- Source: 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, ... (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
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 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,
j
=
i
+
d
i
(
mod
p
)
,
2
≤
d
i
≤
p
−
2.
{\displaystyle j=i+d_{i}\quad ({\bmod {\,}}p),\quad 2\leq d_{i}\leq p-2.}
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:
If a di > p/2, replace it by di − p;
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
d
p
−
1
−
i
=
−
d
i
(
mod
p
)
,
i
=
0
,
1
,
…
p
/
2
−
1
{\displaystyle d_{p-1-i}=-d_{i}\quad ({\bmod {\,}}p),\quad i=0,1,\ldots p/2-1}
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
== 6 vertices
== 8 vertices
== 10 vertices
== 12 vertices
=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
3-jm symbol
6-j symbol
References
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Massot, J.-N.; El-Baz, E.; Lafoucriere, J. (1967). "A general graphical method for angular momentum". Reviews of Modern Physics. 39 (2): 288–305. Bibcode:1967RvMp...39..288M. doi:10.1103/RevModPhys.39.288.
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Danos, M.; Fano, U. (1998). "Graphical analysis of angular momentum for collision products". Physics Reports. 304 (4): 155–227. Bibcode:1998PhR...304..155D. doi:10.1016/S0370-1573(98)00020-9.
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