Graphs whose positive semi-definite matrices have nullity at most two

Let G=(V,E) be a undirected graph containing n vertices, and let be the set of all Hermitian n×n matrices M=(mi,j) with mi,j≠0 if i and j are connected by one edge of G, with if i and j are connected by at least two edges, with mi,j=0 if i≠j, and i and j are not connected by an edge of G, and with mi,i for i=1,…,n a real number. What is the largest nullity attained by any positive semi-definite matrix ? In this paper we characterize, for t=1 and 2, those graphs G for which the maximum nullity is not greater than t.