More chemical signature example

A neat and tidy example of generating a structure from atoms and signatures occurred to me : adenine. It is known that this molecule forms readily from HCN under conditions similar to those thought to exist on the early Earth.

Anyway, the point is that the structure is almost exactly 6 C-N units (ignoring hydrogens and multiple bonds). A C-N is very similar to a height-1 signature, and the height-1 signatures for adenine are shown in the figure below:

Each signature (a-e) is mapped to an atom in the original structure, and then a table is shown of the maximum number of compatible bonds possible between each pair of signatures. Note that the table is not symmetric, as the compatibility operation is not commutative. Also, notice that the diagonal is not all 0 - bonds can form between atoms that both have b as a target signature.

Although these target signatures are smaller than the diameter of the adenine molecular graph, they still seem to define it completely. However, they are not guaranteed to produce only adenine as a solution - there may be other molecules with the same signature of (3a + 2b+3c+d+e).

Edit: Hmmm. That's not actually adenine, is it? In fact, it doesn't seem to be in ChEBI either. I probably should check my examples before posting them to the world.

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The Gale-Ryser Theorem

This is a small aside. While reading a paper by Grüner, Laue, and Meringer on generation by homomorphism they mentioned the Gale-Ryser (GR) theorem. As it turns out, this is a nice small theorem closely related to the better known Erdős-Gallai (EG). So, GR says that given two partitions of an integer ( p and q) there exists a (0, 1) matrix A iff p* dominates q such that the row sum vector r(A) = p and the column sum vector c(A) = q . As with most mathematics, that's quite terse and full of terminology like 'dominates' : but it's relatively simple. Here is an example: The partitions p and q are at the top left, they both sum to 10. Next, p is transposed to get p* = [5, 4, 1] and this is compared to q at the bottom left. Since the sum at each point in the sequence is greater (or equal) for p* than q , the former dominates. One possible matrix is at the top left with the row sum vector to the right, and th...

1,2-dichlorocyclopropane and a spiran

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Adamantane, Diamantane, Twistane

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 . 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 accordin...

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