Minimal Ranks of Pattern Matrices

The minimal rank of a pattern P, denoted by mr(P), is defined as the minimum of ranks of all its real realization matrices. The maximal triangle size of P, denoted by MT(P), is del tried as the maximal number of the orders of all the triangle sub-patterns of P. A pattern P is called a critical pattern if mr(P)=MT(P). A square pattern P is called r-regular if every line of P has exactly r nonzero entries. In this paper, it is proved that an r-regular pattern is critical pattern for r=1, 2,...,n-1.