The Mathematics of Networks Chapter 7

Trees A tree is a graph that –Is connected –Has no circuits 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 Not a Tree: Has circuit, is not connected

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 Y Two different paths joining X and Y make a circuit => Not a Tree

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

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

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

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

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

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