The number of P-vertices for acyclic matrices with given nullity

In this paper, we completely characterize those trees on n vertices for which there is a singular matrix with nullity k and the number of P-vertices is n - k -1 or n - k - 2. The characterization of acyclic matrices, with rank r and the number of P-vertices is r -1, or with odd rank r and the number of P-vertices is r - 2, was first investigated in Fonseca et al. (2021) [10]. Here we introduce a unified method to revisit those results, and further cover the unknown case with even rank r and the number of P-vertices being r - 2. In the end, a continuity problem is fully solved.& COPY; 2023 Elsevier B.V. All rights reserved.