The inertia set of a signed graph

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 6 = 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