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, d ̄ k,k=1,…,n , satisfy the dominance ordering (a) per(A)= d ̄ n(A)⩾ d ̄ n−1(A)⩾⋯⩾ d ̄ 2(A)⩾ d ̄ 1(A)= det(A). The classical Hadamard–Marcus inequalities assert that for an n×n psd matrix A=[a ij] , (b) per(A)= d ̄ n(A)⩾∏ i=1 na ii⩾ d ̄ 1(A)= det(A). In view of the Hadamard–Marcus inequalities, it is natural to ask where the term ∏ i=1 na ii 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 κ(A) such that (c) d ̄ κ(A)(A)⩾∏ i=1 na ii⩾ d ̄ κ(A)−1(A). Heyfron [10] (see also [11,17]) established for all n×n psd A that κ(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 κ(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⌉⩾κ(A)⩾⌈(n+1)/2⌉. Both these bounds on κ(A) are tight and the coefficient 1/3 for the term in m in the upper bound cannot be lowered to 1/4.