Trees with integer reciprocal eigenvalue property

Let G be a graph and A(G) be the adjacency matrix of G. Let m≥2 be a positive integer. Then G is said to have the property m-(R) if mλ is an eigenvalue of G whenever λ is an eigenvalue of G. Further, if λ and mλ have the same multiplicity for each eigenvalue λ, then we say that G satisfies property m-(SR). In this article, we provide a class of graphs with property m-(SR) using the corona operation. We prove that a tree satisfying property m-(R) must be singular. A result relating the coefficients of the characteristic polynomial of graphs with property m-(SR) is obtained. Using this, we construct classes of trees satisfying property m-(SR). A caterpillar tree Pk(n1,…,nk) is obtained from a path Pk on vertices 1,…,k by attaching ni pendant vertices to vertex i, for each i=1,…,k. Let Pn denote the class of all such caterpillar trees on n vertices, where each ni is a positive integer. Observing that property m-(R) and property m-(SR) are equivalent within the class Pn, we characterize all trees in Pn that satisfy property m-(SR) for prime m≥7.