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 u 1 u 2... u n such that each { u i, u i+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 CC CC CC 2 : 1 0 : 1 1 : 2 Diels-Alder reaction

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: C n H m 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 = (4 n + m )/2 for every tree e = v – 1 4 n + m = 2 n + 2 m – 2  m = 2 n + 2  a formula C n H m represents an alkane only if m = 2 n + 2 methane CH 4 ethane C 2 H 6 propane C 3 H 8

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. Name Wiener index (W) Randić index  Hosoya index (Z) Boiling point/ o C (17) methylamine112-6 ethylamine41,414316,5 n-propylamine101,914549 isopropylamine91,732433 n-butylamine202,414877 isobutylamine192,27769 sec-butylamine182,27763 tert-butylamine162546 n-pentylamine352,91413104 isopentylamine332,0631196 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 C n H 2n+2 and alkyl radicals C n H 2n+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 x 1 a x 2 b x 3 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 x i -s with sums of A i with A ranging through possible substituents

20. 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 23 4 56

21. 1 23 4 5 6 1 symmetry consisting od 6 1-cycles: 1· x 1 6 1 2 3 45 6 2 symmetries (left and right rotation for 60°) consisting od 1 6-cycle: 2· x 6 1 1 2 34 5 6 2 symmetries (left and right rotation for 120°) consisting od 2 3-cycles: 2· x 3 2

22. 3 symmetries (diagonals as mirrors) consisting od 2 1-cycles and 2 2-cycles: 3· x 1 2 · x 2 2 4 symmetries (1 rotation for 180° and 3 mirror-operations with mirrors = bisectors of oposite pages) consisting od 3 2-cycles: 4· x 2 3 1 65 4 3 2 4 32 1 6 5 summing the terms  cycle index

23. substitute x i = H i + Cl i 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