The inverse eigenvalue and inertia problems for minimum rank two graphs

Abstract

Let G be an undirected graph on n vertices and let S(G) be the set of all real sym- metric n×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let mr(G) denote the minimum rank of all matrices in S(G), and mr+(G) the minimum rank of all positive semidefinite matrices in S(G). All graphs G with mr(G) = 2 and mr+(G) = k are characterized; it is also noted that mr+(G) = α(G) for such graphs. This charac- terization solves the inverse inertia problem for graphs whose minimum rank is two. Furthermore, it is determined which diagonal entries are required to be zero, are required to be nonzero, or can be either for a rank minimizing matrix in S(G) when mr(G) = 2. Collectively, these results lead to a solution to the inverse eigenvalue problem for rank minimizing matrices for graphs whose minimum rank is two.