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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

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
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 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
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
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
Graphs in chemistry
  • molecular (structural) graphs(often: hydrogen-supressed)
  • degree of a vertex = valence of atom
slide7

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
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 c n h m
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
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…)
slide11

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|

slide14
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
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
slide16

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
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
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

p lya enumeration method
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
example

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
slide21

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

slide22

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

slide23

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
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