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

Abstract

ABSTRACT The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose entry (for ) is nonzero whenever 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 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.