The algorithmic steps leading to unique SMILES generation is discussed below:
(I) Initializing Rank of the Graph Vertices—The rank initialization of the vertices is achieved using combined invariants which in turns are combinations of
several individual atomic invariants. A total of 6 such atomic (node) invariants
in the order of their priority are produced below:
(i) Number of connections
(ii) Number of non-hydrogen bonds
(iii) Atomic Number
(iv) Sign of Charge
(v) Absolute Charge
(vi) Number of attached hydrogen atoms.
It may be noted that the number of invariants can be varied based on the
desired distinguishing properties [27]. The combined invariant will be the
number obtained by successively concatenating the individual invariants such
that higher priority invariants are to the left of lower priority invariants in the
decimal system. For example, a methyl carbon (CH 3 ) in a molecule will have
the individual invariants 1, 01, 06, 0, 0, 3 listed in the order of their priority
while the combined invariant will be 10106003. The distinct combined
invariants in the molecule are then sorted and mapped to their position in
increasing order, hereafter referred to as consecutive ranks. The mapped
position becomes the initial ranks of the atoms. For example, in case of
n-Pentane, i.e. (C 1 –C 2 –C 3 –C 4 –C 5 ), where the subscripts denote the vertex
labels, the combined invariants are 10106003–20206002–20206002–
20206002–10106003 while the initial rank is 1–2–2–2–1.
(II) Extended Connectivity through an Unambiguous Function using Product of
Primes—The initial rank will not be able to identify the vertex symmetries. In
the case of n-Pentane, vertices 2 and 4 are equivalent in terms of vertex
symmetry while vertex 3 is not equivalent to them but is still initially ranked the
same. To resolve this, rank of an atom is replaced by the result of an operation
of a given function over its neighbours. This result is a representation of
extended connectivity. A simple and elegant function is the product of primes
corresponding to the rank of the neighbouring atoms. For example, in the
n-Pentane case discussed so far, the updated rank of vertex 2 will now be prime
number corresponding to rank of vertex 1 multiplied by prime number corresponding to the rank of vertex 3, i.e. 1st prime  2nd prime ¼ 2  3 ¼ 6, as
ranks of vertices 1 and 3 are 1 and 2, respectively. Similarly, the rank of vertex
3 will be updated to 2nd prime  2nd prime ¼ 9. Subsequently, the revised
rank will become 3–6–9–6–3 which can be remapped to consecutive ranks 1–
2–3–2–1. This procedure of rank update is repeated and is stopped when the
updated rank for each atom of the molecule remains same as the previous rank.
It may be noted that in the end, the connectivity symmetrical vertices will be
ranked the same.
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(I) Initializing Rank of the Graph Vertices—The rank initialization of the vertices is achieved using combined invariants which in turns are combinations of
several individual atomic invariants. A total of 6 such atomic (node) invariants
in the order of their priority are produced below:
(i) Number of connections
(ii) Number of non-hydrogen bonds
(iii) Atomic Number
(iv) Sign of Charge
(v) Absolute Charge
(vi) Number of attached hydrogen atoms.
It may be noted that the number of invariants can be varied based on the
desired distinguishing properties [27]. The combined invariant will be the
number obtained by successively concatenating the individual invariants such
that higher priority invariants are to the left of lower priority invariants in the
decimal system. For example, a methyl carbon (CH 3 ) in a molecule will have
the individual invariants 1, 01, 06, 0, 0, 3 listed in the order of their priority
while the combined invariant will be 10106003. The distinct combined
invariants in the molecule are then sorted and mapped to their position in
increasing order, hereafter referred to as consecutive ranks. The mapped
position becomes the initial ranks of the atoms. For example, in case of
n-Pentane, i.e. (C 1 –C 2 –C 3 –C 4 –C 5 ), where the subscripts denote the vertex
labels, the combined invariants are 10106003–20206002–20206002–
20206002–10106003 while the initial rank is 1–2–2–2–1.
(II) Extended Connectivity through an Unambiguous Function using Product of
Primes—The initial rank will not be able to identify the vertex symmetries. In
the case of n-Pentane, vertices 2 and 4 are equivalent in terms of vertex
symmetry while vertex 3 is not equivalent to them but is still initially ranked the
same. To resolve this, rank of an atom is replaced by the result of an operation
of a given function over its neighbours. This result is a representation of
extended connectivity. A simple and elegant function is the product of primes
corresponding to the rank of the neighbouring atoms. For example, in the
n-Pentane case discussed so far, the updated rank of vertex 2 will now be prime
number corresponding to rank of vertex 1 multiplied by prime number corresponding to the rank of vertex 3, i.e. 1st prime  2nd prime ¼ 2  3 ¼ 6, as
ranks of vertices 1 and 3 are 1 and 2, respectively. Similarly, the rank of vertex
3 will be updated to 2nd prime  2nd prime ¼ 9. Subsequently, the revised
rank will become 3–6–9–6–3 which can be remapped to consecutive ranks 1–
2–3–2–1. This procedure of rank update is repeated and is stopped when the
updated rank for each atom of the molecule remains same as the previous rank.
It may be noted that in the end, the connectivity symmetrical vertices will be
ranked the same.
86
Md.I. H. Rizvi et al.