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The minimum rank of a simple graph  G  is defined to be the smallest possible rank over all symmetric real matrices whoseentry (for) is nonzero wheneveris 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 in  G . This article compares the minimum rank with the clique covering number of  G  and 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 on  n  vertices that preserve the minimum rank.

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