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A signed graph is a pair (G, Σ), where G = (V, E) is a graph (in which parallel edges are permitted, but loops are not) with V = {1, . . . , n} and Σ ⊆ E. By S(G, Σ) we denote the set of all symmetric V × V matrices A = [ai,j] with ai,j < 0 if i and j are connected by only even edges, ai,j > 0 if i and j are connected by only odd edges, ai,j ∈ R if i and j are connected by both even and odd edges, ai,j = 0 if i = j and i and j are non-adjacent, and ai,i ∈ R for all vertices i. The stable inertia set of a signed graph (G, Σ) is the set of all pairs (p, q) for which there exists a matrix A ∈ S(G, Σ) with p positive and q negative eigenvalues which has the Strong Arnold Property. In this paper, we study the stable inertia set of (signed) graphs.

The recursive inverse eigenvalue problem for matrices is studied, where for each leading principle submatrix an eigenvalue and associated left and right eigenvectors are assigned. Existence and uniqueness results as well as explicit formulas are proven, and applications to nonnegative matrices, Z-matrices, M-matrices, symmetric matrices, Stieltjes matrices and inverse M-matrices are considered.

A sign pattern matrix is a matrix whose entries are from the set {+, −, 0}. The minimum rank of a sign pattern matrix A is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of A. It is shown in this paper that for any m×n sign pattern A with minimum rank n − 2, rational realization of the minimum rank is possible. This is done using a new approach involving sign vectors and duality. It is shown that for each integer n ≥ 9, there exists a nonnegative integer m such that there exists an n × m sign pattern matrix with minimum rank n − 3 for which rational realization is not possible. A characterization of m × n sign patterns A with minimum rank n − 1 is given (which solves an open problem in Brualdi et al.[6]), along with a more general description of sign patterns with minimum rank r, in terms of sign vectors of certain subspaces. A number of results on the maximum and minimum numbers of sign vectors of k-dimensional subspaces of R n are obtained. In particular, it is shown that the maximum number of sign vectors of 2-dimensional subspaces of R n is 4n + 1. Several related open problems are stated along the way.