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Unformatted text preview: Graphs Mariusz Bajger COMP2781/8781 School of Computer Science, Engineering and Mathematics May 17, 2011 1/29 Reading and Exercises Reading Epp, Chapter 10 Exercises Sec. 10.1 (Ex. 135), 10.2 (all), 10.3 (Ex. 121), 10.5 (Ex. 129) and 10.7 (all except Prim algorithm exercises) Good learning strategy: BE ACTIVE! I Regularly revise lectures I Solve the suggested exercises I Be critical when reading textbook/lectures I Ask your colleagues, ask the lecturer, dont be shy! I It is OK to ask for help any time 2/29 Modeling with Graphs I Air transport system I Computer networks I Telephone I Electric power I Gas pipeline I The Internet I Electronic circuits I and many more ... By wellorganizing things we have a good chance of answering some vital questions: Which routes to choose to minimize the cost of connection? 3/29 Examples of graphs Undirected, unweighted, 5 vertices, 4 edges Directed, unweighted, 5 vertices, 3 edges Directed, unweighted, 5 vertices, 3 edges Directed, weighted, 5 vertices, 3 edges 4/29 More applications I Hypertext : Web documents = vertices, hyperlinks = edges (used by search engines) I Maps : towns = vertices, routes = edges I Circuits : transistors/resistors = vertices, wires = edges I Transactions : customers = vertices, cash transfer = edges I Matching : students apply for positions in selective institutions, vertices = students + institutions, edges = applications Find best matching: student  position I Networks : computers = vertices, edges = connections What is the critical set of connections? I Image Analysis : pixels as vertices, edges between similar pixels I Program compilers : modules=vertices, edge if two modules are connected e.g. methods invoking another method 5/29 Application  Image Segmentation 6/29 Definitions and notation Definition Undirected graph G = ( V , E ) consists of the set of vertices V and the the set of edges E such that each edge e E is associated with an unordered pair of vertices { v , w } V called endpoints . If v = w the edge is called a loop . We say that an edge connects its endpoints. Vertices connected by an edge are called adjacent . An edge is said to be incident on each of its endpoints. A vertex on which no edges are incident is called isolated . Graph G is directed (digraph) if the edges are directed, that is, associated with ordered pairs ( v , w ). A graph with no loops and no parallel edges is called a simple graph . 7/29 Complete graphs A simple graph with n vertices and exactly one edge connecting each pair of distinct vertices is denoted by K n and called a complete graph ....
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 Spring '08
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 Algebra

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