On reciprocal eigenvalue property of weighted trees

An undirected unweighted graph G is called nonsingular if its adjacency matrix A(G) is nonsingular. A graph is said to have property (R) if with every eigenvalue λ of A(G), 1/λ is also an eigenvalue. If, further, the multiplicity of λ and 1/λ as eigenvalues of A(G) are the same, then G is said to have property (SR). In a previous paper the classes of unweighted trees with property (R) and (SR) where shown to be the same and that this common class coincides with the class of corona trees, that is, trees which have been obtained from smaller trees by adding a pendent edge to each vertex.In this paper we continue the study of nonsingular trees in two ways. First we characterize the set of all graphs whose inverses are nonsingular trees. Second, we investigate the extent of property (R) for weighted trees and characterize the property for all trees with 8 vertices or more, under some conditions on the weights.