lecture19

lecture19 - COMPSCI 102 Discrete Mathematics for Computer...

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Unformatted text preview: COMPSCI 102 Discrete Mathematics for Computer Science Graphs II Lecture 18 Recap Theorem: Let G be a graph with n nodes and e edges The following are equivalent: 1. G is a tree (connected, acyclic) 3. G is connected and n = e + 1 4. G is acyclic and n = e + 1 5. G is acyclic and if any two non-adjacent points are joined by an edge, the resulting graph has exactly one cycle 2. Every two nodes of G are joined by a unique path The number of labeled trees on n nodes is n n-2 Cayleys Formula A graph is planar if it can be drawn in the plane without crossing edges Eulers Formula If G is a connected planar graph with n vertices, e edges and f faces, then n e + f = 2 A coloring of a graph is an assignment of a color to each vertex such that no neighboring vertices have the same color Graph Coloring Spanning Trees A spanning tree of a graph G is a tree that touches every node of G and uses only edges from G Every connected graph has a spanning tree Finding Optimal Trees Trees have many nice properties (uniqueness of paths, no cycles, etc.) We may want to compute the best tree approximation to a graph If all we care about is communication , then a tree may be enough. We want a tree with smallest communication link costs Finding Optimal Trees Problem: Find a minimum spanning tree , that is, a tree that has a node for every node in the graph, such that the sum of the edge weights is minimum 4 8 7 9 6 11 9 5 8 7 Tree Approximations Finding an MST: Kruskals Algorithm Create a forest where each node is a separate tree Make a sorted list of edges S While S is non-empty:...
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lecture19 - COMPSCI 102 Discrete Mathematics for Computer...

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