A Combinatorial Formula for Principal Minors of a Matrix with Tree-metric Exponents and Its Applications

2013/11/28 by Hiroshi Hirai, Hirai, Hiroshi, Akihiro Yabe +1
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.1311.7238

16 pages

arxiv created 2014/11/15 · arxiv updated 2014/11/18

Abstract

Let T be a tree with a vertex set \ 1,2,…, N \. Denote by d_ij the distance between vertices i and j. In this paper, we present an explicit combinatorial formula of principal minors of the matrix (t^d_ij), 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.

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