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.

1 0 4 π θα,n,j λα,n,j

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

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

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

The function ηα is defined by   ηα(x) = 2 arctan(κα cot(x / 2)),   where  κα = 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 θα,n,j are the abscissas of the intersection points.

π 0 π θα,n,j

The next figure shows the components of an eigenvector vα,n,j associated to λα,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 vα,n,j are equal to 1.

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

     vα,n,j,p = sin(p θα,n,j) − (1 − α) sin((p − 1) θα,n,j) + α sin((n − pθα,n,j).

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

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