How to find the eigen values of the graph having vertex set as $\{1,2,.......n\}$ and edge set as $\{(l,l+1)\}$ $ \cup (1,n)$ ? where $1\le l \le n$.
Here I am considering the Laplacian matrix of the graph.
Please give some hints . I am totally clueless.
This is a cycle graph. I expect here the edges are undirected type. Hence your Laplacian matrix is $$L=\begin{bmatrix}2 & -1 & 0 & \cdots & 0 & -1\\ -1 & 2 & -1 & \cdots & 0 & 0\\0 & -1 & 2 & \cdots & 0 & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & \cdots & 2 & -1\\ -1 & 0 & 0 & \cdots & -1 & 2\end{bmatrix}.$$ This is a circulant matrix and for a circulant matrix the eigenvalues are already obtained. Using that you can get its eigenvalues. Otherwise, observe that $L=2I_n-\mathcal{P}_n-\mathcal{P}_n^T,$ where $\mathcal{P}_n$ is the full-cycle permutation matrix. As ${\rm det}(\mathcal{P}_n-\lambda I_n)=(-1)^n(\lambda^n-1)$, the eigenvalues of $\mathcal{P}_n$ are the primitive $n$th roots of unity ($1, \omega, \ldots, \omega^{n-1}$ where $\omega=e^{\frac{2\pi\mathtt{i}}{n}}$). Further $\mathcal{P}_n^T=\mathcal{P}_n^{n-1}$. So the eigenvalues of $L$ are given by \begin{align*}2-\omega^k+\omega^{n-k}&=2-\omega^k-\omega^{-k}\\&=2-2\cos\frac{2\pi k}{n}\\&=4\sin^2\frac{\pi k}{n}, ~~~~\text{ for }k=1, \ldots, n.\end{align*}