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1. 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)

2. 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?

3. 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

4. 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

5. 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

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

7. 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

8. 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

9. 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

10. 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…)

11. 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|

12. topological indices and boiling points of several primary amines

14. 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

15. 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

16. 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.

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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