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 (λ is an eigenvalue of A(T) if and only if $\frac{1}{\lambda }$ 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.

2006年表明，在非奇异树T（其邻接矩阵A（T）是非奇异的）中，冠冕树（通过取任何树T然后在T的每个顶点添加一个新的悬垂顶点而获得的树）是唯一满足倒数特征值属性（λ是A ( T )的特征值当且仅当$\frac{\mathrm{1个}}{\lambda }$是A ( T )的特征值，其中它们的重数允许不同)。一个普遍的问题仍然悬而未决。是否存在一棵至少具有一个零特征值且其非零特征值满足倒数特征值属性的树？在这篇文章中，我们表明不存在至少有两个顶点的树。证明是图形乘积的一个漂亮应用。