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The Mathematics of Networks

The Mathematics of Networks. Chapter 7. Trees. A tree is a graph that Is connected Has no circuits. Tree. Tree. Tree. Not a Tree. Not a Tree: Has a circuit. Not a Tree: Has several circuits. Not a Tree. Not a Tree: Has no circuit, but is not connected. Not a Tree.

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The Mathematics of Networks

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  1. The Mathematics of Networks Chapter 7

  2. Trees • A tree is a graph that • Is connected • Has no circuits Tree Tree Tree

  3. Not a Tree Not a Tree: Has a circuit Not a Tree: Has several circuits

  4. Not a Tree Not a Tree: Has no circuit, but is not connected

  5. Not a Tree Not a Tree: Has circuit, is not connected

  6. Properties of Trees Property 1: • If a graph is a tree, there is one and only one path joining any two vertices. • Conversely, if there is one and only one path joining any two vertices of a graph, the graph must be a tree X Two different paths joining X and Y make a circuit => Not a Tree Y

  7. Properties of Trees Property 2: • In a tree, every edge is a bridge. • Conversely, if every edge of a connected graph is a bridge, then the graph must be a tree. Tree: every edge is a bridge. If we erase any edge, the graph will be disconnected

  8. Properties of Trees Property 3: • A tree with N vertices must have (N-1) edges. • However, if a graph has N vertices and (N-1) edges, it will not be a tree always. The graph has 10 vertices and 9 edges, but not a tree because the graph is not connected

  9. Properties of Trees Property 4: • A connected graph with N vertices and (N-1) edges must be a tree Tree: Connected graph has 6 vertices and 5 edges

  10. Disconnected Graph • Graph with five vertices and less than four edges are disconnected

  11. Connected Graph • Graph with five vertices and four edges are – just enough to connect

  12. Not a Tree any more • Graph with five vertices and more than four edges are – circuits begin to form

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