On trees with perfect matchings
type: Item
instance of: scholarly article
title: On trees with perfect matchings
zbMATH Open document ID: 1043.05037
DOI: 10.1016/S0024-3795(02)00454-8
author: Jason J. Molitierno
author: M. Neumann
published in: Linear Algebra and its Applications
publication date: 2003-03-11T00:00:00Z
review text: The main results of the paper are upper bounds on the algebraic connectivity of trees with perfect matchings. A tree is said to have a perfect matching if it has a spanning forest whose components are paths on two vertices only. The authors investigate also connections between the trees, Laplacian matrices of such trees, their group generalized inverses and eigenvalue properties of the matrices. The paper is divided into three sections. In the first section, ``Introduction'', the authors recall definitions and present main results of the paper. The second section, ``Trees with a perfect matching'', concerns mentioned upper bounds (Theorem 2.8) and the representation of Laplacian matrices (Theorem 2.11) of the trees. The section contains a misprint (on page 77, line 9 from above, read ``Claim 2.2''). The third section, ``The group inverse of the Laplacian matrix'', presents results concerning the connection between the maximal diagonal entry of the group (generalized) inverse of the Laplacian matrix of a tree with a perfect matching and the pendant vertices of the tree. This result refines a result by \textit{S. J. Kirkland}, \textit{M. Neumann} and \textit{B. L. Shader} [SIAM J. Matrix Anal. Appl. 18, 827--841 (1997; Zbl 0889.15004)]. The authors of the paper under review use the refinement to narrow down the set of pendant vertices of a tree with a perfect matching which can correspond to the maximal diagonal entry in the group inverse of its Laplacian matrix.
Mathematics Subject Classification ID: 05C05
Mathematics Subject Classification ID: 05C40
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 05C70
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 1880064
zbMATH Keywords: tree
zbMATH Keywords: perfect matching
zbMATH Keywords: algebraic connectivity
zbMATH Keywords: group generalized inverse
reviewed by: Nikolaj M. Glazunov
MaRDI profile type: Publication
cites work: Q4326384
cites work: Q4002435
cites work: Q5682350
cites work: The Laplacian Spectrum of a Graph
cites work: Bounds on the largest eigenvalues of trees with a given size of matching
cites work: Distances in Weighted Trees and Group Inverse of Laplacian Matrices
cites work: Bounds on the subdominant eigenvalue involving group inverse with applications to graphs
cites work: On a bound on algebraic connectivity: the case of equality
cites work: Laplacian matrices of graphs: A survey
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description: scientific article; zbMATH DE number 1880064