Tag Archives | facts I wish I knew in school

## more on matrix-tree formula

Fix a weighted graph $$\Gamma=(V,E), w:E\to \mathbb{R}$$.

The Laplacian $$L$$ of $$\Gamma$$ is the symmetric matrix

$$L_{u,v}=\left\{ \begin{array}{ll} -w(uv)&\mbox{ if }u\neq v,\\ \sum_{u’\neq u} w(uu’)& \mbox{ if }u=v\\ \end{array}\right..$$

(Here we view the weights $$w$$ as formal variables.)

As we all know, any principal minor of $$L$$ equals the sum of the weights of spanning trees of $$\Gamma$$.

Another way to define the principal minor is as the determinant of the restriction the quadratic form given by $$L$$ to any of the coordinate hyperplanes.…