TY - JOUR T1 - Matrices totally positive relative to a tree, II AU - Costas-Santos, R.S. AU - Johnson, C.R. JO - Linear Algebra and its Applications VL - 505 SP - 1 EP - 10 PY - 2016 DA - 2016/09/15/ SN - 0024-3795 DO - https://doi.org/10.1016/j.laa.2016.04.021 UR - https://www.sciencedirect.com/science/article/pii/S0024379516301203 KW - Graph KW - Neumaier conclusion KW - Spectral theory KW - Sylvester's identity KW - Totally positive matrix KW - Totally positive relative to a tree AB - If T is a labelled tree, a matrix A is totally positive relative to T, principal submatrices of A associated with deletion of pendent vertices of T are P-matrices, and A has positive determinant, then the smallest absolute eigenvalue of A is positive with multiplicity 1 and its eigenvector is signed according to T. This conclusion has been incorrectly conjectured under weaker hypotheses. ER -