4258760.jpg
4313[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 GraphY12W139EE4495991.jpg
4343[−2, 6, 2, −4, −2, 3, 3, 6, −3, −3, 2, 4]
[−2, 2, 5, −2, −5, 3, 3, −5, −3, −3, 2, 5]
GraphY12W139EE4412975.jpg
4323[2, 4, −2, 6, 2, −4, −2, 4, 2, 6, −2, −4]
[2, 5, −2, 2, 6, −2, −5, 2, 3, −2, 6, −3]
GraphY12W139EE4487532.jpg
4323[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]
GraphY12W140EE4312097.jpg
44123[3, −3, 5, −3, −5, 3, 3, −5, −3, −3, 3, 5] GraphY12W142EE4231141.jpg
4323[4, 2, 4, −2, −4, 4; –]
[3, 5, 2, −3, −2, 5; –]
[6, 2, −3, −2, 6, 3]2
GraphY12W141EE4400528.jpg
4323[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]
GraphY12W137EE4272638.jpg
3313[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]
GraphY12W133EE4237675.jpg
3323[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]
GraphY12W131EE4219745.jpg
4343[2, 4, −2, −5, 5]2
[−5, 2, 4, −2, 6, 3, −4, 5, −3, 2, 6, −2]
GraphY12W135EE4348153.jpg
4323[−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]
GraphY12W137EE4285630.jpg
3323[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]
GraphY12W134EE4348061.jpg
3323[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]
GraphY12W131EE4211275.jpg
4323[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]
GraphY12W133EE4316541.jpg
4313[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]
GraphY12W134EE4232276.jpg
3323[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]
GraphY12W135EE4267156.jpg
4343[2, 4, −2, 5, 3, −4; –]
[5, −3, 2, 5, −2, −5; –]
[3, 6, 3, −3, 6, −3, 2, 6, −2, 2, 6, −2]
GraphY12W138EE4374286.jpg
4323[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]
GraphY12W136EE4361258.jpg
3323[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]
GraphY12W134EE4334214.jpg
33123[−5, 3, 3, 5, −3, −3, 4, 5, −5, 2, −4, −2] GraphY12W134EE4279794.jpg
3323[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]
GraphY12W131EE4205815.jpg
43123[−4, 5, 2, −4, −2, 5; –] Dürer graph Y12W135EE4325057.jpg
3343[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]
GraphY12W136EE4360342.jpg
3343[6, −2, 6, 4, 6, 4, 6, −4, 6, −4, 6, 2]
[5, 6, −3, 3, 5, −5, −3, 6, 2, −5, −2, 3]
GraphY12W133EE4223739.jpg
3343[4, −2, 4, 6, −4, 2, −4, −2, 2, 6, −2, 2]
[5, −2, 5, 6, 2, −5, −2, −5, 2, 6, −2, 2]
GraphY12W135EE4443130.jpg
33243[6, −2, 2]4 Truncated tetrahedron GraphY12W138EE4576235.jpg
33123 Tietze's Graph Y12W129EE4170908.jpg
33363[2, 6, −2, 6]3 GraphY12W135EE4426200.jpg
44244[−3, 3]6
[3, −5, 5, −3, −5, 5]2
G6, 2, Y6 GraphY12W144EE4227027.jpg
3444[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]
GraphY12W134EE4169366.jpg
3484[−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
GraphY12W132EE4128733.jpg
3424[−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]
GraphY12W132EE4134305.jpg
3424[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]
Y12W130EE4102128.jpg
3484[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 Y12W134EE4166461.jpg
34164[6, −5, 5]4
[3, 4, −4, −3, 4, −4]2
GraphY12W130EE4116056.jpg
3424[−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]
GraphY12W132EE4128805.jpg
3444[4, −3, 4, 5, −4, 4; –]
[4, 5, −5, 5, −4, 5; –]
[−5, −3, 4, 5, −5, 4; –]
GraphY12W128EE4061559.jpg
3424[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]
GraphY12W130EE4093704.jpg
3444[6, −3, 5, 6, −5, 3, 6, −5, −3, 6, 3, 5]
[3, −4, 5, −3, 4, 6, 4, −5, −4, 4, −4, 6]
GraphY12W130EE4099207.jpg
3484[5, 6, 6, −4, 5, −5, 4, 6, 6, −5, −4, 4] GraphY12W128EE4072559.jpg
35164[4, −5, 4, −5, −4, 4; –] GraphY12W126EE4034891.jpg
3444[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]
GraphY12W134EE4155455.jpg
4484[3, 5, 5, −3, 5, 5; –]
[−3, 5, −3, 5, 3, 5; –]
[5, −3, 5, 5, 5, −5; –]
Y12W136EE4145861.jpg
34484[5, −5, −3, 3]3
[−5, 5]6
Franklin graph Y12W132EE4105212.jpg
34244[6]12
[6, 6, −3, −5, 5, 3]2
Y12W138EE4225614.jpg
35184[6, −5, −4, 4, −5, 4, 6, −4, 5, −4, 4, 5] GraphY12W126EE4040388.jpg

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 2n vertices defines a class of quantum mechanical 3n-j 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 j, the vertices are labelled with a handedness representing the order of the three j (of the three edges) in the 3-jm symbol, and the graph represents a sum over the product of all these numbers assigned to the vertices.

There are 1 (6-j), 1 (9-j), 2 (12-j), 5 (15-j), 18 (18-j), 84 (21-j), 607 (24-j), 6100 (27-j), 78824 (30-j), 1195280 (33-j), 20297600 (36-j), 376940415 (39-j) etc. of these (sequence A175847 in the 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 (sequence A111916 in the OEIS ).

See also

References