A combinatorial formula for principal minors of a matrix with tree-metric exponents and its applications

Let T   be a tree with a vertex set {1,2,…,N}{1,2,…,N}. Denote by dijdij the distance between vertices i and j  . In this paper, we present an explicit combinatorial formula of principal minors of the matrix (tdij)(tdij), and its applications to tropical geometry, study of multivariate stable polynomials, and representation of valuated matroids. We also give an analogous formula for a skew-symmetric matrix associated with T.