Diagonal entry restrictions in minimum rank matrices

Let F be a field, let G be a simple graph on n vertices, and let S (G) be the class of all F -valued symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. For each graph G, there is an associated minimum rank class, M R (G) consisting of all matrices A ∈ S (G) with rankA = mr (G), the minimum rank among all matrices in S (G). Although no restrictions are applied to the diagonal entries of matrices in S (G), this work explores when the diagonal entries corresponding to specific vertices of G must be zero or nonzero for all matrices A ∈ M R (G). These vertices are denoted as nil or nonzero, respectively. Vertices whose corresponding diagonal entries are not similarily restricted for all matrices in M R (G) are called neutral. The minimum rank of a graph following an edge-subdivision is determined by the existence of a nil vertex, and several relations between diagonal restrictions and the rank-spread parameter are found. This is followed by the rather different approach of using the graph parameter Ẑ to identify nil and nonzero vertices. The nil, nonzero and neutral vertices of trees are classified in terms of rank-spread. Finally, it is shown that except for K3, 2-connected graphs with maximum nullity 2 have all neutral vertices and, moreover, the graphs with maximum nullity 2 that have nil or nonzero vertices are completely classified.