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MAT 2720 Discrete Mathematics

MAT 2720 Discrete Mathematics. Section 8.5 Representations of Graphs. http://myhome.spu.edu/lauw. Goals. In order to use computers to analyze graphs, we need formal representations Represent Graphs with Matrices Incidence Matrix Adjacency Matrix. Incidence Matrix. Adjacency Matrix.

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MAT 2720 Discrete Mathematics

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  1. MAT 2720Discrete Mathematics Section 8.5 Representations of Graphs http://myhome.spu.edu/lauw

  2. Goals • In order to use computers to analyze graphs, we need formal representations • Represent Graphs with Matrices • Incidence Matrix • Adjacency Matrix

  3. Incidence Matrix

  4. Adjacency Matrix i-jth entry = no. of edges incident on vertex i and j

  5. Question… Is there a meaning for An ?

  6. Answer… Yes…when the graph is simple.

  7. Example 1

  8. Example 1

  9. Example 1

  10. Example 1 No. of path from vertex a to vertex c …. … with length 2

  11. Example 1 No. of path from vertex c to vertex c …. … with length 2

  12. Example 1 … with length 4 No. of path from vertex a to vertex e ….

  13. Theorem If A is the adjacency matrix of a simple graph, the ijth entry of An is equal to the number of paths of length n from vertex i to vertex j, n=1,2,… Proof: Induction on n

  14. MAT 2720Discrete Mathematics Section 8.6 Isomorphisms of Graphs http://myhome.spu.edu/lauw

  15. Goals • Develop the notion of graphs with “different” structures. • Develop the notion of graphs with the “same” structure. • Isomorphisms of Graphs

  16. Example 1 The following 2 graphs have the “same” structure.

  17. Ideas 1. No. of vertices and edges are the same.

  18. Ideas 1. No. of vertices and edges are the same. 2. The connections between vertices are the same.

  19. Definition Version 1

  20. Definition Version 2

  21. Observations f is 1-1, onto guarantees the two graphs have the same number of vertices.

  22. Observations g is 1-1, onto guarantees the two graphs have the same number of edges.

  23. Observations g((x,y))=(f(x),f(y)) guarantees the connections between vertices are the same.

  24. Observations g((x,y))=(f(x),f(y)) guarantees the connections between vertices are the same.

  25. To prove that 2 graphs are isomorphic, it suffices to … … exhibit a pair of isomorphisms f and g.

  26. Example 1

  27. Example 1 It remains to show that

  28. One Last Observation … What about their adjacent matrices? How are they related?

  29. One Last Observation … What about their adjacent matrices? How are they related?

  30. Theorem Two graphs are isomorphic if and only if for some ordering of their vertices, their adjacent matrices are equal.

  31. Facts • To show that 2 graphs are isomorphic could be difficult (either use the definition or adjacent matrices). • To show that they are NOT isomorphic is relatively easier.

  32. Invariants If 2 graphs are isomorphic, they have some common properties (invariants). 1. Same no. of vertices and edges. 2. (v)=(f(v)) (“has a vertex of degree k”) 3. Length of corresponding simple cycles are the same (“has a simple cycle of length k”).

  33. Invariants 2 graphs are NOT isomorphic if they do not share any one of the invariants above.

  34. Example 2

  35. Example 3

  36. Example 4

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