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Decision Maths

Decision Maths. Graphs. Graphs. A graph is just a diagram made up of “dots” and “lines”. These are all graphs. The dots are called “ nodes ” or “ vertices ” (singular is vertex) The lines are called “ edges ” or “ arcs ”. Definitions 1.

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Decision Maths

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  1. Decision Maths Graphs

  2. Graphs • A graph is just a diagram made up of “dots” and “lines”. These are all graphs. • The dots are called “nodes” or “vertices” (singular is vertex) • The lines are called “edges” or “arcs”

  3. Definitions 1 • An edge with the same vertex at each end is called a loop. • The degree or order of a vertex is the number of edges incident on it. • Question – For any graph the total of the orders of its verticies is even, why? • A simple graph is one in which there are no loops, and in which there is no more than one edge connecting any pair of vertices.

  4. A walk is a sequence of edges in which the end of one edge (except the last) is the beginning of the next. A trail is a walk in which no edge is repeated. Definitions 3

  5. Definitions 4 • A path is a trail in which no vertex is repeated. • A graph is connected if there exists a path between every pair of vertices.

  6. Definitions 5 • A cycle is a closed path if the end of the last edge is the start of the first. • A Hamiltonian cycle is a cycle which visits every vertex once and only once.

  7. Definitions 6 • A tree is a simple connected graph with no cycles. A tree Not trees

  8. Definitions 7 • A Digraph (Directed Graph) is a graph in which at least one edge has a direction associated with it. • A complete graph is a simple graph in which every pair of vertices is connected by an edge.

  9. Definitions 8 • An incidence matrix is a way of representing the number of edges between nodes in a matrix. The graph below is represented by the matrix next to it.

  10. Definitions 9 • Two graphs are Isomorphic if one can be stretched, twisted or otherwise distorted into the other. • Which two graphs below are Isomorphic? • If two graphs are isomorphic then the labels on them must correspond to each other.

  11. Definitions 10 • A planar graph is one which can be drawn without any edges crossing. • Which graph(s) below is Planar? • Draw two examples of Planar graphs.

  12. Definitions 11 • A bipartite graph is one in which the vertices fall into two sets and in which each edge has a vertex from one set at one end and from the other set at its other end.

  13. Question 1 • X = { 2,3,4,5,6} Draw a graph to represent the relationship ‘share a common factor other than 1’

  14. Question 2 • X = { London, Oxford, Birmingham, Cambridge, Leicester} • Let X x X be the set of all possible pairs from the set X. (there exists a road between the two towns) • X x X = { (London, Oxford) (London, Birmingham) (London, Cambridge) (London, Leicester) (Oxford, London) (Oxford, Birmingham) (Birmingham, London) (Birmingham, Oxford) (Birmingham, Leicester) (Cambridge, London) (Leicester, London) (Leicester, Birmingham) } • Drawa graph to show the set X x X.

  15. Question 2

  16. Question 3 – Ex 2A q13 pg 54 • Each node represents a section of land. • And each arc is the route over the bridges.

  17. Eulerian • A graph is called Eulerian or traversable if each can be traced once and only once, without lifting pencil from paper. • A graph is traversable if it has no odd vertices or just two odd vertices. • Prove that the graph below is traversable.

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