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L19_IntroGraphs_print

L19_IntroGraphs_print - COMP170 Discrete Mathematical Tools...

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1 COMP170 Discrete Mathematical Tools for Computer Science Discrete Math for Computer Science K. Bogart, C. Stein and R.L. Drysdale Section 6.1, pp. 309-320 Intro to Graphs Version 2.0: Last updated, May 13, 2007 Slides c 2005 by M. J. Golin and G. Trippen
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2 Graphs The Degree of a Vertex Connectivity Cycles Trees Basic Definitions
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3 Graphs Fundamental topic in discrete math and CS. Important because it’s used to model many common situations and to naturally describe many algorithms. Example Map of some cities in eastern US. with communication lines existing between certain pairs of these cities.
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4 What is the minimum number of links needed to send a message from B to NO ? 3 : B – CH – ME – NO . Which city/cities has/have the most communication links em- anating from it/them? A : 6 links. What is the total number of communication links? 20 links.
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5 consists of a set of vertices V , | V | = n , and a set of edges E , | E | = m . Each edge has two endpoints . An edge joins its endpoints, two endpoints are adjacent if they are joined by an edge. When a vertex is an endpoint of an edge, we say that the edge and the vertex are incident to each other. Graph G
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6 More Examples: Vertices: biological species Edges: species have a common ancestor Vertices: people Edges: people attend same school Vertices: MTR stations Edges: direct connection Vertices: Web sites Edges: A link from one site to another How Google models the Internet!
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7 More Graphs: Simple Graph (a, b, c): at most one edge joining each pair of distinct vertices (versus multiple edges (d)) and no edges joining a vertex to itself (= loop ). Complete Graph K n (b, c): graph with n vertices that has an edge between each pair of vertices.
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8 A path in a graph is an alternating sequence of ver- tices and edges such that it starts and ends with a vertex, each edge joins the vertex before it in the sequence to the vertex after it in the sequence, and no vertex appears more than once in the sequence. Length of a path = # of edges on path
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9 Example Path from Boston to New Orleans is B { B,CH } CH { CH,ME } ME { ME,NO } NO . Since the 2 nd endpoint of an edge is the 1 st endpoint of the following edge, we usually just write the succes- sive endpoints, e.g., B,CH,ME,NO. This path has length 3 .
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10 The distance between two vertices is the length of the shortest path between them. dist(CI, W) = 1 dist(CI, B) = 2 dist(CI, NO) = 2 Examples:
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11 Proof sketch: Just delete cycles (loops).
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