Sign patterns that require almost unique rank

A sign pattern matrix is a matrix whose entries are from the set {+,-,0}. For a real matrix B, sgn(B) is the sign pattern matrix obtained by replacing each positive (respectively, negative, zero) entry of B by + (respectively, −, 0). For a sign pattern matrix A, the sign pattern class of A, denoted Q(A), is defined as {B:sgn(B)=A}. The minimum rank mr(A) (maximum rank MR(A)) of a sign pattern matrix A is the minimum (maximum) of the ranks of the real matrices in Q(A). Several results concerning sign patterns A that require almost unique rank, that is to say, the sign patterns A such that MR(A) = mr(A) + 1, are established and are extended to sign patterns A for which the spread is d=MR(A)-mr(A). A complete characterization of the sign patterns that require almost unique rank is obtained.