The poster is not really asking about classic cubic graphs. These are
are given in GraphData up to a reasonable size, and are enumerated athttp://www.research.att.com/~njas/sequences/A002851
Instead, the question is about connected nondirected trivalent multigraphs
Cubic multigraphs are less popular. Apparently, no-one in the world
has yet programmed their enumeration.
Without the connected requirement, there are 9 such graphs on 6 nodes.
If self-loops are allowed, the counts go higher.
Directed graphs and directed multigraphs also lack many
enumerations. Connectedness and self-loops change the enumeration.
Nine edges are needed. One programming method would be to start with
Select[GraphData["Connected", 6], Max[GraphData[#, "Degrees"]] < 4 &]
Graphs containing edges having degree 1-3 or 1-2 can be eliminated. Similarly,
graphs with degree 2 vertices surrounded by degree 3 vertices can be
For each remaining graph, add edge multiples to reach nine in all possible ways,
and select the cubic cases.
--Ed Pegg Jr