Graph theory as a method of improving chemistry and mathematics curricula

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Graph theory as a method of improving chemistry and mathematics curricula. Franka M. Brückler , Dep t. of Mathematics, University of Zagreb (Croatia) Vladimir Stilinović , Dep t. of Chemistry, University of Zagreb (Croatia). Problem(s).

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### Graph theory as a method of improving chemistry and mathematics curricula

Franka M. Brückler,

Dept. of Mathematics, University of Zagreb (Croatia)

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
• 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

0 : 1

C

C

2 : 1

2 : 1

C

Diels-Alder reaction

C

1 : 2

0 : 1

C

C

2 : 1

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…)

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 ability

for 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|

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

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.

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.

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 method

for trees

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 x1ax2bx3c ..., 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

2

3

Example

1

4

6

5

• 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

2

3

1

4

5

6

2

1

6

3

5

4

6

1

5

2

4

3

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

2 symmetries (left and right rotation

for 60°) consisting od 1 6-cycle: 2·x61

2 symmetries (left and right rotation

for 120°) consisting od 2 3-cycles: 2·x32

6

3

5

2

1

4

4

1

3

6

2

5

3 symmetries (diagonals as mirrors)

consisting od 2 1-cycles and

2 2-cycles: 3·x12 ·x22

4 symmetries (1 rotation

for 180° and 3 mirror-operations with

mirrors = bisectors of oposite pages)

consisting od 3 2-cycles: 4·x23

summing the terms  cycle index

substitute xi = Hi + Cli into Z(G) 

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