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Vertex Edge Graphs

Vertex Edge Graphs. Suppose you are a snowplow driver. You want to leave the garage , plow each street only once and then return to the garage. Is this always possible?. Consider this situation…. Garage. Next situation….

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Vertex Edge Graphs

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  1. Vertex Edge Graphs

  2. Suppose you are asnowplow driver. You want to leave the garage, plow each street only once and then return to the garage. Is this always possible?

  3. Consider this situation… Garage

  4. Next situation…

  5. Problems such as these can be addressed using a type of graph called a vertex-edge graph

  6. A vertex-edge graph is made up of two parts: • Points called VERTICES • (each point is called a VERTEX) • and segments or curves • called EDGES

  7. Points A and B are vertices, the curve from A to B is an edge. A B Points D and E are vertices, the segment from D to E is an edge D E

  8. Point J is an EVEN VERTEX K J L If there is an even numberof edges attached to a vertex, the vertex is called an even vertex.

  9. A Point F is an ODD VERTEX H F G If there is an odd numberof edges attached to a vertex, the vertex is called an odd vertex.

  10. Any edge that connects a vertex to itself is called a LOOP. K J The path from K back onto itself can either be clockwise or counterclockwise.

  11. The degree of a vertex is the number of edges connected to that vertex. Loops are counted twice, once for the clockwise route and the other for the counter-clockwise route.

  12. C 2 4 3 D 3 A B E 2 Find the degree of each vertex:

  13. D B C A E Any route that connects one vertex to another vertexwithout repeating any edges is called a PATH. ABC

  14. D B C A E AEC

  15. D B C A E AECD

  16. B C A E Any path that begins and ends at the same vertex is called a CIRCUIT. ABCEA

  17. I H F G Any graph that has at least one path connecting any two vertices is called a CONNECTED GRAPH.

  18. A circuit (beginning and ending at the same vertex) or a path (beginning at one vertex and ending at another) where each edge is used exactly once is called an Euler circuit or Euler path.

  19. These graphs are named after the Swiss mathematician Leonhard Euler.

  20. C D 2 4 4 2 2 4 E B F A To find the Adjacency Matrix of a vertex edge?

  21. C D 3 4 2 3 2 B A E

  22. HW: Worksheet: Paths and Circuits

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