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

Connectivity

The count of simple cubic graphs is 1, 2, 5, 19,... for 2, 4, 6, 8,... vertices . A classification according to edge connectivity

Connectivity (graph theory)

In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements which need to be removed to disconnect the remaining nodes from each other. It is closely related to the theory of network flow problems...

 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

Angular momentum coupling

In quantum mechanics, the procedure of constructing eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling. For instance, the orbit and spin of a single particle can interact through spin-orbit interaction, in which case the...

 (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 first column of the tables below.

LCF notation

The LCF notation is a notation by Joshua Lederberg

Joshua Lederberg

Joshua Lederberg ForMemRS was an American molecular biologist known for his work in microbial genetics, artificial intelligence, and the United States space program. He was just 33 years old when he won the 1958 Nobel Prize in Physiology or Medicine for discovering that bacteria can mate and...

, Coxeter

Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, was a British-born Canadian geometer. Coxeter is regarded as one of the great geometers of the 20th century. He was born in London but spent most of his life in Canada....

 and Frucht

Robert Frucht

Robert Wertheimer Frucht was a German-Chilean mathematician; his research specialty was graph theory and the symmetries of graphs....

, for the representation of cubic graph

Cubic graph

In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs....

s that are Hamiltonian

Hamiltonian path

In the mathematical field of graph theory, a Hamiltonian path is a path in an undirected graph that visits each vertex exactly once. A Hamiltonian cycle is a cycle in an undirected graph that visits each vertex exactly once and also returns to the starting vertex...

.

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

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

Distance (graph theory)

In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path connecting them. This is also known as the geodesic distance...

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

1. If a , replace it by ;
2. avoid repetition of a sequence of 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 ";-". The LCF notation , for example, and would at that stage be condensed to .

4 nodes

connectivity	LCF	names Gallery of named graphs Some of the finite structures considered in graph theory have names, sometimes inspired by the graph's topology, and sometimes after their discoverer. A famous example is the Petersen graph, a concrete graph on 10 vertices that appears as a minimal example or counterexample in many different...
4	[2]^4	K4 Complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.-Properties:...

6 nodes

connectivity	LCF	names Gallery of named graphs Some of the finite structures considered in graph theory have names, sometimes inspired by the graph's topology, and sometimes after their discoverer. A famous example is the Petersen graph, a concrete graph on 10 vertices that appears as a minimal example or counterexample in many different...
3	[2,3,-2]^2	Wn,2, prism graph Y3
4	[3]^6	K3,3

8 nodes

connectivity	LCF	names Gallery of named graphs Some of the finite structures considered in graph theory have names, sometimes inspired by the graph's topology, and sometimes after their discoverer. A famous example is the Petersen graph, a concrete graph on 10 vertices that appears as a minimal example or counterexample in many different...
2	[2,2,-2,-2]^2
3	[4,-2,4,2]^2 or [2,3,-2,3;-]
3	[2,4,-2,3,3,4,-3,-3]
4	[-3,3]^4	cubical graph
4	[4]^8 or [4,-3,3,4]^2	Wagner graph Wagner graph In the mathematical field of graph theory, the Wagner graph is a 3-regular graph with 8 vertices and 12 edges. It is the 8-vertex Möbius ladder graph.-Properties:...

10 nodes

connectivity	LCF	names Gallery of named graphs Some of the finite structures considered in graph theory have names, sometimes inspired by the graph's topology, and sometimes after their discoverer. A famous example is the Petersen graph, a concrete graph on 10 vertices that appears as a minimal example or counterexample in many different...
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
2	[4,2,3,-2,-4,-3,2,2,-2,-2]
2	[3,3,3,-3,-3,-3,2,2,-2,-2]
2	[2,3,-2,2,-3,-2,2,2,-2,-2]
2	[2,2,-2,-2,5]^2
3	[2,3,-2,5,-3]^2
3	[2,-4,-2,5,2,4,-2,4,5,-4]
3	[5,3,5,-4,-3,5,2,5,-2,4]
3	[-4,3,3,5,-3,-3,4,2,5,-2]
3	[3,-3,5,-3,2,4,-2,5,3,-4]
3	[-4,4,2,5,-2]^2
3	[5,-2,2,4,-2,5,2,-4,-2,2]
3	[2,5,-2,5,5]^2
3	[5,-3,-3,3,3]^2
4	[5,-4,4,-4,4]^2
4	[5,-4,4,5,5]^2
4	[5]^10
4	[-4,4,-3,5,3]^2
4		Petersen graph Petersen graph In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named for Julius Petersen, who in 1898 constructed it...

