s6_1 - CHAPTER 6 Introduction to Graph Theory 1....

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Unformatted text preview: CHAPTER 6 Introduction to Graph Theory 1. Introduction to Graphs 1.1. Simple Graphs. Definition 1.1.1 . A simple graph ( V,E ) consists of a nonempty set represent- ing vertices, V , and a set of unordered pairs of elements of V representing edges, E . A simple graph has no arrows, no loops, and cannot have multiple edges joining vertices. Discussion Graphs offer a convenient way to represent various kinds of mathematical objects. Essentially, any graph is made up of two sets, a set of vertices and a set of edges. Depending on the particular situation we are trying to represent, however, we may wish to impose restrictions on the type of edges we allow. For some problems we will want the edges to be directed from one vertex to another; whereas, in others the edges are undirected. We begin our discussion with undirected graphs . The most basic graph is the simple graph as defined above. Since the edges of a simple graph are undirected, they are represented by unordered pairs of vertices rather than ordered pairs . For example, if V = { a,b,c } , then { a,b } = { b,a } would represent the same edge. Exercise 1.1.1 . If a simple graph G has 5 vertices, what is the maximum number of edges that G can have? 1.2. Examples. Example 1.2.1 . V = { v 1 ,v 2 ,v 3 ,v 4 } E = {{ v 1 v 2 } , { v 1 ,v 3 } , { v 2 ,v 3 } , { v 2 ,v 4 }} 178 1. INTRODUCTION TO GRAPHS 179 v 1 v 2 v 3 v 4 Example 1.2.2 . V = { a,b,c } , E = {{ a,b } , { b,c } , { a,c }} a b c 1.3. Multigraphs. Definition : A multigraph is a set of vertices, V , a set of edges, E , and a function f : E {{ u,v } : u,v V and u 6 = v } . If e 1 ,e 2 E are such that f ( e 1 ) = f ( e 2 ), then we say e 1 and e 2 are multiple or parallel edges ....
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s6_1 - CHAPTER 6 Introduction to Graph Theory 1....

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