Working on a physics problem, I've encountered some structured cyclic tridiagonal $n\times n$ matrices. They're all of the following form: $ \tiny \begin{bmatrix} \alpha_1 & \frac{\sqrt{\alpha_1\alpha_2}}2 & 0 & \cdots &\cdots &\cdots &\cdots &0 & \frac{\sqrt{\alpha_n\alpha_1}}2 \\ \frac{\sqrt{\alpha_1\alpha_2}}2 & \alpha_2 & \frac{\sqrt{\alpha_2\alpha_3}}2 &0 & \cdots &\cdots &\cdots &\cdots &0 \\ 0 & \ddots&\ddots&\ddots &0 & \cdots &\cdots &\cdots &0 \\ \vdots &0 &\frac{\sqrt{\alpha_{k-2}\alpha_{k-1}}}2 & \alpha_{k-1} & \frac{\sqrt{\alpha_{k-1}\alpha_k}}2 &0 & \cdots& \cdots&\vdots \\ 0& \cdots & 0 &\frac{\sqrt{\alpha_{k-1}\alpha_k}}2 & \alpha_k &\frac{\sqrt{\alpha_k\alpha_{k+1}}}2 & 0 & \cdots & 0\\ \vdots & \cdots & \cdots & 0 &\frac{\sqrt{\alpha_k\alpha_{k+1}}}2 & \alpha_{k+1} &\frac{\sqrt{\alpha_{k+1}\alpha_{k+2}}}2 &0 & \vdots \\ 0 & \cdots& \cdots& \cdots& 0 &\ddots &\ddots &\ddots & 0 \\ 0 & \cdots& \cdots& \cdots& \cdots & 0&\frac{\sqrt{\alpha_{n-2}\alpha_{n-1}}}2 & \alpha_{n-1}&\frac{\sqrt{\alpha_{n-1}\alpha_n}}2\\ \frac{\sqrt{\alpha_n\alpha_1}}2& 0 & \cdots& \cdots& \cdots& \cdots & 0&\frac{\sqrt{\alpha_{n-1}\alpha_n}}2 & \alpha_n \end{bmatrix} $ i.e. they obey $M_{k,k+1}=M_{k+1,k} = \tfrac12\sqrt{\alpha_k\alpha_{k+1}}$ with $M_{k,k}=\alpha_k$ and $k=n+1$ is remapped to $k=1$.
I am interested in the eigenvalues of such a a matrix, or at least its characteristic polynomial, but I was not able to simplify the problem further than this, even is the geometric means on the second diagonals let me hope there is a solution to this problem.
P.S. One interesting case for me is when $\alpha_k$ is the binomial coefficient $\displaystyle \binom{n}{k}$, but I don't think it simplifies the problem.