Hook Immanantal Inequalities for Hadamardʹs Function Original Research Article

For an n×n positive semi-definite (psd) matrix A, Peter Heyfron showed in [9] that the normalized hook immanants, image , satisfy the dominance ordering image The classical Hadamard–Marcus inequalities assert that for an n×n psd matrix A=[aij], image 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 image Heyfron [10] (see also [11,17]) established for all n×n psd A that image 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 ngreater-or-equal, slanted2 vertices with a maximum matching of size m, we show that left ceilingn/2+m/3right ceilinggreater-or-equal, slantedκ(A)greater-or-equal, slantedleft ceiling(n+1)/2right ceiling. 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.