On nonsingular trees and a reciprocal eigenvalue property

Abstract

Let G be a simple connected graph. Let A(G) be the adjacency matrix of G. We give a combinatorial description of A(G)−1 of a bipartite graph G with a unique perfect matching. As a corollary, we obtain the combinatorial description of the inverse of a nonsingular tree. A graph is said to have property (R) if is an eigenvalue of G whenever λ is an eigenvalue of G. Further, if λ and have the same multiplicity, for each eigenvalue λ then it is said to have the property (SR). We characterize all trees with property (R) and show that it is exactly the class of all trees with property (SR). The class of trees with property (R) is also identical to the class of corona trees, namely, the trees which can be obtained from other trees by adding pendants to all vertices. Other equivalent conditions for a tree to have property (R) are also given.