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A330730 Determinant of the adjacency matrix for the n^2 X n^2 sudoku graph. 1
1, 0, -5103, 321958994845368320000000000000, 485935624939288938823190812356750274771920395378553038379099149324247264303267002105712890625 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The sudoku graph has n^4 vertices, corresponding to the cells of an n^2 by n^2 sudoku grid. Two vertices are adjacent if and only if the cells belong to the same row, column, or n by n block. Consequently, the adjacency matrix is n^4 by n^4.

All of the eigenvalues of the adjacency matrix are integers. The spectrum (for n>1) is

       -1-n with multiplicity 2n^3-4n^2+2n,

         -1 with multiplicity n^4-2n^3+n^2,

   n^2-2n-1 with multiplicity n^2-2n+1,

    n^2-n-1 with multiplicity 2n^2-2n,

  2n^2-2n-1 with multiplicity 2n-2, and

  3n^2-2n-1 with multiplicity 1.

LINKS

Table of n, a(n) for n=0..4.

Torsten Sander, Sudoku graphs are integral, The Electronic Journal of Combinatorics, 16(1), 25 (2009).

FORMULA

a(n) = (-1-n)^(2n^3-4n^2+2n) * (n^2-2n-1)^(n^2-2n+1) * (n^2-n-1)^(2n^2-2n) * (2n^2-2n-1)^(2n-2) * (3n^2-2n-1).

EXAMPLE

For n = 2 the adjacency matrix (with determinant -5103) is:

  [[0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0],

   [1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0],

   [1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0],

   [1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1],

   [1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0],

   [1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0],

   [0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0],

   [0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1],

   [1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0],

   [0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0],

   [0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1],

   [0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1],

   [1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1],

   [0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1],

   [0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1],

   [0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0]]

MAPLE

A330730 := proc(n) (-1-n)^(2*n^3-4*n^2+2*n) * (n^2-2*n-1)^(n^2-2*n+1) * (n^2-n-1)^(2*n^2-2*n) * (2*n^2-2*n-1)^(2*n-2) * (3*n^2-2*n-1); end;

MATHEMATICA

A330730[n_] := (-1-n)^(2n^3-4n^2+2n) (n^2-2n-1)^(n^2-2n+1) (n^2-n-1)^(2n^2-2n) (2n^2-2n-1)^(2n-2) (3n^2-2n-1); Array[a, 0, 5]

PROG

(Python) a330730 = lambda n: (-1-n)**(2*n**3-4*n**2+2*n) * (n**2-2*n-1)**(n**2-2*n+1) * (n**2-n-1)**(2*n**2-2*n) * (2*n**2-2*n-1)**(2*n-2) * (3*n**2-2*n-1)

CROSSREFS

Sequence in context: A223400 A135842 A224492 * A219329 A025398 A025397

Adjacent sequences:  A330727 A330728 A330729 * A330731 A330732 A330733

KEYWORD

sign,easy

AUTHOR

David Radcliffe, Dec 28 2019

STATUS

approved

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Last modified February 18 02:57 EST 2020. Contains 332006 sequences. (Running on oeis4.)