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CSE 321 Discrete Structures Winter 2008 Lecture 25 Graph Theory

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Graph formalism –G = (V, E) –Vertices –Edges Directed Graph –Edges ordered pairs Undirected Graph –Edges sets of size two

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Graph examples Communication Networks Road networks

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Social networks Community Graph –Linked In, Face Book Transactions –Ebay Authorship –Erdos Number

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The web graph

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Page Rank Determine the value of a page based on link analysis Model of randomly traversing a graph –Weighting factors on nodes –Damping (random transitions)

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Graph terminology NeighborhoodDegree

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Degree sequence Find a graph with degree sequence –3, 3, 2, 1, 1 Find a graph with degree sequence –3, 3, 3, 3, 3

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Handshake Theorem

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Directed Degree Theorem

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Special Graphs Complete Graphs K n Cycle C n Hypercube Q n Mesh M n,m

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Bipartite Graphs

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2-coloring A graph is two colorable iff all cycles have even length

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Graph Representations Adjacency Lists Adjacency Matrices Incidence Matrices

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Graph Connectivity

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Strong connectivity vs. Weak Connectivity

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Strongly Connected Components

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Counting Paths Let A be the Adjacency Matrix. What is A 2 ? d c b e a

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Graph Isomorphism I Are these two graphs the same? ad cb w x y z

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Graph Isomorphism II Are these graphs the same?

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Graph Isomorphism III Are these graphs the same?

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