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. The edges in Σ are called odd edges and the other edges of E even. 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 adjacent and all edges between i and j are even, ai,j>0 if i and j are adjacent and all edges between i and j are odd, ai,j∈R if i and j are connected by 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.