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Section 1.2 Isomorphisms

Section 1.2 Isomorphisms. By Christina Touhey and Sarah Graham. a. b. 1. 2. 3. f. 4. e. 6. 5. d. c. Section 1.2 Isomorphisms. Isomorphism-A one-to-one correspondence of vertices that preserves adjacency Two graphs G , are isomorphic if

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Section 1.2 Isomorphisms

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  1. Section 1.2 Isomorphisms By Christina Touhey and Sarah Graham

  2. a b 1 2 3 f 4 e 6 5 d c Section 1.2 Isomorphisms • Isomorphism-A one-to-one correspondence of vertices that preserves adjacency • Two graphs G , are isomorphic if • There exists a one-to-one correspondence between vertices in G and . • If and only if the corresponding pair of vertices are adjacent in and a pair of vertices are adjacent in G.

  3. Recall from Section 1.1 • Degree of a vertex- number of edges incident to the vertex have the same degree • In-degree- number of edges pointed in toward the vertex • Out-degree-number of edges point away from the vertex a b c d

  4. Section 1.2 Isomorphisms • Complement- a graph, with the same set of vertices but now with edges between exactly those pairs of vertices not linked in G G

  5. Section 1.2 Isomorphisms • Subgraph- a graph formed by a subset of vertices and edges of a larger graph. • If the two larger subgraphs are isomorphic then the corresponding subgraphs must also be isomorphic • Complete graph- a graph in which each vertex is adjacent to all the other vertices, Kn. K3 K5

  6. An approach to checking isomorphism: • Count the vertices. The graphs must have an equal number. • Count the edges. The graphs must have an equal number. • Check vertex degrees. Each graph must have the same amount of vertices with equal degree. • Check the subgraphs for isomorphism. If the subgraphs are not isomorphic the larger graphs are not as well. • Check for a one-to-one correspondence between vertices of one graph and vertices of the other.

  7. 1 4 e 5 h 8 f g 7 6 b Example 1 (p. 21-22) Are the two graphs isomorphic? a d 2 3 c Both graphs have 8 vertices and 10 edges. b, d, f, h, 3, 4, 7, 8 all have degree 2 The other vertices have degree 3. Taking a subgraph of degree 2, we find: b, d, f, h have no common edges there are edges (4,3) and (8,7), the subgraph is not isomorphic. Note: If the subgraphs are not isomorphic than the larger graphs are not isomorphic. However, if the subgraphs are isomorphic the larger graphs are not necessarily isomorphic.

  8. d a b c e g f 5 6 7 3 4 For the class to try: Are these pairs of graphs isomorphic? 3 a e 5 1 #1 c d b f 6 2 Isomorphic: (a-1, b-5, c-4, d-3, e-2, f-6) 4 2 1 #2 Not Isomorphic: The vertices when matched up are not adjacent. There is no one-to-one correspondence.

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