Table of simple cubic graphs

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

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

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. 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.)

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

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