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 imm anantal 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 matric es A. Motivated by a conjecture of R. Merris, it was shown by the auth ors that (1) may be improved to (d) over bar(k-1)(L(T)) less than or e qual 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 t ree T. The proof of (2) relied on rather involved recursive relations for weighted matchings in the tree T as well as identities of hook cha racters. In this work, we circumvent this tedium with a new proof usin g the notion of vertex orientations. This approach makes (2) immediate ly 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)).