Extremes Of Permanents Of I-1-Matrices And Upper Bound For Permanents Of Non-negative Integral Matrices

In this paper, we discuss the following problems of the permanents:1. Extremes of permanents for (0, -1)-matrix, (-1, 1)-matrix, (0, -1, 1)-matrix which have some restricted conditions;2. The upper bounds of permanents for some (0, 1)-matrices and fully indecomposable matrices with nonnegative integer entries.We firstly investigate the values of permanents of three classes of matrices:(1) S(m×n, 1) is the set of all m×n (0, -1)-matrices with exactly l 0's;(2) F(m×n, k) is the set of all m×n (-1, 1)-matrices with exactly k -1's;(3) G(m×n, k, l) is the set of all m×n (0, -1, 1)-matrix with exactly k -1's and l 0's.We determine the maximum, the second largest, the minimum, the second smallest values of the permanent function and find the matrices which attain some extremes values for S(m×n, l) and F(n×n, k). For G(m×n, k, l) we determine the minimum values of the permanent function and find the matrices which attain these extremes values under condition max{k + l, m}≤n; we also determine the maximum values of the permanent function and find the matrices which attain these extremes values for G(n×n, k, l) under the conditions: 0 and -1 are not on a same line and . Secondly, we give some new upper bounds of permanents for some (0, 1)-matrices and fully indecomposable matrices with nonnegative integer entries. Concretly, we generalize some upper bounds of the permanents for the n×n (0, 1)-matrices. We also genenralize some results of the permanents for fully indecomposable matrices with nonnegative integer entries to some new bounds. Some further research problems are also put forward in this paper.