Invariants for incidence matrix of a tree

For an oriented tree, we compute several graph invariants, including the minimal norm of the generalized inverse and the norm of the Moore–Penrose inverse of its incidence matrix. We present equivalent characterizations of these invariants in terms of the minimal ratios of vector norms related to the oriented tree’s incidence matrix. We also provide a method to find the optimal vertex cut and edge cut with respect to these invariants.