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=ij 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=uv is a downer edge for λ,A if and only if u and v are both downer vertices for λ,A; and e=uv 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.