I was wondering whether there exists any kind of literature on the the powers of the discrete Laplacian, in particular the the discrete bi-Laplacian, possibly with weights on the edges.
In particular I wanted to ask the following question:
given some uniformly bounded, symmetric weights one can easily see that the one has the operator inequality
$$-\Delta_{w}\geq c(-\Delta)$$
where $-\Delta_{w}$ denotes the weighted discrete Laplacian and $-\Delta$ thee "free" one. $c$ is a positive constant given by the uniform bound on the weights. The inequality should be seen as an operator inequality in the sense of the quadratic Dirichlet form.
Obviously, $x^2$ is not an operator monotone function. But maybe there are some special properties of the discrete Laplacian, so that
$$(-\Delta_{w})^2\geq c(-\Delta)^2$$
could be true?