Hook immanantal inequalities for trees explained
type: Item
instance of: scholarly article
title: Hook immanantal inequalities for trees explained
zbMATH Open document ID: 0895.15006
DOI: 10.1016/S0024-3795(97)00369-8
author: Onn Chan
author: Tao Kai Lam
published in: Linear Algebra and its Applications
publication date: 1998-08-31T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 1129383
zbMATH Keywords: positive semidefinite Hermitian matrix
zbMATH Keywords: hook partition
zbMATH Keywords: hook immanant
zbMATH Keywords: vertex orientation
zbMATH Keywords: Laplacian matrix of a tree
zbMATH Keywords: immanantal inequalities
reviewed by: Arbind Kumar Lal
MaRDI profile type: Publication
cites work: Immanant Inequalities for Laplacians of Trees
cites work: Hook immanantal inequalities for Laplacians of trees
cites work: Q4002278
cites work: Q3669422
cites work: Immanants of combinatorial matrices
cites work: Immant dominance orderings for hook partitions
cites work: The Laplacian permanental polynomial for trees
cites work: Laplacian matrices of graphs: A survey
cites work: Q5339348
cites work: Q4001520
Wikidata QID: Q126351602
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description: scientific article; zbMATH DE number 1129383