Another graph-theoretic application.

Given an undirected finite graph $G$, the Colin de Verdière graph invariant $\mu(G)$ is defined as the maximum multiplicity of the second smallest eigenvalue of a matrix $M\in O_G$, provided that this eigenvalue is structurally stable. Here, $O_G$ is an interesting (for $G$) subset of the symmetric matrices called the Schrödinger-like operators on $G$.

It turns out that this invariant is very nice in the following sense: if $H$ is a minor of $G$, then the invariant verifies $\mu(H) \leq\mu(G)$. A trick used to prove this result is to put weights on the edges of $G$ to turn it into a metric space (with the obvious distance of smallest sum of weight in a path). Then, any minor $H$ can be obtained as a Gromov-Hausdorff limit of $G$ with suitable weights (contracted edges see their weight go to zero and deleted edges have their weight go to infinity), and conversely, any sequence of weights on the edges of $G$ that converges gives a minor.

Now that I wrote it down, it does not seem like such a powerful remark, but, as I recall, this shift in point of view is (at the very least) very convenient to prove the result.

Another graph-theoretic application.

Given an undirected finite graph $G$, the Colin de Verdière graph invariant $\mu(G)$ is defined as the maximum multiplicity of the second smallest eigenvalue of a matrix $M\in O_G$, provided that this eigenvalue is structurally stable. Here, $O_G$ is an interesting (for $G$) subset of the symmetric matrices called the Schrödinger-like operators on $G$.

It turns out that this invariant is very nice in the following sense: if $H$ is a minor of $G$, then the invariant verifies $\mu(H) \leq\mu(G)$. A trick used to prove this result is to put weights on the edges of $G$ to turn it into a metric space (with the obvious distance of smallest sum of weight in a path). Then, any minor $H$ can be obtained as a Gromov-Hausdorff limit of $G$ with suitable weights (contracted edges see their weight go to zero and deleted edges have their weight go to infinity), and conversely, any sequence of weights on the edges of $G$ that converges gives a minor.

Now that I wrote it down, it does not seem like such a powerful remark, but, as I recall, this shift in point of view is (at the very least) very convenient to prove the result.