grafovi - curricula Franka M Brckler...

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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) Vladimir Stilinović Dept. of Chemistry, University of Zagreb (Croatia)
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    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?
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    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
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    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
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    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
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    Graphs in chemistry molecular (structural) graphs   (often: hydrogen-supressed)  • degree of a vertex = valence of atom
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     reaction graphs  –  union of the molecular graphs of the  supstrate and the product C C C C C C 2 : 1 2 : 1 2 : 1 0 : 1 0 : 1 1 : 2 Diels-Alder reaction
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    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
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