A hollow matrix described by a graph G is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in G. For a given graph G, the determination of all possible spectra of matrices associated with G is the hollow inverse eigenvalue problem for G. Solutions to the hollow inverse eigenvalue problems for paths and complete bipartite graphs are presented. Results for related subproblems such as possible ordered multiplicity lists, maximum multiplicity of an eigenvalue, and minimum number of distinct eigenvalues are presented for additional families of graphs.@en

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 and the other edges of E even. If there are parallel edges, then only two edges in each parallel class are permitted, one of which is even and one of which is odd. By S(G,Σ) we denote the set of all symmetric n×n matrices A=[ai,j] with ai,j<0 if i and j are connected by an even edge, ai,j>0 if i and j are connected by an odd edge, ai,j∈R if i and j are connected by both an even and an odd edge, ai,j=0 if i≠j and i and j are non-adjacent, and ai,i∈R for all vertices i. The maximum nullity M(G,Σ) of a signed graph (G,Σ) is the maximum nullity attained by any A∈S(G,Σ). Arav et al. gave a combinatorial characterization of 2-connected signed graphs (G,Σ) with M(G,Σ)=2. In this paper, we give a complete combinatorial characterization of the signed graphs (G,Σ) with M(G,Σ)=2.@en