1. Trees with the reciprocal eigenvalue property.
- Author
-
Barik, Sasmita, Mondal, Debabrota, and Pati, Sukanta
- Subjects
- *
EIGENVALUES , *TREES - Abstract
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 1 λ 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. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF