So, given my previous posts on what Faulon's signatures are, here is an explanation of how they are used in the structure enumeration algorithm that I am almost finished implementing.
The core test in this algorithm is for compatible bonds. Two atoms are only joined if : a) they have compatible target signatures and b) there are less than the target number of bonds already. A target signature here is just a signature that is set on the atom for it to match, like a pattern.
The first of these tests is illustrated here:
The equation to the right of the graph defines part of the condition for a compatible bond. The tau terms are just target signatures, as shown on the upper right. The tricky term is h-1στ(y)(z) which means 'a signature starting from the neighbour z of y in the subgraph defined by τ(y)'. This requires using a target signature (τ(y)) as if it was a subgraph - shown on the bottom left for the target b and the neighbour n.
The same process is repeated on the bottom right of the figure for b and m - which matches the height - 1 target signature for c. This should make sense, since the atoms labelled with b in the graph are attached to both a and c - so the signature for b must be compatible with both. It is easy to check that a and c are not compatible, and cannot therefore be bonded.