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Integer Invariants of Abelian Cayley Graphs Deelan Jalil James Madison University July 26, 2013 Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 1 / 29 Cube example. . Deelan Jalil (2013) Integer Invariants of

  1. Integer Invariants of Abelian Cayley Graphs Deelan Jalil James Madison University July 26, 2013 Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 1 / 29

  2. Cube example. . Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 2 / 29

  3. Cube example. (0 , 0 , 0) (0 , 0 , 1) (0 , 1 , 0) (0 , 1 , 1) (1 , 0 , 0) (1 , 0 , 1) (1 , 1 , 0) (1 , 1 , 1) (0 , 0 , 0)   (0 , 0 , 1)     (0 , 1 , 0)     (0 , 1 , 1)     (1 , 0 , 0)     (1 , 0 , 1)     (1 , 1 , 0)   (1 , 1 , 1) Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 3 / 29

  4. Cube example. (0 , 0 , 0) (0 , 0 , 1) (0 , 1 , 0) (0 , 1 , 1) (1 , 0 , 0) (1 , 0 , 1) (1 , 1 , 0) (1 , 1 , 1) (0 , 0 , 0)  0 1 1 0 1 0 0 0  (0 , 0 , 1) 1 0 0 1 0 1 0 0     (0 , 1 , 0) 1 0 0 1 0 0 1 0     (0 , 1 , 1) 0 1 1 0 0 0 0 1     (1 , 0 , 0) 1 0 0 0 0 1 1 0     (1 , 0 , 1) 0 1 0 0 1 0 0 1     (1 , 1 , 0) 0 0 1 0 1 0 0 1   (1 , 1 , 1) 0 0 0 1 0 1 1 0 Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 4 / 29

  5. Cube example. Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 5 / 29

  6. Cube example. 1 0 0 0 0 0 0 0   0 1 0 0 0 0 0 0     0 0 1 0 0 0 0 0     0 0 0 1 0 0 0 0     0 0 0 0 1 0 0 0     0 0 0 0 0 1 0 0     0 0 0 0 0 0 3 0   0 0 0 0 0 0 0 3 Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 6 / 29

  7. Smith normal form. Given any integer matrix A , we can perform row and column operations so that:   0 0 · · · 0 d 1 0 d 2 0 · · · 0     0 0 · · · 0 d 3 A = where d 1 | d 2 , d 2 | d 3 , ... , d n − 1 | d n .    . . . .  ... . . . .   . . . .   0 0 0 · · · d n Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 7 / 29

  8. Smith normal form. Given any integer matrix A , we can perform row and column operations so that:   0 0 · · · 0 d 1 0 d 2 0 · · · 0     0 0 · · · 0 d 3 A = where d 1 | d 2 , d 2 | d 3 , ... , d n − 1 | d n .    . . . .  ... . . . .   . . . .   0 0 0 · · · d n Swap any two rows/columns. Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 7 / 29

  9. Smith normal form. Given any integer matrix A , we can perform row and column operations so that:   0 0 · · · 0 d 1 0 d 2 0 · · · 0     0 0 · · · 0 d 3 A = where d 1 | d 2 , d 2 | d 3 , ... , d n − 1 | d n .    . . . .  ... . . . .   . . . .   0 0 0 · · · d n Swap any two rows/columns. Multiply any row/column by a nonzero integer. Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 7 / 29

  10. Smith normal form. Given any integer matrix A , we can perform row and column operations so that:   0 0 · · · 0 d 1 0 d 2 0 · · · 0     0 0 · · · 0 d 3 A = where d 1 | d 2 , d 2 | d 3 , ... , d n − 1 | d n .    . . . .  ... . . . .   . . . .   0 0 0 · · · d n Swap any two rows/columns. Multiply any row/column by a nonzero integer. Add a integer multiple of one row/column to another. Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 7 / 29

  11. Cube example part 2. (0 , 0 , 0) (0 , 0 , 1) (0 , 1 , 0) (0 , 1 , 1) (1 , 0 , 0) (1 , 0 , 1) (1 , 1 , 0) (1 , 1 , 1) (0 , 0 , 0)  0 1 1 0 1 0 0 0  (0 , 0 , 1) 1 0 0 1 0 1 0 0     (0 , 1 , 0) 1 0 0 1 0 0 1 0     (0 , 1 , 1) 0 1 1 0 0 0 0 1     (1 , 0 , 0) 1 0 0 0 0 1 1 0     (1 , 0 , 1) 0 1 0 0 1 0 0 1     (1 , 1 , 0) 0 0 1 0 1 0 0 1   (1 , 1 , 1) 0 0 0 1 0 1 1 0 Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 8 / 29

