After cubane, the thought occurred to look at other regular hydrocarbons. If only there was some sort of classification of chemicals that I could use look up similar structures. Oh wate, there is.

The carbons with three carbon neighbours (degree-3, in the simple graph) have signature (a) and the degree-2 carbons have signature (b). Atoms of one type are only connected to atoms of another - the graph is bipartite.

This structure is also 3-colored by signatures (if that's the right terminology...) but is not bipartite with respect to degree. Or, to put it more simply, there are carbons with two carbon neigbours connected together. It seems quite clear from looking at the colored structure on the right that atoms with the same color are similar in the context of the structure.

Anyway, adamantane is not as regular as cubane, but it is highly symmetrical, looking like three cyclohexanes fused together. The vertices fall into two different types when colored by signature:

The carbons with three carbon neighbours (degree-3, in the simple graph) have signature (a) and the degree-2 carbons have signature (b). Atoms of one type are only connected to atoms of another - the graph is bipartite.

Adamantane connects together to form diamondoids (or, rather, this class have adamantane as a repeating subunit). One such is diamantane, which is no longer bipartite when colored by signature:

It has three classes of vertex in the simple graph (a and b), as the set with degree-3 has been split in two. The tree for signature (c) is not shown. The graph is still bipartite according to the degree of each vertex.

This structure is also 3-colored by signatures (if that's the right terminology...) but is not bipartite with respect to degree. Or, to put it more simply, there are carbons with two carbon neigbours connected together. It seems quite clear from looking at the colored structure on the right that atoms with the same color are similar in the context of the structure.

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- David