In Spectra of Simple Graphs , it is given that an unweighted path graph with $n$ vertices has eigenvalues $$ 2\cos(\pi j/(n+1)), j = 1,\cdots, n. $$ All multiplicities are 1. Assuming the edges of the path are weighted in some order, for example, $w_{1,2}\geq w_{2,3}\cdots \geq w_{n-1,n}$ where $w_{j-1,j}$, $j=2,\cdots, n$ is the weight of the edge between vertex $j-1$ and $j$; is there some general expression for the eigenvalues of the weighted path graph?
Thanks in advance.