Specifically, we have a finite graph $G=(V,E)$, with graph Laplacian $$\Delta(x,y) = \begin{cases} \deg_G(x) \text{ if } x=y,\\ -\alpha(x,y) \text{ if } x\neq y; \end{cases}$$ where $\alpha(x,y)$ is the number of edges from $x$ to $y$ in $G$. Consider the new graph $G'$ formed by identifying two vertices $a$, $b$ and contracting them to a single vertex $c$. Here, we allow multiple edges but not loops, so in the contraction we will remove any edges from $a$ to $b$ but not multiple edges occurring. It seems like we should be able to bound the determinant of the new Laplacian formed by this in terms of $\Delta$ but I’ve been having trouble getting anything.
I’d like to do this without the assumption of $G$ having bounded degree to be able to extend this to infinite graphs if possible.