1. CSE 222 Systems Programming Graph Theory Basics Dr. Jim Holten

2. CSE 222 2 01/21/09 Introduction Basic Definitions Concepts and Algorithms Applications

3. CSE 222 3 01/21/09 Basic Definitions Nodes aka Vertices Links aka Edges aka Arcs

4. CSE 222 4 01/21/09 Basic Definitions Subgraphs Undirected vs Directed Graphs Paths, Distances, and Graph Diameter Connected Graphs Components of a Graph Fully connected graph or subgraph

5. CSE 222 5 01/21/09 Undirected Graph

6. CSE 222 6 01/21/09 Directed Graph

7. CSE 222 7 01/21/09 Basic Definitions Node Attributes aka Labels, Indices, Weights, Categories, etc. Link Attributes aka Labels, Indices, Weights, Categories, etc. A Semantic Graph is a graph partitioned based on node or link attribute values

8. CSE 222 8 01/21/09 Semantic Graph Authors Documents Concepts

9. CSE 222 9 01/21/09 More Definitions A Hypernode replaces a collection of nodes and their included links. A Hypergraph is any graph that includes one or more hypernodes.

10. CSE 222 10 01/21/09 Example of Hypergraph Authors Documents Concepts

11. CSE 222 11 01/21/09 Hypergraph of Social Network

12. CSE 222 12 01/21/09 Some Concepts A cluster is a subgraph made up of set of highly interconnected nodes. A critical node is a node on a unique path between two nodes or subgraphs of interest. A critical link is a link on a unique path between two nodes or subgraphs of interest.

13. CSE 222 13 01/21/09 Related Algorithms Finding clusters is defined many ways –Fully connected subgraph or just better than some threshold on connectedness? –Is a node only connected to a node in the cluster to be included in the cluster? Finding critical paths, nodes, or links

14. CSE 222 14 01/21/09 Related Concepts Projections of nodes or subgraphs are mappings via directed links. The domain subgraph includes the projection's "from" nodes. The range subgraph includes the projection's "to" nodes. Basis Node Set is a minimal set of nodes that have a path to every other node

15. CSE 222 15 01/21/09 Projection Mapping

16. CSE 222 16 01/21/09 Subset Projections

17. CSE 222 17 01/21/09 Related Concepts There are multiple ways of looking at a single collection of nodes and links. Are multiple graphs just a single multi- component graph? Interdependencies among multiple graphs can be useful.

18. CSE 222 18 01/21/09 Interdependencies

19. CSE 222 19 01/21/09 Applications Software Organization Activity Sequence Organization Interactions between processes

20. CSE 222 20 01/21/09 Software Organization Flow charts Data interdependencies Interactions between processes

21. CSE 222 21 01/21/09 Data Organization Chart

22. CSE 222 22 01/21/09 Processes Organization Chart

23. CSE 222 23 01/21/09 Activity Sequence Organization Gantt and Pert charts Project resource planning charts Storyboarding

24. CSE 222 24 01/21/09 Storyboard of the Software Development Process

25. CSE 222 25 01/21/09 Network Modeling Applications Communications (Internet, etc.) Energy distribution (electric power, natural gas, etc.) Transportation (highways, railways, waterways, etc) Social associations Financial and product flows

26. CSE 222 26 01/21/09 Infrastructure Networks

27. CSE 222 27 01/21/09 Critical Infrastructure Interdependencies

28. CSE 222 28 01/21/09 Physics Models Mesh Models –Space segmented into volumes –Volumes surrounded by faces and edges –Edges meet at vertices Model element type dependencies form a graph of interdependent element sets

29. CSE 222 29 01/21/09 Mesh Models

30. CSE 222 30 01/21/09 Element Type Dependencies

31. CSE 222 31 01/21/09 Summary Graphs are a significant factor in the present and the future of computer science!