When I did some research, I have not found the analytical expression about the eigenvalues and eigenvectors of Laplacian matrix in a chain graph while I only found those in a cycle graph. The Laplcian matrix is as follows.
$$L=\begin{bmatrix}1 & -1 & 0 & 0 & 0\\-1 & 2 & -1 & 0 & 0\\0 & -1 & 2 & -1 & 0\\ 0 & 0 & -1 & 2 & -1\\0 & 0 & 0 & -1 & 1\end{bmatrix}$$
It seems to have some relationship with the Fourier series, but I still found no paper in which the writer gave this analytical expression about the eigenvalues and eigenvectors of Laplacian matrix in a chain graph. Could you give me some advice on it?



The eigenvectors and eigenvalues of Laplacian matrix in a chain graph
You can find it in [this paper](https://ieeexplore.ieee.org/abstract/document/5717507?casa_token=wzm-Ny2TJyYAAAAA:yoCoqQc-JPp7yPxO9AhtR43atzGbV0L61c4o1teX4A3QxPGlJEv0_traXxbEmajwKjBfgOstKZs).


In case that link doesn't work, search at Google Scholar for "On the observability of path and cycle graphs".


