1. Can you connect the dots as shown without taking your pen off the page or drawing the same line twice.

3. Graphs terminology

4. Objectives To understand the key words: node, arc, order of node, trail, path, closed trail, cycle, connected. To know what Eulerian and semi-Eulerian graphs are and be able to identify them.

5. Maths Algorithms Homework also Ex 1B p6 and Ex 1C p8 Next Lesson: More Graph theory and terminology. Leonhard Euler 1707-1783: Swiss mathematician and general genius who invented graph theory!

6. Nodes/vertices and arcs/edges The order/degree of a node is the number of arcs which meet at the node.

7. Hmmm... Given all arcs must start and end at a node what can you say about the sum of the orders of all nodes in a graph?

8. A walk is a sequence of arcs such that the end node of one arc is the start node of the next arc... If a walk does not travel an arc more than once then it is called a trail (or route).

9. A path is a trail which passes through a node once only.

10. A closed trail is a trail where the first node and the last node are the same.

11. A cycle is a closed trail where only the first and last nodes are the same.

12. So... A walk that only uses each of its arcs once is a trail... A trail that has the same start and end node is a closed trail. A path is a trail that only visits each node once. A cycle is a closed path (allowing the path to visit its start node twice at the start and finish). So given this increasing specialism what do you suppose a Hamiltonian Cycle is? A cycle that visits every node

13. http://www.cut-the- knot.org/Curriculum/Combinatorics/GraphPractice.shtml http://www.cut-the- knot.org/Curriculum/Combinatorics/GraphPractice.shtml

14. A connected graph is one where, for any two nodes, a path can be found connecting the two nodes. 

15. An Eulerian graph is a connected graph which has a closed trail containing every arc precisely once

16. A semi-Eulerian graph is a connected graph which has a trail containing every arc precisely once (you can draw it without going back over a line or taking your pen off the page)

17. Some graphs are neither Eulerian nor semi-Eulerian.

18. Eulerian Not Eulerian

19. Eulerian Semi-Eulerian

20. Eulerian Graphs All nodes in an Eulerian Graph have even order. This means the Graph is fully traversable (there is a closed trail that uses every edge). Recall a trail never repeats and edge. You could draw it without taking your pen off the page and will start and end at the same vertex. A semi-Eulerian Graph has two nodes of odd degree (one pair since odd nodes must come in pairs). This means the graph is semi- traversable (there is a trail that uses every edge). Note that the trail is not closed so the start and end points are different (the odd nodes). You could draw it without taking your pen off the page but will start and end at different nodes.

21. Eulerian, Semi Eulerian OR Neither

22. Euler’s Problem http://www.youtube.com/wa tch?v=CnU1ybtwgw8

24. Complete Graphs Complete graphs are graphs where every node is connected to every other. They are denoted K n where n is the number of nodes

25. More Thinking When is a complete graph Eulerian?

26. Simple Graphs Contains no loop arcs (connects a node to itself) or multiple arcs between the same pair  

27. A planar graph is one which can be drawn in a plane in such a way that arcs only meet at nodes. Can you draw K 4 without letting any lines cross? Is K 4 planar? Can you draw K 5 without letting any lines cross? Is K 5 planar?

28. Bipartite Graphs Bipartite Graphs are graphs that have two sets of vertices usually drawn in a vertical line

29. Bipartite Graphs Vertices cannot be joined to other vertices in that set

30. Bipartite Graphs Vertices cannot be joined to other vertices in that set  

31. Complete Bipartite Graphs The complete Bipartite Graphs have every possible matching they are written K m,n where m,n are the number of nodes in each set K 4,4 K 4,2

32. A Bipartite Graph Puzzle

33. Trees The final piece of graph theory terminology we need to know about are trees. Trees connect nodes but can have multiple branches from each node unlike walks etc.

34. Trees A Tree

35. Spanning Trees A Spanning Tree includes every node

36. Applications - Networks It is possible for graphs to have weighted arcs. These weights can mean distances, time or cost (or several other things) 13 11 12 13 2 3 1 2 3 1 2 5 Once a graph has weights it can be called a network

37. Other Things It is possible for graphs to be represented in a matrix/table (1 represents an arc 0 or - no arc (you can also have weights in here too) B H A D C E F G

38. A B C D E