On 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. In this paper we develop upper bounds on the algebraic connectivity of such trees and we consider other eigenvalue properties of its Laplacian matrix. Furthermore, for trees with perfect matchings, we refine a result, due to Kirkland, Neumann, and Shader, concerning the connection between the maximal diagonal entry of the group inverse of the Laplacian matrix of a (general) tree and the pendant vertices of the tree, and use this refinement to narrow down the set of the 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.
Molitierno, J. and M. Neumann. "On Trees with Perfect Matchings." Linear Algebra and its Applications 362 (2003 Mar 15): 75-85.