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1.5 Graph Theory

1.5 Graph Theory. Graph Theory. The Branch of mathematics in which graphs and networks are used to solve problems. Graph. In Graph theory, graphs are unlike Cartesian graphs. A graph (or network) consists of line segments and nodes. Usually referred to as vertices and edges. Map Coloring.

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1.5 Graph Theory

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  1. 1.5 Graph Theory

  2. Graph Theory • The Branch of mathematics in which graphs and networks are used to solve problems.

  3. Graph • In Graph theory, graphs are unlike Cartesian graphs. • A graph (or network) consists of line segments and nodes. • Usually referred to as vertices and edges.

  4. Map Coloring • Determine the smallest number of colors needed for each map such that all neighboring countries have different colors.

  5. Maps with networks • Represent each map with networks

  6. Adjacent • If two vertices are connected by an edge, they are considered to be adjacent.

  7. Vertex Degree • The number of edges that begin or end at a vertex is the degree of that vertex.

  8. More Terms • Any connected sequence of vertices is called a Path. • If the path begins and ends at the same vertex, the path is called a circuit.

  9. Circuit?

  10. Connected • A network is Connected if and only if there is at least one path connecting each pair or vertices. • A Complete Network is a network with an edge between every pair.

  11. Traceable • In a Traceable network, all vertices are connected to at least one other vertex and all the edges can be travelled exactly once in a continuous path.

  12. Done with definitions, Example Time! People of the town believes that it was impossible to tour the town, crossing eachbridge exactly once, regardless of where tour started. We they correct?

  13. Euler • Good ole Leonard Euler showed that • A network is traceable if it has only vertices of even degree of exactly 2 vertices of odd degree. • If the network has two vertices of odd degree, the tracing path must begin at one of the odd degree vertices and end at the other.

  14. Traceable and Degree • Determine if the network is traceable

  15. Planar • A network is planar if it can drawn on a 2-dimensional surface so that no edges cross.

  16. Graphic Design • A graphic designer is working on a logo representing the different tourist regions in Ontario. What is the minimum number of colors required for the design shown below to have all adjacent areas colored differently?

  17. Assignment • Pg 49 #’s 1,2,6,7,9,11

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