Eigenvalues and eigenvectors of the laplacian matrix associated to the cycle graph with one weighted edge (interactive visualization)

This program illustrates some ideas and results from our paper.
Sergei M. Grudsky, Egor A. Maximenko, Alejandro Soto-González (2022):
Eigenvalues of the laplacian matrices of the cycles with one weighted edge.
Linear Algebra Appl. 653, 86–115. DOI: 10.1016/j.laa.2022.07.011.

Please, choose the value of the parameter α and the order n of the matrix.

α = 0.3

n = 8

The next picture shows the graph of the generating function g(x) = (2 sin(x / 2))².
The ordinates of the yellow points are the eigenvalues of the corresponding n×n matrix.

The eigenvalues λ_α,n,1, …, λ_α,n,n are numbered in the ascending order. Each eigenvalue λ_α,n,j is computed as g(θ_α,n,j).

If j is odd, then θ_α,n,j is just (j − 1) π / n.

If j is even, then θ_α,n,j is the unique solution of the main equation: n x − (j −1) π = η_α(x).

The function η_α is defined by η_α(x) = 2 arctan(κ_α cot(x / 2)), where κ_α = α / (1 − α).

The next figure shows the left-hand side and the right-hand side of the main equation for the even values of j.
The numbers θ_α,n,j are the abscissas of the intersection points.

The next figure shows the components of an eigenvector v_α,n,j associated to λ_α,n,j.
Recall that n = 8 (the value of n can be changed above).
Choose the index j of the eigenvalue/eigenvector pair:

j = 1

For j = 1, all components of v_α,n,j are equal to 1.

For j ≥ 2 and p = 1, …, n, the p-st component of v_α,n,j is computed as

v_α,n,j,p = sin(p θ_α,n,j) − (1 − α) sin((p − 1) θ_α,n,j) + α sin((n − p) θ_α,n,j).

The thin gray curve is the graph of the smooth function defined by the same formula, with a continuous variable p.