Graphs whose minimal rank is two: The finite fields case

Let F be a finite field, G = (V, E) be an undirected graph on n vertices, and let S(F,G) be the set of all symmetric n× n matrices over F whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let mr(F,G) be the minimum rank of all matrices in S(F,G). If F is a finite field with pt elements, p = 2, it is shown that mr(F,G) ≤ 2 if and only if the complement of G is the join of a complete graph with either the union of at most (pt+1)/2 nonempty complete bipartite graphs or the union of at most two nonempty complete graphs and of at most (pt − 1)/2 nonempty complete bipartite graphs. These graphs are also characterized as those for which 9 specific graphs do not occur as induced subgraphs. If F is a finite field with 2t elements, then mr(F,G) ≤ 2 if and only if the complement of G is the join of a complete graph with either the union of at most 2t +1 nonempty complete graphs or the union of at most one nonempty complete graph and of at most 2t−1 nonempty complete bipartite graphs. A list of subgraphs that do not occur as induced subgraphs is provided for this case as well.