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Cayleys Enumeration on the Structural Isomers of AlkanesMatthew P. Yeager

Also: topoisomers, isotopomers, nuclear isomers, spin isomers

Significance of IsomersIsomers 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:ProtonsNeutrons

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-, respectivelyAn 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:

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 derivativesCarbon 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 TheoryConsider the molecular formula of a carbon-backbone compound:

C4H10What is its molecular structure?

Every carbon must bond to another carbonNumber of H = 2 x (Number of C) + 2

So, how about?Butane

Genesis of Chemical Graph TheoryButane (CH3CH2CH2CH3) 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 (CnH2n+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 vertexEvery single bond or lone pair is an edge1,2 - dichloropropanepseudograph representation

Arthur Cayley (1875)Using chemical principals, Cayley made generalizations that would limit the enumeration alkane isomers (CnH2n+2):

Alkanes are trees:Only single bonds; no double / triple bonds, or lone pairsAcyclic

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 EnumerationSo how many structural isomers exist for pentane (C5H12)?That is, how many unique trees are there with 5 nondistinct vertices?pentaneisopentane(methylbutane)neopentane(dimethylpropane)

Cayleys ApproachCayley enumerated trees of valency 4 by counting the number of centered and bicentered H-depleted graphs for any quantity of nodesCentered: a tree of diameter 2m contains a unique node at the midpoint, called a centerBicentered: 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 Th, the number of tertiary rooted trees with n nodes and height at most h

Find C2h, the number of centered 4-valent trees with n nodes and diametere 2h

Find Cn, the number of centered 4-valent trees with n nodes

For bicentered trees, the approach is a little easier:

Let Bn be the total number of bicentered k-valent trees with n nodes

We now want to find B2h+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 FunctionsAfter the lengthy derivation, we receive:

for the centered trees, and

for the bicentered trees

Generating FunctionsExpansion yields:C(z) = z + z3 + z4 + 2z5 + 2z6 + 6z7 + 9z8 + 20z9 + 37z10 + B(z) = z2 + z4 + z5 + 3z6 + 3z7 + 9z8 + 15z9 + 38z10 +

C(z) + B(z) = z + z2 + z3 + 2z4 + 3z5 + 5z6 + 9z7 + 18z8 + 35z9 + 75z10 + Computational techniques must be applied due to the rapidly-increasing isomers (consider n=22, with 2,278,658 alkane isomers!)

n123456789 10 11 centered10112269 20 37 86bicentered01011339 15 38 73 total1112359 18 35 75 159

Side note: AnnulenesHydrocarbons with chemical formula CnHnExamples:

Hydrogen-depleted representations are regular graphs of degree 3 (cubic graphs)

1,3 - cyclobutadienebenzene

Without any knowledge of chemistry, can we remark on the annulenes with odd n?Mathematically impossible by graph theoryThe number of vertices of odd degree must be evenCannot be synthesized into a stable structure

cyclopentadiene (radical)bicyclo[2.2.1]hexa-2,5-diene (radical)

Other applicationsThis was just the beginning, since then:Redfield-Plyas 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 productAdjacency matrices Fundamental in quantum theoryNMR SpectroscopyTopological studiesInsight into properties of (bio)macromolecules

ReferencesBalaban, 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 Cayleys Enumeration of Alkanes (or 4-Valent Trees). J. Integer Seq. 1999, 2: 99.1.1

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