A note on universally optimal matrices and field independence of the minimum rank of a graph

For a simple graph G on n vertices, the minimum rank of G over a field F, written as mr^F (G), is defined to be the smallest possible rank among all n × n symmetric matrices over F whose (i, j)th entry (for i ≠ j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. A symmetric integer matrix A such that every off-diagonal entry is 0, 1, or - 1 is called a universally optimal matrix if, for all fields F, the rank of A over F is the minimum rank of the graph of A over F. Recently, Dealba et al. [L.M. Dealba, J. Grout, L. Hogben, R. Mikkelson, K. Rasmussen, Universally optimal matrices and field independence of the minimum rank of a graph, Electron. J. Linear Algebra 18 (2009) 403-419] initiated the study of universally optimal matrices and established field independence or dependence of minimum rank for some families of graphs. In the present paper, more results on universally optimal matrices and field independence or dependence of the minimum rank of a graph are presented, and some results of Dealba et al. [5] are improved.