Hook Immanantal Inequalities for Hadamard's Function

AbstractFor 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(a)per(A)=d̄n(A)⩾d̄n−1(A)⩾⋯⩾d̄2(A)⩾d̄1(A)=det(A). The classical Hadamard–Marcus inequalities assert that for an n×n psd matrix A=[aij],(b)per(A)=d̄n(A)⩾∏i=1naii⩾d̄1(A)=det(A). In view of the Hadamard–Marcus inequalities, it is natural to ask where the term ∏i=1naii 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)d̄κ(A)(A)⩾∏i=1naii⩾d̄κ(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