Bicycles

How many rings has the structure (a)?

(a)

diphenyl ether (functional class) 1,1′-oxydibenzene (multiplicative) phenoxybenzene (substitutive)

Why, there are two, you’ll say. Anybody can see that. And you’ll be right.

What about (b) then?

(b)

norbornane (trivial) bicyclo[2.2.1]heptane (von Baeyer)

You may recall that we defined trail as a walk in which all edges are distinct, and cycle as a trail in which the only repeated vertices are the first and last vertices. Let’s number the carbon atoms in (b) like this:

So we can see that there are three different cycles in (b): 1–2–3–4–5–6–1; 1–2–3–4–7–1; and 1–7–4–5–6–1.

On the other hand, chemists often look for the minimum number of cycles required to describe a ring system, which is equivalent to the minimum number of edges (i.e. bonds) we need to remove from the graph (structure) to turn it from cyclic to acyclic [1—3]. Think about it while you cut up the six-pack plastic rings. This number μ, variously known as circuit rank, cyclomatic number, Frèrejacque number or nullity, is defined as

μ = |E| − |V| + |C|

where |E| is the number of edges (size), |V| is number of vertices (order) and |C| is the number of connected components of the graph. For one-component graph, |C| = 1. Thus, for (a) μ = 14 − 13 + 1 = 2 and for (b) μ = 8 − 7 + 1 = 2. In other words, both structures are bicyclic.

There are several possible scenarios for mutual arrangement of two cycles which result in very different names.

1. Two rings have no atoms in common and are linked to each other via at least one atom as in (a).
2. Two rings have no atoms in common and are directly linked to each other via a single or double bond as in (c). Such structures are known as ring assemblies. If the ring components are identical, a variant of multiplicative nomenclature is used; otherwise, the structure is named substitutively.
3. Two rings have one atom in common as in (d). Such arrangements are named according to spiro nomenclature [4].
4. Two mancude rings have two atoms in common as in (e). Such systems are named using fused ring nomenclature [5].
5. Two saturated rings have two or more atoms in common as in (b). Such systems are named using von Baeyer nomenclature [6].

(c)	(d)	(e)

4,4′-bipyridine (ring assembly) vetispirane (trivial) 6,10-dimethyl-2-(propan-2-yl)spiro[4.5]decane (spiro + substitutive) 1-benzothiophene (fused ring) benzo[b]thiophene (fused ring)

Edward W. Godly calls the scenarios II—V “rings in close association” [7, p. 6] and the scenario I, surprise surprise, “rings not in close association” [7, p. 55].

References

1. Zamora, A. (1979) An algorithm for finding the smallest set of smallest rings. Journal of Chemical Information and Computer Sciences 16, 40—43.
2. Downs, G.M., Gillet, V.J., Holliday, J.D. and Lynch, M.F. (1989) Review of ring perception algorithms for chemical graphs. Journal of Chemical Information and Computer Sciences 29, 172—187.
3. García-Domenech, R., Gálvez, J., de Julián-Ortiz, J.V. and Pogliani, L. (2008) Some new trends in chemical graph theory. Chemical Reviews 108, 1127—1169.
4. Moss, G.P. (1999) Extension and revision of the nomenclature for spiro compounds (IUPAC Recommendations 1999). Pure and Applied Chemistry 71, 531-558.
5. Moss, G.P. (1998) Nomenclature of fused and bridged fused ring systems (IUPAC Recommendations 1998). Pure and Applied Chemistry 70, 143—216.
6. Moss, G.P. (1999) Extension and revision of the von Baeyer system for naming polycyclic compounds (including bicyclic compounds) (IUPAC Recommendations 1999). Pure and Applied Chemistry 71, 513—529.
7. Godly, E.W. Naming Organic Compounds: A Systematic Instruction Manual, 2^nd Ed. Ellis Horwood, Hemel Hempstead, 1995.