A minimax-condition for the characteristic center of a tree

Abstract

Let L be the Laplacian matrix of a tree. We present a graph-theoretic interpretation of the cofactors of order 2 of L. From this, we deduce a description for the inverse of the rooted Laplacian, reflecting the geometry of a branch. Defining the thickness of a branch as the Perron root of this matrix, we present a minimax-characterization of the characteristic center of the tree based on the thickness of its branches.