Fig. 6

Construction of non-isomorphic multigraphs with valency 4 in the proof of Prop. 48. The first three isomers are a cycle (with loops), a cycle with a single triple-bond indicating an “origin”, and a graph with an additional double bond. In the third graph, the asymmetric arrangement of the double and triple bonds implies an unambiguous ordering of the remaining vertices (numbered from 1 to n). Non-isomorphic graphs are obtained converting a pair of loops into a double bound. Since each vertex has at most one bond in addition to the cycle, the resulting graphs correspond to Kleitman’s “irreducible diagrams” [84]. If crossings of bonds are excluded, the resulting induced subgraphs with vertex set \(\{1,\dots ,n\}\) are isomorphic to RNA secondary structures on sequences of n monomers. The number \(S_n\) of secondary structures grows asymptotically \(\sim 2.6^n\) [85]