On a conjecture for fully indecomposable exponent and Hall exponent

Abstract

Let r, n be integers, − n < r < n. An n×n Boolean matrix A is called r-indecomposable if it contains no k×l zero submatrix with k + l = n − r + 1. If A is primitive, then there is a smallest positive integer, e r (A), such that A e r (A) is r-indecomposable, and there is also a smallest positive integer, e* r (A), such that A m is r-indecomposable for all . The integers e r (A) and e* r (A) are called the exponent and the strict exponent of r-indecomposability of A, respectively. The 0-indecomposable exponent and the 1-indecomposable exponent of A are also called the Hall exponent and the fully indecomposable exponent of A, respectively. In this article, we obtain bounds on these exponents, and prove that the Brualdi–Liu's conjecture about the fully indecomposable exponent and the Hall exponent is true for certain subsets of all primitive matrices with intersections of cycles.