12 nodes

connectivity	LCF	names Gallery of named graphs Some of the finite structures considered in graph theory have names, sometimes inspired by the graph's topology, and sometimes after their discoverer. A famous example is the Petersen graph, a concrete graph on 10 vertices that appears as a minimal example or counterexample in many different...
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
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
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
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
2	[4,2,3,-2,-4,-3]^2
2	[4,2,3,-2,-4,-3,3,3,3,-3,-3,-3]
2	[4,2,3,-2,-4,-3,2,3,-2,2,-3,-2]
2	[3,3,3,-3,-3,-3]^2
2	[3,3,3,-3,-3,-3,2,3,-2,2,-3,-2]
2	[2,3,-2,2,-3,-2]^2
2	[6,2,3,-2,3,-3,6,-3,2,2,-2,-2]
2	[6,3,3,4,-3,-3,6,-4,2,2,-2,-2]
2	[4,2,3,-2,-4,-3,5,2,2,-2,-2,-5]
2	[3,3,3,-3,-3,-3,5,2,2,-2,-2,-5]
2	[2,3,-2,2,-3,-2,5,2,2,-2,-2,-5]
2	[5,2,4,-2,3,-5,-4,-3,2,2,-2,-2]
2	[3,4,4,-3,3,-4,-4,-3,2,2,-2,-2]
2	[2,4,-2,4,2,-4,-2,-4,2,2,-2,-2]
2	[2,2,-2,-2,-5,5]^2
2	[4,5,3,4,-4,-3,-5,-4,2,2,-2,-2]
2	[5,2,-3,-2,6,-5,2,2,-2,-2,6,3]
2	[2,4,-2,3,3,-4,-3,-3,2,2,-2,-2]
2	[2,2,-2,-2,5,3,5,3,-3,-5,-3,-5]
2	[2,2,-2,-2,6,6]^2
2	[6,-2,2,2,-2,-2,6,2,2,-2,-2,2]
2	[-2,3,-3,2,-3,-2,2,2,-2,-2,2,3]
2	[2,2,-2,-2,5,2,5,-2,2,-5,-2,-5]
2	[-2,-2,2,2]^3
3	[2,3,-2,3,-3,3,-3;-]
3	[2,3,-2,3,-3,4,-3,3,3,-4,-3,-3]
3	[2,3,-2,3,-3,4,-3,4,2,-4,-2,-4]
3	[2,4,-2,3,4,-4,-3,3,-4,2,-3,-2]
3	[2,4,-2,3,5,-4,-3,3,3,-5,-3,-3]
3	[2,4,-2,3,6,-4,-3,2,3,-2,6,-3]
3	[-5,2,-3,-2,6,3,3,5,-3,-3,6,3]
3	[-4,2,-3,-2,6,2,3,-2,4,-3,6,3]
3	[3,4,4,-3,4,-4,-4,3,-4,2,-3,-2]
3	[3,4,4,-3,5,-4,-4,3,3,-5,-3,-3]
3	[6,3,3,6,-3,-3]^2
3	[2,5,-2,2,4,-2,-5,3,-4,2,-3,-2]	Frucht graph Frucht graph In the mathematical field of graph theory, the Frucht graph is a 3-regular graph with 12 vertices, 18 edges, and no nontrivial symmetries. It was first described by Robert Frucht in 1939....
3	[2,5,-2,2,5,-2,-5,3,3,-5,-3,-3]
3	[2,5,-2,2,6,-2,-5,2,3,-2,6,-3]
3	[3,5,3,-3,4,-3,-5,3,-4,2,-3,-2]
3	[3,5,3,-3,5,-3,-5,3,3,-5,-3,-3]
3	[3,6,4,-3,6,3,-4,6,-3,2,6,-2]
3	[-4,4,2,-4,-2,-4;-]
3	[-4,4,-3,3,6,-4,-3,2,4,-2,6,3]
3	[-5,2,6,-2,5,6,4,5,6,-5,-4,6]
3	[-5,2,4,-2,6,3,-4,5,-3,2,6,-2]
3	[-4,2,3,-2,5,-3,5,3,4,-5,-3,-5]
3	[2,6,-2,3,4,5,-3,6,-4,2,-5,-2]
3	[2,-5,-2,4,5,6,4,-4,5,-5,-4,6]
