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Discrete Structures – CNS2300

Discrete Structures – CNS2300. Text Discrete Mathematics and Its Applications (5 th Edition) Kenneth H. Rosen Chapter 8 Graphs. Section 8.3. Representing Graphs and Graph Isomorphism. Isomorphic Graphs.

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Discrete Structures – CNS2300

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  1. Discrete Structures – CNS2300 Text Discrete Mathematics and Its Applications (5th Edition) Kenneth H. Rosen Chapter 8 Graphs

  2. Section 8.3 Representing Graphs and Graph Isomorphism

  3. Isomorphic Graphs When there is a one-to-one correspondence between their vertex sets of two graphs that preserves edges we say that the two graphs are isomorphic.

  4. Representing Graphs A way to represent a graph with no multiple edges is to use an adjacenylist which specifies the vertices that are adjacent to each vertex of the graph.

  5. e g a b d Vertex Adjacent Vertices a b,c b a, c, d c a, b d b, e, f, g e d,g f d, g f c Simple Graph Example

  6. b c d Vertex Adjacent Vert a a, f b a, f,e c b, d d b e c, e f b, c, e e a f Directed Graph, Adjacency List

  7. Adjacency Matrices The adjacency matrix A [aij] for a graph G is

  8. e g a b d f c é ù 0 1 1 0 0 0 0 ê ú 1 0 1 1 0 0 0 ê ú ê ú 1 1 0 0 0 0 0 ê ú 0 1 0 0 1 1 1 ê ú ê ú 0 0 0 1 0 0 1 ê ú 0 0 0 1 0 0 1 ê ú ê ú 0 0 0 1 1 1 0 ë û

  9. e g a b d f c

  10. b c d e a f

  11. Incidence Matrices Let G = (V,E) be an undirected graph. Suppose that v1, v2, …, vn are the vertices and e1, e2, …, em are the edges of G. Then the incidence matrix with respect to this ordering of V and E is the nxm matrix M = [mij] where

  12. e1 e2 e3 e4 e5 e6 v1 v2 v3 v4 v5 v1 v2 v3 e6 e3 e4 e5 e1 e2 v4 v5

  13. e6 a e3 b d e7 e4 e1 e8 e5 c e1 e2 e3 e4 e5 e6 e7 e8 e9 a b c d e2 e9

  14. Isomorphisms of Graphs The simple graphs G1=(V1,E1) and G2=(V2,E2) are isomorphic if there is a one-to-one and onto function f from V1 to V2 with the property that a and b are adjacent in G1 if and only if f(a) and f(b) are adjacent in G2, for all a and b in V1. Such a function f is called an isomorphism.

  15. Isomorphisms of Graphs Two graphs are isomorphic when there is a one-to-one correspondence between vertices of the two graphs that preserves the adjacency relationship.

  16. r w a b c d e x z y r b w x a c z y e d Examples

  17. a d r w c x z y b e r w a b x y e c d z More Examples

  18. finished

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