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
} Constitute the atomic nucleus
 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:
C4H10
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 (CH3CH2CH2CH3) 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 (CnH2n+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 (CnH2n+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
(C5H12)?
– 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 CC 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 kvalent 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 + z3 + z4 + 2z5 + 2z6 + 6z7 + 9z8 + 20z9 + 37z10 +
…
9
10 + …
B(z)
= z21+ z4 2+ z5 +3 3z6 4+ 3z7 5+ 9z86+ 15z
n
7 + 38z
8
9
10
11
centered
1
0
1
1
2
2
6
9
20
37
86
bicentered
0
1
0
1
1
3
3
9
15
38
73
total
1
1
1
2
3
5
9
18
35
75
159
C(z) + B(z) = z + z2 + z3 + 2z4 + 3z5
+ 5z6 + 9z7 + 18z8 + 35z9 + 75z10 + …
Computational techniques must be applied due to the rapidlyincreasing isomers (consider n=22, with 2,278,658 alkane isomers!)
Side note: Annulenes
• Hydrocarbons with chemical formula CnHn
• 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 4Valent Trees). J. Integer Seq. 1999, 2: 99.1.1