On edge star sets in trees

Let A be a Hermitian matrix whose graph is G (i.e. there is an edge between the vertices i and j in G if and only if the ( i , j ) entry of A is non-zero). Let λ be an eigenvalue of A with multiplicity m A ( λ ) . An edge e = i j is said to be Parter (resp., neutral, downer) for λ , A if m A ( λ ) − m A − e ( λ ) is negative (resp., 0, positive ), where A − e is the matrix resulting from making the ( i , j ) and ( j , i ) entries of A zero. For a tree T with adjacency matrix A a subset S of the edge set of G is called an edge star set for an eigenvalue λ of A , if | S | = m A ( λ ) and A − S has no eigenvalue λ . In this paper the existence of downer edges and edge star sets for non-zero eigenvalues of the adjacency matrix of a tree is proved. We prove that neutral edges always exist for eigenvalues of multiplicity more than 1. It is also proved that an edge e = u v is a downer edge for λ , A if and only if u and v are both downer vertices for λ , A ; and e = u v is a neutral edge if u and v are neutral vertices. Among other results, it is shown that any edge star set for each eigenvalue of a tree is a matching.