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 – Electrons 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 } Constitute the atomic nucleus  Found around the nucleus in a statistical “cloud”

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: 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 ) ? Isobutane (methylpropane)

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 nodes and diametere 2h –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 be the total number of bicentered k-valent trees with n nodes –We now want to find B 2h+1,n, the number of bicentered k- valent trees with n nodes and diameter 2h+1 –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 z z 10 + … B(z) = z 2 + z 4 + z 5 + 3z 6 + 3z 7 + 9z z z 10 + … n centered bicentered total C(z) + B(z) = z + z 2 + z 3 + 2z 4 + 3z 5 + 5z 6 + 9z z z z 10 + … 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: Hydrogen-depleted representations are regular graphs of degree 3 (cubic graphs)   1,3 - cyclobutadiene benzene

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

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