On edge star sets in trees

Let AA be a Hermitian matrix whose graph is GG (i.e. there is an edge between the vertices ii and jj in GG if and only if the (i,j)(i,j) entry of AA is non-zero). Let λλ be an eigenvalue of AA with multiplicity mA(λ)mA(λ). An edge e=ije=ij is said to be Parter (resp., neutral, downer) for λ,Aλ,A if mA(λ)−mA−e(λ)mA(λ)−mA−e(λ) is negative (resp., 0, positive ), where A−eA−e is the matrix resulting from making the (i,j)(i,j) and (j,i)(j,i) entries of AA zero. For a tree TT with adjacency matrix AA a subset SS of the edge set of GG is called an edge star set for an eigenvalue λλ of AA, if |S|=mA(λ)|S|=mA(λ) and A−SA−S has no eigenvalue λλ. 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=uve=uv is a downer edge for λ,Aλ,A if and only if uu and vv are both downer vertices for λ,Aλ,A; and e=uve=uv is a neutral edge if uu and vv are neutral vertices. Among other results, it is shown that any edge star set for each eigenvalue of a tree is a matching.