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Excursions in Modern Mathematics Sixth Edition

Excursions in Modern Mathematics Sixth Edition. Peter Tannenbaum. Euler Circuits. 5.2 Graphs. Euler Circuits. Vertices - dots Edges - lines The edges do not have to be straight lines. But they have to connect two vertices. Loop - an edge connecting a vertex back with itself.

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Excursions in Modern Mathematics Sixth Edition

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  1. Excursions in Modern MathematicsSixth Edition Peter Tannenbaum

  2. Euler Circuits 5.2 Graphs

  3. Euler Circuits • Vertices- dots • Edges- lines The edges do not have to be straight lines. But they have to connect two vertices. • Loop- an edge connecting a vertex back with itself A graph is a picture consisting of:

  4. Euler Circuits This graph has six vertices A, B, C, D, E, and F and eight edges. The edges can be described by giving the two vertices that are connected by the edge. Thus the edges are AB, AD, BB, BC, BE, CD, CD, and DE

  5. Euler Circuits First, note that the point where edges BE and AD cross is not a vertex– it is just the crossing point of two edges. Second, that vertex F is not connected to any other vertex. Such a vertex is called an isolated vertex.

  6. Euler Circuits Third, note that this graph has a loop, namely the edge BB. Finally, note that it is permissible to have two edges connecting the same two vertices, as in the case with C and D. When a graph has more than one edge connecting the same pair of vertices, it is said to have multiple edges.

  7. Euler Circuits This graph is considered a single graph, even though it consists of two separate, disconnected pieces. Such graphs are called disconnected graph, and the individual pieces are called the components of the graph..

  8. Euler Circuits A Graph with No Edges? Yes, its possible. Without edges, every vertex of the graph is an isolated vertex.

  9. Euler Circuits Graphs A graph is a structure that defines pairwise relationships within a set to objects. The objects are the vertices, and the pairwise relationships are the edges: X is related to Y if and only if XY is an edge.

  10. Euler Circuits 5.3 Graph Concepts and Terminology

  11. Euler Circuits Adjacent vertices. Two vertices are said to be adjacent if there is an edge joining them. Vertices B and E are adjacent; C and D are not. Also because of the loop at E, we can say that Vertex E is adjacent to itself.

  12. Euler Circuits Adjacent edges. Two edges are adjacent if they share a common vertex. AB and AD are adjacent; edges AB and DE are not.

  13. Euler Circuits Odd and even vertices. An odd vertex is a vertex of odd degree; an even vertex is a vertex of even degree. The graph has two even vertices (D and E) and six odd vertices (all the others).

  14. Euler Circuits Degree of a vertex. The degree of a vertex is the number of edges at that vertex. When there is a loop at the vertex, the loop contributes twice. The deg(A) = 3, deg(B) = 5, deg(C) = 3, deg(D) = 2, deg(E) = 4, etc.

  15. Euler Circuits Paths. A path is a sequence of vertices with the property that each vertex in the sequence is adjacent to the next one. The key requirement in a path is that an edge can be part of a path only once.

  16. Euler Circuits Paths (continued). The number of edges in the path is called the length of the path. A, B, E, D. This is a path from vertex A to D, consisting of the edges AB, BE, and ED. The length of this path is 3.

  17. Euler Circuits Circuits. A circuit has the same definition as a path, but has the additional requirement that the trip starts and ends at the same vertex.

  18. Euler Circuits Connected graphs. A graph is connected, if given any two vertices, there is a path joining them. A graph that is not connected is said to be disconnected. A disconnected graph is made up of separate components.

  19. Euler Circuits Bridges. Sometimes in a connected graph there is an edge such that if we were to erase it, the graph would become disconnected—such an edge is called a bridge. BF, FG, and FH are bridges.

  20. Euler Circuits Euler paths. An Euler path is a path that passes through every edge of a graph once and only once. The graph shown in (a) does not have an Euler path; the graph in (b) has several Euler paths. One of them is L,A,R,D,A,R,D,L,A.

  21. Euler Circuits Euler circuits. An Euler circuit is a circuit that passes through every edge of a graph and ends up back where you started. One of them is L,A,R,D,A,R,D,L,A,L. Note that if a graph has an Euler circuit it cannot have an Euler path, and vice versa.

  22. Summary Where edges cross is NOT a vertex Loops described as BB or FF

  23. Summary Circuits and Paths Edges can only be traveled once Circuit is closed trip Path is open trip Euler = edges

  24. Summary Go through example 5.12 on pg. 174 and then complete intro to graph theory questions. When completed bring sheet to Callahan for feedback.

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