Lecture2 - CME 305: Discrete Mathematics and Algorithms...

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CME 305: Discrete Mathematics and Algorithms Instructor: Professor Amin Saberi (saberi@stanford.edu) January 7, 2010 Lecture 2: Graphs, Trees, and Cayley’s Theorem 1 Basic Definitions of Graphs and Trees A graph G ( V,E ) is a set V of vertices and a set E of edges , where E = { ( a,b ) ,a V,binV } . We usually denote | V | as n , and | E | as m . In an undirected graph, an edge is an unordered pair of vertices. An ordered pair of vertices is called a directed edge. If we allow multi-sets of edges, i.e. multiple edges between two vertices, this is known as a multigraph . A self-loop or loop is an edge between a vertex and itself. An undirected graph without loops or multiple edges is known as a simple graph. In this class, we will assume graphs to be simple and finite (i.e. number of vertices is finite) unless otherwise stated. If vertices a and b share an edge e , we say that they are adjacent and write a b . If vertex a is one of edge e ’s endpoints, a is incident to e and we write a e . The degree of a vertex is the number of edges incident to it, denoted d i for vertex v i V . The degree sequence of graph is the sequence { d 1 ,d 2 ,...,d n } . A walk is a sequence of vertices v 1 ....v k such that i 1 ..k - 1, v i v i +1 . A path is a walk where v i 6 = v j , i 6 = j , i.e. a walk that visits each vertex at most once. A closed walk is a walk where v 1 = v k . A cycle is a closed path, i.e. a path combined with the edge ( v k ,v 1 ). A graph is connected if there exists a path between each pair of vertices. A tree is a connected graph with no cycles. A forest is a graph where each connected component is a tree. A node in a forest with degree 1 is called a leaf . 2 Basic Properties of Trees Claim 1 (simple properties of trees) Every tree has at least two leaves (vertices of degree 1), assuming n 2 . The number of edges in a tree of size n is n - 1 Proof: The first property can be seen by starting a path at an arbitrary node, walking to any neighbor that hasn’t been visited before. After at most n steps, we must reach a vertex v where there are no untraversed neighbors. Therefore, v must have only one neighbor and is a leaf, otherwise there is a cycle and the graph is not a tree. We can then traverse a path starting at v
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Lecture2 - CME 305: Discrete Mathematics and Algorithms...

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