Unimodularity and circle graphs

AbstractA property of unimodularity is introduced for antisymmetric integral matrices. It is satisfied by the adjacency matrix of a circle graph provided with a Naji orientation [8]. In a further paper we shall interprete this result in terms of symmetric matroids introduced in [2]. In this communication we give a direct proof by means of techniques used in [1] for an algorithmic solution of the Gauss problem on self-intersecting surves in the plane

A note on the spectra of certain skew-symmetric {1,0,−1}-matrices

AbstractWe characterize skew-symmetric {1,0,−1}-matrices with a certain combinatorial property. In particular, we exhibit several equivalent descriptions of this property. These results allow characterizations of unimodular orientations of the complete graph, of rank 2 chirotopes, and of a class of multipartite oriented graphs