Suppose D is the distance matrix of a tree. Graham and Pollack showed that the determinant of D satisfies an identity that depends only on surprisingly little information of the given tree. I will explain how this identity generalizes to a combinatorial expression for the determinant of any principal submatrix of D. This new identity involves counts of spanning forests and is proved by use of potential-theoretic concepts on graphs. This generalizes further to a determinant identity for effective resistance matrices on an arbitrary finite graph. I will mention a connection to the ultra-log-concave version of Mason's conjecture, and a strengthening for the graphic case.

This is joint work with Farbod Shokrieh and Chenxi Wu. https://arxiv.org/abs/2411.11488