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Unformatted text preview: COMPSCI 102 Introduction to Discrete Mathematics Graphs Lecture 18 (November 3, 2010) A tree is a connected graph with no cycles What’s a tree? Tree Not aTree Not a Tree Tree How Many nNode Trees? 1: 2: 3: 4: 5: Notation In this lecture: n will denote the number of nodes in a graph e will denote the number of edges in a graph Theorem: Let G be a graph with n nodes and e edges The following are equivalent: 1. G is a tree (connected, acyclic) 3. G is connected and n = e + 1 4. G is acyclic and n = e + 1 5. G is acyclic and if any two nonadjacent points are joined by adding a new edge, the resulting graph has exactly one cycle 2. Every two nodes of G are joined by a unique path To prove this, it suffices to show 1 ⇒ 2 ⇒ 3 ⇒ 4 ⇒ 5 ⇒ 1 1 ⇒ 2 1. G is a tree (connected, acyclic) 2. Every two nodes of G are joined by a unique path Proof: (by contradiction) Assume G is a tree that has two nodes connected by two different paths: Then there exists a cycle! 2 ⇒ 3 2. Every two nodes of G are joined by a unique path Proof: (by induction) Assume true for every graph with < n nodes 3. G is connected and n = e + 1 Let G have n nodes and let x and y be adjacent Let n 1 ,e 1 be number of nodes and edges in G 1 Then n = n 1 + n 2 = e 1 + e 2 + 2 = e + 1 x y G 1 G 2 3 ⇒ 4 Proof: (by contradiction) Assume G is connected with n = e + 1, and G has a cycle containing k nodes 3. G is connected and n = e + 1 4. G is acyclic and n = e + 1 k nodes Note that the cycle has k nodes and k edges...
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