The number of P-vertices in a matrix with maximum nullity

Let T be a tree with n≥2 vertices. Set S(T) for the set of all real symmetric matrices whose graph is T. Let A∈S(T) and i∈{1,…,n}. We denote by A(i) the principal submatrix of A obtained after deleting the row and column i. We set mA(0) for the multiplicity of the eigenvalue zero in A (the nullity of A). When mA(i)(0)=mA(0)+1, we say that i is a P-vertex of A. As usual, M(T) denotes the maximum nullity occurring of B∈S(T). In this paper we determine an upper bound and a lower bound for the number of P-vertices in a matrix A∈S(T) with nullity M(T). We also prove that if the integer b is between these two bounds, then there is a matrix E∈S(T) with b P-vertices and maximum nullity.