Is there a mathematical generalization for the number of isomers (structural and configurational) of a given compound?

Question

The original question was:

How do I find the number of structural and configurational isomers of a given compound? is there any formula?

For a specific example, how would I find the number of structural and configurational isomers of a bromo compound $\ce{C5H9Br}$ formed by addition of $\ce{HBr}$ to 2-pentyne

Answer 1

1. Generally speaking, graph-theoretical enumeration aims at counting chemical compounds as graphs (2D structures). In other words, it is concerned with constitutional (or structural) isomers.

   • The most famous method for combinatorial enumeration of graphs is based on Polya's theorem: G. Polya and R. C. Read,
     Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds (Springer, 1987).
   • G. Polya, R. E. Tarjan, and D. R. Woods, Notes on Introductory Combinatorics (Birkhauser, 1983) Chapter 6.
   • An introduction to the practical uses of Polya's theorem from a chemical viewpoint has appeared: O. E. Polansky, Polya's method for the Enumeration of Isomers (MATCH Commun. Math. Comput. Chem. 1, 11--31 (1975)). free access.
   • Many examples of chemical application of Polya's theorem have been described in the following book: A. T. Balaban (Ed.), Chemical Applications of Graph Theory (Academic Press, 1976).

2. To accomplish enumeration of 3D structures (concerned with configurational isomers or stereoisomers), in contrast, it is necessary to take chirality/achirality of substituents into explicit consideration.

   • Symmetry-itemized enumeration of chemical compounds as 3D structures can be accomplished by adopting Fujita's unit-subduced-cycle-index (USCI) approach: S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry (Springer 1991).
   • Gross enumeration without symmetry-itemization can be accomplished by adopting Fujita's proligand method: S. Fujita, Combinatorial Enumeration of Graphs, Three-Dimensional Structures, and Chemical Compounds (University of Kragujevac, Kragujevac, 2013).
   • For the enumeration of alkanes ($\ce{C_nH_{2n+2}}$) and monosubstituted alkane ($\ce{C_nH_{2n+1}X}$) as 3D structures, I would like to introduce an account article, which is freely available: S. Fujita, Numbers of Alkanes and Monosubstituted Alkanes. A Long-Standing Interdisciplinary Problem over 130 Years (Bull. Chem. Soc. Japan, 83, 1--18 (2010)). The following article is also available freely: S. Fujita, Numbers of Achiral and Chiral Monosubstituted Alkanes Having a Given Carbon Content and Given Numbers of Asymmetric and Pseudoasymmetric Centers (Bull. Chem. Soc. Jpn., 81, 193--219 (2008)).

3. To grasp perspectives of the theoretical foundations of stereochemistry and of the state-of-the-art chemical enumeration, see also the following recent book: S. Fujita, Mathematical Stereochemistry (De Gruyter, Berlin, 2015)

Answer 2

As I said in my comment, the mathematical way to solve this kind of problem relies on Graph Theory. This article (Applications of Graph Theory in Chemistry, preprint) reviews the problem and further more.

If you google 'isomer' and 'graph theory' you will find anything you want about this exciting topic. But for generic question, there is no generic answer. It is about building all possible graphs or find out inherent properties inside them. You cannot answer to this question unless it is well-defined and properly bounded.

If you have such question, please rewords it and post it on Mathematical Exchange. This should give you fine grained answer. If you just wanted to understand how it works, welcome to the fabulous world of Graph Theory, you will have to learn a bit to answer it by yourself.

Answer 3

A useful tool for finding answers to specific questions along these lines is the Online Encyclopedia of Integer Sequences. It catalogues many sequences and provides a search engine, so if you compute by hand the number of isomers for five or six small cases then it may give you some suggestions for continuations and references.

For example, if you're counting non-stereoisomeric monosubstituted alkanes then you might get counts 1, 2, 3, 5, 8, 14; and one of the (at present) 66 search results is "Number of monosubstituted alkanes C(n-1)H(2n-1)-X with n-1 carbon atoms that are not stereoisomers" with a table with 3000 entries and references to papers by Blair and Henze; and Polyá.