On trees with perfect matchings
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.
scientific article; zbMATH DE number 1880064
scientific article