A sign pattern (matrix) is a matrix whose entries are from the set {+, −, 0}. The minimum rank (resp., rational minimum rank) of a sign pattern matrix

This paper studies lower and upper bounds for the number of vertices of the polytope of stochastic tensors (i.e. triply stochastic arrays of dimension n). By using known results on polytopes (i.e. the Upper and Lower Bound Theorems), we present some new lower and upper bounds. We show that the new upper bound is tighter than the one recently obtained by Chang et al. [Ann Funct Anal. 2016;7(3):386–393] and also sharper than the one in Linial and Luria’s [Discrete Comput Geom. 2014;51(1);161–170]. We demonstrate that the analog of the lower bound obtained in such a way, however, is no better than the existing ones.

The base set of primitive zero‐symmetric sign pattern matrices with zero diagonal is {1, 2, …, 2n − 1}. In this paper, the primitive zero‐symmetric sign pattern matrices with zero diagonal attaining the maximal base 2n − 1 are characterized.