Combinatorial properties of generalized M-matrices

Abstract

An M_â¨-matrix has the form A = sI â B with s ⥠Ï(B) and B^k is entrywise nonnegative for all sufficiently large integers k. In this paper, the existence of a preferred basis for a singular M_â¨- matrix A = sI â B with index(B) ⤠1 is proven. Some equivalent conditions for the equality of the height and level characteristics of A are studied. The well structured property of the reduced graph of A is discussed. Also the possibility of the existence of preferred basis for another generalization of M-matrices, known as GM-matrices, is studied.