Universally Optimal Matrices and Field Independence of the Minimum Rank of a Graph∗

The minimum rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose ijth entry (for i != j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. We define a universally optimal matrix to be an integer matrix A such that every off-diagonal entry of A is 0, 1, or −1, and for all fields F , the rank of A is the minimum rank over F of its graph. We use universally optimal matrices to establish field independence of minimum rank for numerous graphs. We also provide examples verifying lack of field independence for other graphs.