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Graphs and Isomorphisms

Graphs and Isomorphisms. Backyards of Old Houses in Antwerp in the Snow Van Gogh. Discrete Structures (CS 173) Madhusudan Parthasarathy, University of Illinois. Administrative. How was the exam? Midterm graded by Friday next week (hopefully)

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Graphs and Isomorphisms

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  1. Graphs and Isomorphisms Backyards of Old Houses in Antwerp in the Snow Van Gogh Discrete Structures (CS 173) Madhusudan Parthasarathy, University of Illinois

  2. Administrative • How was the exam? • Midterm graded by Friday next week (hopefully) • Remember: homework this week and discussions this week continue…

  3. Proof with one-to-one Let A, B be subsets of reals. Claim: Any strictly increasing function from A to B is one-to-one. Definition: is one-to-oneiff Definition: is strictly increasingiff overhead

  4. Permutations Ordered selection Suppose I have 6 gems, and you get to choose 1. How many different combinations of gems can you choose? Suppose I have gems and want to put them in a row from left to right. How many different ways can I arrange them? Suppose I have 6 gems and want to put three of them in a row from left to right. How many different ways can I arrange them? Unordered selection Suppose I have 6 gems, and you get to choose 2. How many different combinations of gems can you choose? Suppose I have gems, and you choose . How many combinations?

  5. Permutations Suppose with and . How many different one-to-one functions can I create? How many ways can I rearrange the letters in “nan”? How many ways can I rearrange the letters in “yellowbelly”?

  6. Graphs • How to represent graphs? • What are the properties of a graph? • Degrees, special types • When are two graphs isomorphic, having the same structure?

  7. Fastest path from Chicago to Bloomington?

  8. Fastest path from Chicago to Bloomington?

  9. Fastest path from Chicago to Bloomington? start 1 2 3 end 4

  10. Fastest path from Chicago to Bloomington? start C 30 1 20 2 4 15 1 2 35 120 3 110 4 60 90 end B 3

  11. Other applications of graphs • Modeling the flow of a network • Traffic, water in pipes, bandwidth in computer networks, etc.

  12. Basics of graphs Graph = (V, E) Terminology: vertex/node, edge, neighbor/adjacent, directed vs. undirected, simple graph, degree of a node overhead

  13. Degrees and handshaking theorem Loops count twice overhead

  14. Types of graphs : complete graph with nodes How many edges does each type have? overhead

  15. Types of graphs : cycle graph with nodes : wheel graph with nodes How many edges does each type have? overhead

  16. Isomorphism An isomorphism from to is a bijection such that any pair of nodes and are joined by an edge iff and are joined by an edge Two graphs are isomorphicif there is an isomorphism between them overhead

  17. Isomorphism examples An isomorphism from to is a bijection such that any pair of nodes and are joined by an edge iff and are joined by an edge overhead

  18. Isomorphism examples An isomorphism from to is a bijection such that any pair of nodes and are joined by an edge iff and are joined by an edge overhead

  19. Requirements for graphs to be isomorphic overhead

  20. Requirements for two graphs to be isomorphic • Same number of nodes and edges • Same number of nodes of degree • Every subgraph in the first must have a matching subgraph in the second

  21. Automorphism: an isomorphism from a graph to itself • Automorphisms identify symmetries in the graph • How many different automorphisms? overhead

  22. Small graphs without non-trivial automorphism? overhead

  23. Isomorphism is an equivalence relation: reflexive, symmetric, and transitive

  24. Things to remember • A graph is defined by a set of nodes and a set of edges that connect them • Be able to identify types of graphs and degrees of nodes • Be able to identify isomorphisms (or lack thereof) between graphs

  25. Next week: more graphs and induction

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