The number of P-vertices of singular acyclic matrices: An inverse problem

Let A be a real symmetric matrix. If after we delete a row and a column of the same index, the nullity increases by one, we call that index a P-vertex of A. When A is an n × n singular acyclic matrix, it is known that the maximum number of P-vertices is n − 2. If T is the underlying tree of A, we will show that for any integer number k ∈ {0, 1, . . . , n − 2}, there is a (singular) matrix whose graph is T and with k P-vertices. We will provide illustrative examples.