Hook immanantal inequalities for trees explained (Q1381277)

Let \(\overline d_k\) denote the normalized hook immanant corresponding to the partition \((k, 1^{n-k})\) of \(n.\) It was proved by \textit{P. Heyfron} [Linear Multilinear Algebra 24, No. 1, 65-78 (1988; Zbl 0678.15009)] that the family of immanantal inequalities \[ \det A = \overline d_1(A) \leq \overline d_2(A) \leq \cdots \leq \overline d_n(A)= \text{per }A \] holds for all positive semidefinite Hermitian matrices \(A.\) The authors improve the above bound to \[ \overline{d}_{k-1}(L(T)) \leq \frac{k-2}{k-1} \overline{d}_k (L(T)) \] for \(2 \leq k \leq n\) whenever \(L(T)\) is the Laplacian matrix of a tree \(T.\) Heyfron's proof technique for the above result relied too much on recursive relations and it didn't give any insight into the correctness of the result. In this paper, the authors give an easy and elegant proof using the notion of vertex orientations. This also helps them to obtain an improved bound \[ 0 \leq \frac{1}{k-1} [ \overline{d}_k(L(T)) - \overline{d}_k(L(S(n))) ] \leq \frac{k-2}{k-1} \overline{d}_k (L(T)) - \overline{d}_{k-1} (L(T)) \] where \(S(n)\) is the star on \(n\) vertices.