A198637 Coefficient table for the characteristic polynomials of the adjacency matrices of the cycle graphs C_n.

1, 0, 1, -4, 0, 1, -2, -3, 0, 1, 0, 0, -4, 0, 1, -2, 5, 0, -5, 0, 1, -4, 0, 9, 0, -6, 0, 1, -2, -7, 0, 14, 0, -7, 0, 1, 0, 0, -16, 0, 20, 0, -8, 0, 1, -2, 9, 0, -30, 0, 27, 0, -9, 0, 1, -4, 0, 25, 0, -50, 0, 35, 0, -10, 0, 1, -2, -11, 0, 55, 0, -77, 0, 44, 0, -11, 0, 1, 0, 0, -36, 0, 105, 0, -112, 0, 54, 0, -12, 0, 1

Offset: 0

Author

Wolfdieter Lang, Nov 08 2011

Keywords

• sign
• easy
• tabl

Comments

The proof for the row polynomials C(n,x), n>=2, follows by repeated expansion of the determinant, using the Chebyshev S-polynomials recurrence. For n=0 one defines C(0,x):=1, and for n=1 one has C(1,x)=x.

Modulo signs and first terms, essentially the same as A123343. - Eric W. Weisstein, Apr 05 2017

Examples

			The table begins
n\m  0   1   2   3   4   5   6   7    8  9  10 ...
0:   1
1:   0   1
2:  -4   0   1
3:  -2  -3   0   1
4:   0   0  -4   0   1
5:  -2   5   0  -5   0   1
6:  -4   0   9   0  -6   0   1
7:  -2  -7   0  14   0  -7   0   1
8:   0   0 -16   0  20   0  -8   0    1
9:  -2   9   0 -30   0  27   0  -9    0  1
10: -4   0  25   0 -50   0  35   0  -10  0   1
...
C(4,x) = -4*x^2 - x^4, with zeros 2, 0, -2, 0.
C(5,x) =-2 + 5*x - 5*x^3 + x^5, with zeros 2, phi-1, -phi, -phi and  phi-1, with the golden section  phi:=(1+sqrt(5))/2.
The adjacency matrix for C_1 is [[0]],
for C_2 it is [[0,2],[2,0]], and for C_3 it is [[0,1,1],[1,0,1],[1,1,0]].
		

Links

• Eric Weisstein's Mathworld: Adjacency Matrix
• Eric Weisstein's Mathworld: Characteristic Polynomial
• Eric Weisstein's Mathworld: Cycle Graph

Crossrefs

Cf. A127672.

Cf. A123343 (essentially the same sequence).

Programs

Mathematica

Flatten[{{1}, {0, 1}, Table[(-1)^n CoefficientList[CharacteristicPolynomial[AdjacencyMatrix[CycleGraph[n]], x], x], {n, 2, 10}]}] (* Eric W. Weisstein, Apr 05 2017 *)
Flatten[{{1}, {0, 1}, Table[CoefficientList[2 (ChebyshevT[n, x/2] - 1), x], {n, 2, 10}]}] (* Eric W. Weisstein, Apr 05 2017 *)

Formula

a(n,m)=[x^m]C(n,x), with C(0,x):=1, C(1,x)=x, and

C(n,x) = 2*(T(n,x/2)-1) = R(n,x)-2 , for n>=2, with Chebyshev's T-polynomial or its monic integer version R(n,x) (usually called Chebyshev C-polynomials) with coefficient table A127672, from which a formula for a(n,m) follows. Only the column m=0 differs.