Determinant of discrete Laplacian

Question

It can easy be shown by induction that the determinant of the $(N-1)\times (N-1)$ matrix $$\begin{pmatrix} 2 & -1 & & \\ -1 & 2 & \ddots & \\ & \ddots & \ddots & -1\\ & & -1 & 2 \end{pmatrix}$$ equals $N$. (Up to a factor) this matrix corresponds to the representing matrix of the discrete Laplacian on an interval with Dirichlet boundary conditions, on an equidistant partition.

Now it the partition is not equidistant, say it is $0 = \tau_0 < \tau_1 < \dots < \tau_N$ with increments $\Delta_j := \tau_j - \tau_{j-1}$, then (up to a factor) we end up with the matrix $$\begin{pmatrix} \Delta_1 + \Delta_2 & -\Delta_2 & & \\ -\Delta_2 & \Delta_2 + \Delta_3 & \ddots & \\ & \ddots & \ddots & -\Delta_{N-1}\\ & & -\Delta_{N-1} & \Delta_{N-1}+\Delta_N \end{pmatrix}.$$ Is the determinant of this matrix still explicitly computable? If not, can we at least compute some (suitably normalized) limit of the determinant as the mesh of the partition goes to zero?

\edit: user35593 answered in the comments that the determinant is equal to $$ \prod_{j=1}^N {\Delta_j}\sum_{i=1}^N\frac{1}{\Delta_i},$$ which is indeed easy to check by induction. Thank you very much!

Answer 1

You already have the answer and proof, so let me just use this post to advertise some basic graph theory :). Take a cycle graph on $N$ vertices, and weigh each edge by $\Delta_i$, $i=1,2,\dots N$. Then your matrix is a cofactor of the Laplacian matrix of this graph. By Kirchhoff's theorem, this is the (weighted) count of the spanning trees of your graph. An $N$-cycle has $N$ spanning trees, and their weights are $\frac{\prod_{i=1}^N \Delta_i}{\Delta_j}$, when you omit the $j$th edge. Therefore the cofactor is equal to $\prod_{i=1}^N \Delta_i \left(\sum_{j=1}^N\frac{1}{\Delta_j}\right)$.