Z. Du, C.M. da Fonseca (2023) The number of P-vertices for acyclic matrices with given nullity, Discrete Mathematics, Vol. 346, pp.113-592, 10.1016/j.disc.2023.113592
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.