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)
Dec 24, 2015
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