On signed digraphs with all cycles negative

It is known that signed graphs with all cycles negative are those in which each block is a negative cycle or a single line. We now study the more difficult problem for signed diagraphs. In particular we investigate the structure of those diagraphs whose arcs can be signed (positive or negative) so that every (directed) cycle is negative. Such diagraphs are important because they are associated with qualitatively nonsingular matrices. We identify certain families of such diagraphs and characterize those symmetric diagraphs which can be signed so that every cycle is negative.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/25544/1/0000086.pd

Sign-nonsingular matrices and even cycles in directed graphs

AbstractA sign-nonsingular matrix or L-matrix A is a real m× n matrix such that the columns of any real m×n matrix with the same sign pattern as A are linearly independent. The problem of recognizing square L-matrices is equivalent to that of finding an even cycle in a directed graph. In this paper we use graph theoretic methods to investigate L-matrices. In particular, we determine the maximum number of nonzero elements in square L-matrices, and we characterize completely the semicomplete L-matrices [i.e. the square L-matrices (aij) such that at least one of aij and aij is nonzero for any i,j] and those square L-matrices which are combinatorially symmetric, i.e., the main diagonal has only nonzero entries and aij=0 iff aji=0. We also show that for any n×n L-matrix there is an i such that the total number of nonzero entries in the ith row and ith column is less than n unless the matrix has a completely specified structure. Finally, we discuss the algorithmic aspects