Graph theory as a method of improving chemistry

and mathematics curricula

Franka M. Brückler, Dept. of Mathematics, University of

Zagreb (Croatia)Vladimir Stilinović,

Dept. of Chemistry, University of Zagreb (Croatia)

Problem(s)

• school mathematics: dull? too complicated? to technical?

• various subjects taught in school: to separated from each other? from the real life?

• possible solutions?

Fun in school

• fun and math/chemistry - a contradiction?

• can you draw the picture traversing each line only once? – Eulerian tours

• is it possible to traverse a chessboard with a knight so that each field is visited once? – Hamiltonian circuits

Graphs• vertices (set V) and edges (set E) –

drawn as points and lines• the set of edges in an (undirected) graph

can be considered as a subset of P(V) consisting of one- and two-member sets

• history: Euler, Cayley

Basic notions• adjacency – u,v adjacent if {u,v} edge• vertex degrees – number of adjacent vertices• paths – sequences u1u2...un such that each

{ui,ui+1} is and edge + no multiple edges• circuits – closed paths• cycles – circuits with all vertices appearing

only once• simple graphs – no loops and no multiple

edges• connected graphs – every two vertices

connected by a path• trees – connected graph without cycles

Graphs in chemistry• molecular (structural) graphs (often:

hydrogen-supressed) • degree of a vertex = valence of atom

• reaction graphs – union of the molecular graphs of the supstrate and the product

C C

C C

CC

2 : 1

2 : 1

2 : 1

0 : 1

0 : 11 : 2

Diels-Alder reaction

Mathematical trees grow in chemistry

• molecular graphs of acyclic compounds are trees

• example: alkanes• basic fact about trees: |V| = |E| + 1• basic fact about graphs: 2|E| = sum

of all vertex degrees

5–isobutyl–3–isopropyl–2,3,7,7,8-pentamethylnonan

Alkanes: CnHm• no circuits & no multiple bonds tree• number of vertices: v = n + m• n vertices with degree 4, m vertices wit

degree 1• number of edges: e = (4n + m)/2• for every tree e = v – 1• 4n + m = 2n + 2m – 2 m = 2n + 2 • a formula CnHm represents an alkane only

if m = 2n + 2

methane CH4 ethane C2H6 propane C3H8

Topological indices• properties of substances depend not only of their

chemical composition, but also of the shape of their molecules

• descriptors of molecular size, shape and branching• correlations to certain properties of substances

(physical properties, chemical reactivity, biological activity…)

Ev}{u,e )()(

1)(

vdudG

2/|E|

0k

)k(p)G(Z

Wiener index – 1947.sum of distances between all pairs of vertices in a H-supressed graph; only for trees; developed to determine parrafine boiling points

Randić index – 1975. Good correlation abilityfor many physical &biochem properties

Hosoya index – p(k) is the number of ways for choosing k non-adjacent edges from the graph; p(0)=1, p(1)=|E|

1)j(d),i(d)G(Vj,i

ijd21

)G(W

NameWiener index(W)

Randić index

Hosoya index(Z)

Boiling point/oC (17)

methylamine 1 1 2 -6

ethylamine 4 1,414 3 16,5

n-propylamine 10 1,914 5 49

isopropylamine 9 1,732 4 33

n-butylamine 20 2,414 8 77

isobutylamine 19 2,27 7 69

sec-butylamine 18 2,27 7 63

tert-butylamine 16 2 5 46

n-pentylamine 35 2,914 13 104

isopentylamine 33 2,063 11 96

topological indices and boiling points of several primary amines

-20

0

20

40

60

80

100

120

0 10 20 30 40

W

Bp/C

-20

0

20

40

60

80

100

120

0 1 2 3 4

R

Bp/C

-20

0

20

40

60

80

100

120

140

0 5 10 15

Z

Bp/C

• possible exercises for pupils: • obviously: to compute an index from a

given graph• to find an expected value of the boiling

point of a primary amine not listed in a table, and comparing it to an experimental value. Such an exercise gives the student a perfect view of how a property of a substance may depend on its molecular structure

