Skew-adjacency matrices of tournaments with bounded principal minors

Abstract

Let T be a tournament with n vertices v1,…,vn. The skew-adjacency matrix of T is the n×n zero-diagonal matrix S=[sij] in which sij=−sji=1 if vi dominates vj. It is well-known that the determinant of S is zero or the square of an odd integer. Moreover, the principal minors of S are at most 1 if and only if T is a local order. In this paper, we characterize the class of tournaments for which the principal minors of the skew-adjacency matrix do not exceed 9.