Z. Du, C.M. da Fonseca (2023) The characterization of the minimal weighted acyclic graphs, Linear Algebra and its Applications, Vol. 672, pp.75-92, 10.1016/j.laa.2023.04.019
Abstract. In this paper, we completely characterize all the trees on n vertices with diameter d for which there is a symmetric matrix with nullity n-d and n-d-1, respectively. These characterizations cover all recent results proved for the standard 0-1 adjacency matrices. Here a new technique is developed for the general case, breaking the limitation of star complement technique for the standard 0-1 adjacency matrices.