A380149 Characteristic polynomial of the tesseract graph: a(n) = n^6*(n^2-16)*(n^2-4)^4.

0, -1215, 0, -3189375, 0, 27348890625, 978447237120, 15920336210625, 163074539520000, 1214314872035265, 7134511104000000, 34856907746165505, 146828238520320000, 547377978676010625, 1841813423998894080, 5678883183381890625, 16238028554439229440, 43474602051830210625, 109846357522513920000

Offset: 0

Author

Darío Clavijo, Jan 13 2025

Keywords

• sign
• easy

Comments

Given that the eigenvalues of the adjacency matrix of the tesseract graph are: {4,2,0,-2,-4} and their multiplicities are defined by binomial(4,k) for k = 0..4 which results in {1,4,6,4,1}, and the characteristic polynomial is given by P(x) = Prod_{k=1..j} (x-lambda_k)^m_k with j=5, substitute the eigenvalues and their multiplicities as:

k | eigenvalue(lambda_k) | multiplicity(m_k) | term

---+----------------------+-------------------+---------

1 | 4 | 1 | (x-4)^1

2 | 2 | 4 | (x-2)^4

3 | 0 | 6 | (x)^6

4 | -2 | 4 | (x+2)^4

5 | -4 | 1 | (x+4)^1

This results in the characteristic polynomial:

P(x) = (x-4) * (x-2)^4 * (x)^6 * (x+2)^4 * (x+4).

Also all terms are a(n) = 0 mod 5.

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Characteristic Polynomial.
• Eric Weisstein's World of Mathematics, Tesseract Graph.
• Eric Weisstein's World of Mathematics, Graph Spectrum.
• Wikipedia, Hypercube Graph.

Crossrefs

Cf. A001014, A028347, A028566.

Programs

Mathematica

A380149[n_] := n^6*(n^2 - 16)*(n^2 - 4)^4; Array[A380149, 20, 0] (* Paolo Xausa, Jan 21 2025 *)

Python

a = lambda n: (n**6)*(n**2-16)*(n**2-4)**4
print([a(n) for n in range(0,19)])

Formula

a(n) = -4096*n^6 + 4352*n^8 - 1792*n^10 + 352*n^12 - 32*n^14 + n^16.