Minimum-rank and maximum-nullity of graphs and their linear preservers

The minimum rank of a simple graphGis defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i not equal j) is nonzero whenever (i, j) is an edge in G and is zero otherwise. The sum of the minimum rank of a graph and its maximum nullity (similarly defined) is always the number of vertices inG. This article compares the minimum rank with the clique covering number ofGand the Boolean rank of its adjacency matrix. It does the same analysis for bipartite graphs. Finally, we investigate the linear operators on the set of graphs onnvertices that preserve the minimum rank.