Minimum rank with zero diagonal

Associated with a simple graph G is a family of real, symmetric zero diagonal matrices with the same nonzero pattern as the adjacency matrix of G. The minimum of the ranks of the matrices in this family is denoted mr(0)(G). We characterize all connected graphs G with extreme minimum zero-diagonal rank: a connected graph G has mr(0)(G) \u3c= 3 if and only if it is a complete multipartite graph, and mr0(G) = vertical bar G vertical bar if and only if it has a unique spanning generalized cycle (also called a perfect vertical bar 1,2 vertical bar-factor). We present an algorithm for determining whether a graph has a unique spanning generalized cycle. In addition, we determine maximum zero-diagonal rank and show that for some graphs, not all ranks between minimum and maximum zero-diagonal ranks are allowed