The maximum number of P-vertices of some nonsingular double star matrices

In this short note, we construct a nonsingular matrix A whose graph is a double star of order n ⩾ 4 with n − 2 P-vertices. This example leads to a positive answer, for n ⩾ 6 , to a last open question proposed recently by Kim and Shader regarding the trees for which each nonsingular matrix has at most n − 2 P-vertices.