The connected 3-regular 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,.... 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, 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.
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 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 joined by a chord with, The vector of the integers is a suitable, although not unique, representation of the cubicHamiltonian graph. This is augmented by two additional rules:
If a, replace it by ;
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, and the redundant part is then replaced by a semicolon and a dash "; –". The LCF notation, for example, and would at that stage be condensed to.
Table
4 vertices
diam.
girth
Aut.
connect.
LCF
names
picture
1
3
24
4
4
K4
6 vertices
8 vertices
10 vertices
12 vertices
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.
Each 4-connected simple cubic graph on vertices defines a class of quantum mechanical j symbols. Roughly speaking, each vertex represents a 3-jm symbol, the graph is converted to a digraph by assigning signs to the angular momentumquantum numbers, the vertices are labelled with a handedness representing the order of the three in the 3jm symbol, and the graph represents a sum over the product of all these numbers assigned to the vertices. There are 1, 1, 2, 5, 18, 84, 607, 6100, 78824, 1195280, 20297600, 376940415 etc. of these. If they are equivalent to certain vertex-induced binary trees, they are representations of recoupling coefficients, and are then also known as Yutsis graphs.
