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The (edge-)connected simple cubic graphs counted in A002851 can be classified as 1-connected (containing bridges), 2-connected, and 3-connected. The 3-connected graphs are subdivided in the cases (i) allowing a cut of 3 edges which leaves subgraphs with cycles and (ii) cyclically 4-connected and counted here. (Computed by adding the rows with k>=4 in Brouder's Table 1.)

Each of the non-isomorphic cyclically 4-connected graphs defines a 3n-j symbol of the vector coupling coefficients in the quantum mechanics of SO(3), one 6j symbol, one 9j symbol, two 12j symbols, five 15j symbols etc.

The Yutsis graphs (A111916) are a subset of the cyclically 4-connected graphs, which admit a representation as vertex-induced binary trees.

The value a(8)=576 is found in some earlier literature (e.g., Durr et al.) - R. J. Mathar, Sep 06 2011