Examples• 2-methylbutane• W =

0,5((1+2+2+3)+(1+1+1+2)+(1+1+2+2)+(1+2+3+3)+

(1+2+2+3)) = 18:•

• There are four edges, and two ways of choosing two non adjacent edges so

• Z = p(0) + p(1) + p(2) = 1 + 4 + 2 = 7

270,231

1

12

1

23

1

31

1

R

For isoprene W isn’t defined, since its molecular graph isn’t a tree Randić index is

and Hosoya index is Z = 1 + 6 + 6 = 13. 

270,231

1

12

1

23

1

31

1

R

For cyclohexane W isn’t defined, since its molecular graph isn’t a tree Randić index is

and Hosoya index is Z = 1 + 6 + 18 + 2 = 27. 

322

16

R

Enumeration problems• historically the first application of graph theory

to chemistry (A. Cayley, 1870ies)• originally: enumeration of isomers i.e.

compounds with the same empirical formula, but different line and/or stereochemical formula

• generalization: counting all possible molecules for a given set of supstituents and determining the number of isomers for each supstituent combination (Polya enumeration theorem)

• although there is more combinatorics and group theory than graph theory in the solution, the starting point is the molecular graph

Cayley’s enumeration of trees

• 1875. attempted enumeration of isomeric alkanes CnH2n+2 and alkyl radicals CnH2n+1

• realized the problems are equivalent to enumeration of trees / rooted trees

• developed a generating function for enumeration of rooted trees

• 1881. improved the methodfor trees

...)1()1()1( 1110

10

nn

AAA xAxAAxxx n

Pólya enumeration method • 1937. – systematic method for enumeration• group theory, combinatorics, graph theory• cycle index of a permutation group: sum of all

cycle types of elements in the group, divided by the order of the group

• cycle type of an element is represented by a term of the form x1

ax2bx3

c ..., where a is the number of fixed points (1-cycles), b is the number of transpositions (2-cycles), c is the number of 3-cycles etc.

• when the symmetry group of a molecule (considered as a graph) is determined, use the cycle index of the group and substitute all xi-s with sums of Ai with A ranging through possible substituents

Example

• how many chlorobenzenes are there? how many isomers of various sorts?

• consider all possible permutations of vertices that can hold an H or an Cl atom that result in isomorphic graphs (generally, symmetries of the molecular graph that is embedded with respect to geometrical properties)

• of 6!=720 possible permutations only 12 don’t change the adjacencies

1

2 3

4

56

1

2 3

4

56

1 symmetry consisting od 6 1-cycles: 1· x16

1 2

3

45

6

2 symmetries (left and right rotationfor 60°) consisting od 1 6-cycle: 2· x6

1

1

2

34

5

62 symmetries (left and right rotationfor 120°) consisting od 2 3-cycles: 2· x3

2

3 symmetries (diagonals as mirrors) consisting od 2 1-cycles and 2 2-cycles: 3· x1

2 · x22

4 symmetries (1 rotationfor 180° and 3 mirror-operations withmirrors = bisectors of oposite pages) consisting od 3 2-cycles: 4· x2

3

1

6 5

4

32

4

3 2

1

65

summing the terms cycle index

16

22

21

32

23

61 x2xx3x4x2x

121

)G(Z

substitute xi = Hi + Cli into Z(G)

6542332456 ClHClClH3ClH3ClH3ClHH

i.e. there is only one chlorobenzene with 0, 1, 5 or 6 hydrogen atoms and there are 3 isomers with 4 hydrogen atoms, with 3 hydrogen atoms and with with 2 hydrogen atoms

Planarity and chirality

• planar graphs: possible to embed into the plane so that edges meet only in vertices

• a molecule is chiral if it is not congruent to its mirror image

• topological chirality: there is no homeomorphism transforming the molecule into its mirror image

• if the molecule is topologically chiral then the corresponding graph is non-planar