Cayley’s Enumeration on the Structural Isomers of Alkanes

Matthew P. Yeager

Also: topoisomers, isotopomers, nuclear isomers, spin isomers

Significance of Isomers

• Isomers contain identical molecular formulas, but differ in structural formulas, thereby generating various compounds of different physical properties • Important for many reasons: – Medicine / pharmacokinetics – Manufacturing impurities – Optical activity / polarizability – Biochemistry (amino acids, neurotransmitters, etc…)

Brief Review in Chemistry

• • – Parts of the atom: – Protons – Neutrons

}

Constitute the atomic nucleus Electrons  Found around the nucleus in a statistical “cloud” Electrons, e , surround the nucleus in various energy states, with the outermost state being occupied known as the valence shell.

• The valence number is how many electrons exist in the valence shell when in the ground state.

– s, p, d, and f orbitals may contain up to 2, 8, 18, and 14 e , respectively – An atom with a fully-occupied valence shell is less reactive (more stable), and thus more favorable

• • Molecules are derived from the spatial activities and interactions (bonding) between the valence electrons of different atoms.

There exist two principal types of bonds: 1.

Ionic - Dissimilar overall atomic charges generate attraction 2. Covalent - Composed of two electrons; favorable when it completes the valence states of participating atoms The tendency for atoms to covalently bond is contingent on whether the bond will achieve a full valence

Hydrocarbons and other derivatives

• Carbon naturally contains 4 valence e (exactly one-half of its maximum valence e ), thus making it highly versatile at bonding: Other chemical species behave similarly to satisfy their valence:

Genesis of Chemical Graph Theory

• Consider the molecular formula of a carbon-backbone compound: C 4 H 10 What is it’s molecular structure?

– Every carbon must bond to another carbon – Number of H = 2 x (Number of C) + 2 So, how about?

Butane

•

Genesis of Chemical Graph Theory

Butane (CH 3 CH 2 CH 2 CH 3 ) fits this formula, but what about: Isobutane (methylpropane) • Butane and isobutane are structural isomers; that is, they contain identical molecular formulas, but have different bonding schemes.

• Can we generalize about alkanes (C n H 2n+2 ) ?

Arthur Cayley (1875)

• Although chemists had been trying to count potential isomers for years, Cayley was the first to identify a correspondence between the structural isomers of alkanes / alkyl derivatives and planar graphs • Suppose : – Every nucleus is a vertex – Every single bond or lone pair is an edge  1,2 - dichloropropane pseudograph representation

Arthur Cayley (1875)

• Using chemical principals, Cayley made generalizations that would limit the enumeration alkane isomers (C n H 2n+2 ): – Alkanes are trees: • Only single bonds; no double / triple bonds, or lone pairs • Acyclic – Since hydrogen constitutes all the terminal vertices (leaves), they may be omitted for simplicity (

hydrogen-depleted graphs

) – The degree of all vertices (carbons) must satisfy the valence shell, and therefore cannot exceed 4

Alkane Isomer Enumeration

• So how many structural isomers exist for pentane (C 5 H 12 )?

– That is, how many unique trees are there with 5 nondistinct vertices?

pentane isopentane (methylbutane) neopentane (dimethylpropane)

Cayley’s Approach

• Cayley enumerated trees of valency ≤ 4 by counting the number of “centered” and “bicentered” H-depleted graphs for any quantity of nodes – Centered: a tree of diameter

2m

contains a unique node at the midpoint, called a

center

– Bicentered: a tree of diameter

2m+1

contains a unique pair of nodes called

bicenters

• This enumeration was performed by developing generating functions for both types of trees

• For centered trees, consider the half of the longest C C path of the alkane – Can designate a starting vertex (root) and height (

h

) – Every vertex is tertiary rooted (maximum of 3 edges not connected to the root) – Find

T h ,

the number of tertiary rooted trees with

n

nodes and height at most

h

– Find

C 2h

, the number of centered 4-valent trees with

n

and diametere

2h

nodes – Find

C n ,

the number of centered 4-valent trees with

n

nodes

• For bicentered trees, the approach is a little easier: – Let

B n

nodes be the total number of bicentered

k-valent

trees with

n

– We now want to find

B 2h+1,n

valent trees with

n

, the number of bicentered nodes and diameter

2h+1 k

– Using results from the previous algorithm makes for an easy determination of the generating function of

B(z)

Generating Functions

• After the lengthy derivation, we receive: for the centered trees, and for the bicentered trees

Generating Functions

• Expansion yields:

C(z) = z + z 3 … + z 4 + 2z 5 + 2z 6 + 6z 7 + 9z 8 + 20z 9 + 37z 10 + B(z) = z 2

1

+ z 4

2

+ z 5 + 3z 6 + 3z 7

5

+ 9z 8

6

+ 15z 9 + 38z 10 +

9

…

10

centered 1 0 1 1 2 2 6 9 20 37 bicentered total 0 1 1 1 0 1 1 2 1 3 3 5 3 9 9 18 15 35 38 75

C(z) + B(z) = z + z 2 + z 3 + 2z 4 + 5z 6 + 3z + 9z 7 5 + 18z 8 + 35z 9 + 75z 10 + …

11

86 73 159 Computational techniques must be applied due to the rapidly increasing isomers (consider

n=22

, with 2,278,658 alkane isomers!)

Side note: Annulenes

• Hydrocarbons with chemical formula C n H n • Examples:  1,3 - cyclobutadiene  benzene •

Hydrogen-depleted

representations are regular graphs of degree 3 (

cubic graphs

)

• Without any knowledge of chemistry, can we remark on the annulenes with odd

n

?

– Mathematically impossible by graph theory – The number of vertices of odd degree

must

be even – Cannot be synthesized into a stable structure cyclopentadiene (radical) bicyclo[2.2.1]hexa-2,5-diene (radical)

Other applications

This was just the beginning, since then: • Redfield-Pólya’s Theorem – Highly useful for enumerating any chemical compounds (not just alkanes) • Reaction graphs – Mapping the stepwise, directional (or reversible) reactions (edges) between intermediates (vertices) from the reactant to product • Adjacency matrices – Fundamental in quantum theory • NMR Spectroscopy • Topological studies – Insight into properties of (bio)macromolecules

References

Balaban, Alexandru T. Applications of Graph Theory in Chemistry.

J.

Chem. Inf. Comput. Sci. 1985 , 25:334-343.

Balaban, Alexandru T. Local versus Global (i.e. Atomic versus Molecular) Numerical Modeling of Molecular Graphs. J. Chem. Inf. Comput. Sci. 1994 , 34: 398-402 Balaban, Alexandru T. Chemical Graphs: Looking Back and Glimpsing Ahead. J. Chem. Inf. Comput. Sci. 1995 , 35, 339-350.

Balasubramanian, K. Applications of Combinatorics and Graph Theory to Spectroscopy and Quantum Chemistry. Chem. Rev. 1985 , 85: 599-618.

Garcia-Domenech, R.; Galvez, J.; de Julian-Ortiz, J. V.; Pogliani, L. Some New Trends in Chemical Graph Theory. Chem. Rev. 2008 , 108:1127-1169.

Rains, E. M.; Sloane, N. J. A. On Cayley’s Enumeration of Alkanes (or 4 Valent Trees). J. Integer Seq. 1999 , 2: 99.1.1