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Cayley Digraphs (directed graphs) of Groups

Cayley Digraphs (directed graphs) of Groups. A graphical representation of a group. Why should we study?. Visualize a group Connects groups and graphs Applications in computer science A good review of previous algebra Its fun. Definition:

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Cayley Digraphs (directed graphs) of Groups

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  1. Cayley Digraphs (directed graphs) of Groups A graphical representation of a group.

  2. Why should we study? • Visualize a group • Connects groups and graphs • Applications in computer science • A good review of previous algebra • Its fun

  3. Definition: Let G be a finite group and S a set of generators for G. We define a digraph Cay(S:G), called the Cayley digraph of G with generating set S, as follows: 1. Each element of G is a vertex of Cay(S:G) 2. For x and y in G, there is an arc from x to y iff xs=y for some s in S.

  4. Examples • Z6 • Z3(+)Z2 • D4

  5. That’s great, so what can we do with it?

  6. Hamiltonian Circuits and Paths • A Hamiltonian Circuit exists if you can start at some vertex and move along arcs in such a way that you hit every vertex exactly once before returning to the vertex you started with. • A Hamiltonian Path is defined in the same way without the condition of returning to the starting vertex.

  7. Examples • Cay({(1,0),(0,1)}:Z3(+)Z2 • Cay({R90,H}:D4)

  8. Theorems • Cay({(1,0),(0,1)}:Zm(+)Zn) does not have a Hamiltonian circuit when m and n are relatively prime and greater than one. • Cay({(1,0),(0,1)}:Zm(+)Zn) has a Hamiltonian circuit when n divides m. ex:(Z4(+)Z2)

  9. Theorems (cont.) • Let G be a finite Abelian group, and let S be any (nonempty) generating set for G. Then Cay(S:G) has a Hamiltonian path. ex: Z3(+)Z2

  10. Computer Science • In 1981, Hamiltonian paths in Cayley digraphs were used in an algorithm for creating computer graphics of Escher-type repeating patterns in the hyperbolic plane.

  11. Escher’s Circle Limit 1

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