Abstract
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.
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