Trees with the reciprocal eigenvalue property

It was shown in 2006 that among the nonsingular trees T (whose adjacency matrix A(T) is nonsingular), the corona trees (trees that are obtained by taking any tree T and then adding a new pendant vertex at each vertex of T) are the only ones which satisfy the reciprocal eigenvalue property (lambda is an eigenvalue of A(T) if and only if 1 lambda is an eigenvalue of A(T), where their multiplicities are allowed to be different). A general question remained open. Can there be a tree which has at least one zero eigenvalue and whose nonzero eigenvalues satisfy the reciprocal eigenvalue property? In this note, we show that there are no such trees with at least two vertices. The proof is a beautiful application of the product of graphs.

Automated summary

The study in 2006 showed that only corona trees among the non-singular trees satisfy a specific eigenvalue property, leaving a general question open about trees with reciprocal eigenvalues. However, it was found that there are no such trees with at least two vertices. The proof involves a beautiful application of graph products.