  12. Cube example part 2. (0 , 0 , 0) (0 , 0 , 1) (0 , 1 , 0) (0 , 1 , 1) (1 , 0 , 0) (1 , 0 , 1) (1 , 1 , 0) (1 , 1 , 1) (0 , 0 , 0)  0 − 1 − 1 0 − 1 0 0 0  (0 , 0 , 1) − 1 0 0 − 1 0 − 1 0 0     (0 , 1 , 0) − 1 0 0 − 1 0 0 − 1 0     (0 , 1 , 1) 0 − 1 − 1 0 0 0 0 − 1     (1 , 0 , 0) − 1 0 0 0 0 − 1 − 1 0     (1 , 0 , 1) 0 − 1 0 0 − 1 0 0 − 1     (1 , 1 , 0) 0 0 − 1 0 − 1 0 0 − 1   (1 , 1 , 1) 0 0 0 − 1 0 − 1 − 1 0 Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 9 / 29

  13. Cube example part 2. (0 , 0 , 0) (0 , 0 , 1) (0 , 1 , 0) (0 , 1 , 1) (1 , 0 , 0) (1 , 0 , 1) (1 , 1 , 0) (1 , 1 , 1) (0 , 0 , 0)  3 − 1 − 1 0 − 1 0 0 0  (0 , 0 , 1) − 1 3 0 − 1 0 − 1 0 0     (0 , 1 , 0) − 1 0 3 − 1 0 0 − 1 0     (0 , 1 , 1) 0 − 1 − 1 3 0 0 0 − 1     (1 , 0 , 0) − 1 0 0 0 3 − 1 − 1 0     (1 , 0 , 1) 0 − 1 0 0 − 1 3 0 − 1     (1 , 1 , 0) 0 0 − 1 0 − 1 0 3 − 1   (1 , 1 , 1) 0 0 0 − 1 0 − 1 − 1 3 Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 10 / 29

  14. Cube example part 2. 1 0 0 0 0 0 0 0   0 1 0 0 0 0 0 0     0 0 1 0 0 0 0 0     0 0 0 1 0 0 0 0     0 0 0 0 2 0 0 0     0 0 0 0 0 8 0 0     0 0 0 0 0 0 24 0   0 0 0 0 0 0 0 0 Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 11 / 29

  15. Smith normal form. Given any integer matrix A , we can perform row and column operations so that:   d 1 0 0 · · · 0 0 d 2 0 · · · 0     0 0 · · · 0 d 3 A = where d 1 | d 2 , d 2 | d 3 , · · · , d n − 1 | d n .    . . . .  ... . . . .   . . . .   0 0 0 · · · d n Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 12 / 29

  16. Smith normal form. Given any integer matrix A , we can perform row and column operations so that:   d 1 0 0 · · · 0 0 d 2 0 · · · 0     0 0 · · · 0 d 3 A = where d 1 | d 2 , d 2 | d 3 , · · · , d n − 1 | d n .    . . . .  ... . . . .   . . . .   0 0 0 · · · d n Invariant factors. Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 12 / 29

  17. Smith normal form. Given any integer matrix A , we can perform row and column operations so that:   d 1 0 0 · · · 0 0 d 2 0 · · · 0     0 0 · · · 0 d 3 A = where d 1 | d 2 , d 2 | d 3 , · · · , d n − 1 | d n .    . . . .  ... . . . .   . . . .   0 0 0 · · · d n Invariant factors. Elementary divisors. Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 12 / 29

  18. Invariant factors vs. Elementary divisors 1 0 0 0   0 3 0 0      0 0 6 0    0 0 0 60 Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 13 / 29

  19. Invariant factors vs. Elementary divisors 1 0 0 0   0 3 0 0      0 0 6 0    0 0 0 60 Invariant factors: 1 , 3 , 6 , 60 Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 13 / 29

  20. Invariant factors vs. Elementary divisors 1 0 0 0   0 3 0 0      0 0 6 0    0 0 0 60 Invariant factors: 1 , 3 , 6 , 60 1 0 0 0   0 3 0 0      0 0 3 · 2 0    5 · 3 · 2 2 0 0 0 Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 13 / 29

  21. Invariant factors vs. Elementary divisors 1 0 0 0   0 3 0 0      0 0 6 0    0 0 0 60 Invariant factors: 1 , 3 , 6 , 60 1 0 0 0   0 3 0 0      0 0 3 · 2 0    5 · 3 · 2 2 0 0 0 Elementary divisors: 2 , 2 2 , 3 , 3 , 3 , 5 Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 13 / 29

  22. Back to the title. Integer Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 14 / 29

  23. Back to the title. Integer Invariants Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 14 / 29

  24. Back to the title. Integer Invariants Abelian Cayley Graphs Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 14 / 29

  25. Back to the title. Integer Invariants Abelian Cayley Graphs Graph: edges and vertices Deelan Jalil (2013) Integer Invariants of Abelian Cayley Graphs July 26, 2013 14 / 29

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