9. Graphs - 8/27/09 11 8/27/09 MATH 224 Discrete...

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Unformatted text preview: 8/27/09 11 8/27/09 MATH 224 Discrete Mathematics Formally a graph is just a collection of unordered or ordered pairs, where for example, if {a,b} G if a, b V. Here G is the graph and V is called the set of nodes. So a graph G(V, E) is a set of objects V (of any type), which are called nodes or sometimes vertices. And E is actually the set of pairs, which are called edges or sometimes called links. The edges may be unordered pairs {a,b} or ordered pairs (a,b). With unordered pairs {a,b} and {b, a} are equivalent. But with ordered pairs the order matters so (a,b) is not the same as (b,a). Graphs 8/27/09 22 8/27/09 MATH 224 Discrete Mathematics Undirected Graph 1 3 4 2 5 Edges = {{0,2}, {1,2}, {1,3}, {1,4}, {2,4}, {2,5}, } 8/27/09 33 8/27/09 MATH 224 Discrete Mathematics Directed Graph 1 3 4 2 5 Edges = {(0,2), (1,2), (3,1), (1,4), (2,4), (4,2), } 8/27/09 44 8/27/09 MATH 224 Discrete Mathematics Simple Graphs Multiple edges between pairs of nodes and self loops are not allowed in simple graphs. 1 3 4 2 5 Multiple Edges Self Loop 8/27/09 55 8/27/09 MATH 224 Discrete Mathematics Paths 1 3 4 2 5 Paths are a sequence of nodes, e.g., 0, 2, 5, 3, where no node is repeated and consecutive nodes correspond to an edge. 8/27/09 66 8/27/09 MATH 224 Discrete Mathematics Not a Path (called a walk) 1 3 4 2 5 Note that some nodes are repeated: 1, 2, 4, 1, 3, 5 8/27/09 77 8/27/09 MATH 224 Discrete Mathematics Cycle 1 3 4 2 5 A cycle is a sequence of nodes, e.g., 1, 2, 4, 5, 3, 1, where A cycle is a sequence of nodes, e....
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9. Graphs - 8/27/09 11 8/27/09 MATH 224 Discrete...

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