A245290 Number of normalized graph Laplacian matrices of nonempty labeled graphs of 2n vertices that are separable in C^2 X C^n as density matrices in quantum mechanics.

1, 31, 5119, 9961471, 259577085951, 94554701453852671, 494214691850093043122175, 37747948215762478445361018961919, 42694960288928350006693371507341885702143, 722273364120299921501331975953872089285372151857151

Offset

1,2

Comments

Since separability is not invariant under graph isomorphism, all 2^(n(2n-1))-1 nonzero Laplacian matrices are treated as different. A nonzero Laplacian matrix different from the complete graph is separable in C^2 X C^n if and only if its complement is. Since the complete graph is separable, this means that a(n) is odd for all n.

Links

Table of n, a(n) for n=1..10.

Chai Wah Wu, Conditions for separability in generalized Laplacian matrices and diagonally dominant matrices as density matrices, Physics Letters A, 351 (2006), 18-22.

Chai Wah Wu, Graphs whose normalized Laplacian matrices are separable as density matrices in quantum mechanics, arXiv:1407.5663 [quant-ph], 2014.

Formula

a(n) + A245291(n) = 2^(n*(2*n-1))-1.

a(n) = 2^(n*(n-1))*A229865(n)-1.

Crossrefs

Cf. A245291, A229865.

Sequence in context: A115736 A110848 A214109 * A090681 A297767 A065756

Adjacent sequences: A245287 A245288 A245289 * A245291 A245292 A245293

Keyword

nonn

Author

Chai Wah Wu, Jul 16 2014

Status

approved