Hook immanantal inequalities for trees explained

Let (d) over bar(k) denote the normalized hook immanant corresponding to the partition (k, 1(n-k)) of n. P. Heyfron proved the family of immanantal inequalities det A = (d) over bar(1)(A) less than or equal to (d) over bar(2)(A) less than or equal to ... less than or equal to (d) over bar(n)(A) = per A for all positive semidefinite Hermitian matrices A. Motivated by a conjecture of R. Merris, it was shown by the authors that (1) may be improved to (d) over bar(k-1)(L(T)) less than or equal to k-2/k-1 (d) over bar(k)(L(T)) for all 2 less than or equal to k less than or equal to n whenever L(T) is the Laplacian matrix of a tree T. The proof of (2) relied on rather involved recursive relations for weighted matchings in the tree T as well as identities of hook characters. In this work, we circumvent this tedium with a new proof using the notion of vertex orientations. This approach makes (2) immediately apparent and more importantly provides an insight into why it holds, namely the absence of certain vertex orientations for all trees. As a by-product we obtain an improved bound, 0 less than or equal to 1/k-1[(d) over bar(k)(L(T)) - <(d)over bar(k)(L(S(n)))] less than or equal to k-2/k-1 (d) over bar(k)(L(T)) - (d) over bar(k-1)(L(T)).