A characterization of graphs of competition number m

AbstractIn this note we give a characterization of graphs with competition number less than or equal to m. We also give an alternate proof of a theorem characterizing competition graphs

Some properties of GM-matrices and their inverses

AbstractThe class of GM-matrices is defined by requiring that a positive cycle contain any negative cycle it intersacts. Using the cycle structure, a canonical form is developed for irreducible GM-matrices. Relationships between the signs of the principal minors and the cycle are derived. In special cases, results are obtained on the signs of elements of the inverse of a GM-matrix

Vanishing minor conditions for inverse zero patterns

AbstractWe study the relationship between the zero-nonzero pattern of an invertible matrix and the vanishing minors of the matrix and of its inverse. In particular we show how to determine when a matrix B could be the inverse of a matrix A with a given zero-nonzero pattern. In fact, there is always a set of almost principal minors of B (in one-to-one correspondence with the set of zero entries of A) whose vanishing implies that B-1 has zeros everywhere that A does, provided certain principal minors of B do not vanish

Inverting graphs of rectangular matrices

AbstractThis paper addresses the question of determining the class of rectangular matrices having a given graph as a row or column graph. We also determine equivalent conditions on a given pair of graphs in order for them to be the row and column graphs of some rectangular matrix. In connection with these graph inversion problems we discuss the concept of minimal inverses. This concept turns out to have two different forms in the case of one-graph inversion. For the two-graph case we present a method of determining when an inverse is minimal. Finally we apply the two-graph theorem to a class of energy related matrices

Spectra and inverse sign patterns of nearly sign-nonsingular matrices

AbstractA nearly sign-nonsingular (NSNS) matrix is a real n × n matrix having at least two nonzero terms in the expansion of its determinant with precisely one of these terms having opposite sign to all the other terms. Using graph-theoretic techniques, we study the spectra of irreducible NSNS matrices in normal form. Specifically, we show that such a matrix can have at most one nonnegative eigenvalue, and can have no nonreal eigenvalue z in the sector {z: |arg z| ⩽ κ(n − 1)}. We also derive results concerning the sign pattern of inverses of these matrices

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