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- W3144897784 abstract "A conjecture of Barrett, Butler and Hall may be stated as follows: If n≥3 and A∈{0,1}n×n (the family of n×n 0–1 matrices) is a nonsingular symmetric matrix, then the following two statements are equivalent: (a) All of the principal minors of A of order n−2 are zero; and (b) A−1 is a matrix all of whose entries have the same modulus and all of whose diagonal entries are equal. We show that this conjecture holds if A does not have both a zero and a nonzero principal minor of order n−4 (if n≥5). The aforementioned conjecture was stated in terms of the enhanced principal rank characteristic sequence (epr-sequence) of the matrix A, a sequence defined as follows: ℓ1ℓ2⋯ℓn, where ℓj is either A, S, or N, based on whether all, some but not all, or none of its principal minors of order j are nonzero. It is shown that, if B is an arbitrary symmetric matrix with zero diagonal, then the odd-girth of its graph, as well as whether or not its graph is bipartite or an odd cycle, is deducible from its epr-sequence. For an arbitrary (not necessarily symmetric) nonsingular matrix A∈{0,1}n×n with n≥3, we establish necessary conditions for A−1 to be a matrix all of whose entries have the same modulus." @default.
- W3144897784 created "2021-04-13" @default.
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- W3144897784 date "2021-08-01" @default.
- W3144897784 modified "2023-10-18" @default.
- W3144897784 title "On 0–1 matrices whose inverses have entries of the same modulus" @default.
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- W3144897784 doi "https://doi.org/10.1016/j.laa.2021.03.037" @default.
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