Graphs with reciprocal eigenvalue properties

Abstract

In this paper, only simple graphs are considered. A graph G is nonsingular if its adjacency matrix A(G) is nonsingular. A nonsingular graph G satisfies reciprocal eigenvalue property (property R) if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and G satisfies strong reciprocal eigenvalue property (property SR) if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and they both have the same multiplicities. From the definitions property SR implies property R. Furthermore, for some classes of graphs (for example, trees), it is known that these properties are equivalent. However, the equivalence of these two properties is not yet known for any nonsingular graph. In this article, it is shown that these properties are not equivalent in general.