Hook immanantal inequalities for Hadamard's function

Abstract

For an n × n positive semi-definite (psd) matrix A, Peter Heyfron showed in [9] that the normalized hook immanants, d̄k, k = 1, . . . , n, satisfy the dominance ordering per(A) = d̄n(A) ≥ d̄n-1(A) ≥ ⋯ ≥;> d̄2(A) ≥ d̄1(A) = det(A). (a) The classical Hadamard-Marcus inequalities assert that for an n × n psd matrix A = [aij], per(A) = d̄n(A) ≥ nΠi=1aii ≥ d̄1(A) = det(A). (b) In view of the Hadamard-Marcus inequalities, it is natural to ask where the term Πn i=1 aii sits in the family of descending normalized hook immanants in (a). More specifically, for each n × n psd A one wishes to determine the smallest K (A) such that d̄k(A)(A) ≥ nΠi=1 aii d̄k(A)-1(A). (C) Heyfron [10] (see also [11,17]) established for all n × n psd A that k(A) ≥ min{n -2, 1 + √n-1]. In this work, we focus on the case where A is the Laplacian matrix of a tree T. It is meaningful to seek bounds on k(A) that depend on some topological features of the tree T such as the size of a maximum matching in T. For a tree T on n ≥ 2 vertices with a maximum matching of size m, we show that ⌈n/2 + m/3⌉ ≥ k(A) ≥ ⌈(n + 1)/2⌉. Both these bounds on k (A) are tight and the coefficient 1/3 for the term in m in the upper bound cannot be lowered to 1/4. © 1999 Elsevier Science Inc. All rights reserved.