3	[2,5,-2,4,-5,4,-5,-4,2,-4,-2,5]
3	[3,6,4,-3,4,5,-4,6,-4,2,-5,-2]
3	[3,5,5,-3,-5,4,-5,-5,2,-4,-2,5]
3	[-5,-2,3,-5,5,-3,2,5,-2,-5,5,2]
3	[-4,2,3,-2,6,-3,5,2,4,-2,6,-5]
3	[4,6,6,2,-4,-2]^2
3	[-5,3,3,5,-3,-3,4,5,-5,2,-4,-2]
3	[5,6,-4,3,4,-5,-3,6,-4,2,4,-2]
3	[-4,5,2,-4,-2,5;-]	Dürer graph Dürer graph In the mathematical field of graph theory, the Dürer graph is an undirected graph with 12 vertices and 18 edges. It is named after Albrecht Dürer, whose 1514 engraving Melencolia I includes a depiction of Dürer's solid, a convex polyhedron having the Dürer graph as its skeleton...
3	[2,5,-2,5,3,5,-5,-3,-5,2,-5,-2]
3	[2,-5,-2,3,5,6,-3,3,5,-5,-3,6]
3	[2,6,-2,5,2,5,-2,6,-5,2,-5,-2]
3	[6,-2,2]^4
3		Tietze's Graph Tietze's graph In the mathematical field of graph theory, the Tietze's graph is an undirected cubic graph with 12 vertices and 18 edges, formed by applying a Y-Δ transform to the Petersen graph and thereby replacing one of its vertices by a triangle....
3	[2,6,-2,6]^3
4	[-3,3]^6
4	[3,6,6,-3,6,6]^2
4	[-4,4,-3,3,5,-4,-3,3,4,-5,-3,3]
4	[4,-5,-5,4,-4,6,4,-4,5,5,-4,6]
4	[4,-5,5,6,-4,5,5,-5,5,6,-5,-5]
4	[4,-4,6]^4
4	[6,-5,5]^4
4	[3,-5,5,-3,5,6,4,-5,5,-5,-4,6]
4	[5,-5,-3,4,5,-5,4,-4,5,-5,-4,3]
4	[-5,3,6,-5,-3,4,5,5,6,-4,5,-5]
4	[-3,6,4,-4,4,5,-4,6,-4,3,-5,4]
4	[5,6,6,-4,5,-5,4,6,6,-5,-4,4]
4	[-4,5,-4,4,-5,4,-5,-4,4,-4,4,5]
4	[-3,4,-3,5,3,-4,4,-3,-5,3,-4,3]
4	[-5,3,-3,5,-3,5,3,5,-5,-3,-5,3]
4	[5,-5,-3,3]^3	Franklin graph Franklin graph In the mathematical field of graph theory, the Franklin graph a 3-regular graph with 12 vertices and 18 edges.The Franklin graph is named after Philip Franklin, who disproved the Heawood conjecture on the number of colors needed when a two-dimensional surface is partitioned into cells by a graph...
4	[3,6,6,-3,-5,5]^2
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 . 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 nodes 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 , the vertices are labelled with a handedness representing the order of the three (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 (6j), 1 (9j), 2 (12j), 5 (15j), 18 (18j), 84 (21j), 607 (24j), 6100 (27j), 78824 (30j), 1195280 (33j), 20297600 (36j), 376940415 (39j) etc. of these .

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 .