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On the cardinality of a factor set of binary matrices

LINEAR ALGEBRA AND ITS APPLICATIONS 2017年 534卷 122-134页

作者： Yordzhev, Krasimir South West Univ Neofit Rilski Fac Math & Nat Sci Blagoevgrad Bulgaria

The work considers an equivalence relation in the set of all n x m binary matrices. In each element of the factor-set generated by this relation, we define the concept of canonical binary matrix, namely the minimal element with respect to the lexicographic order. For this purpose, the binary matrices are uniquely represented by ordered n-tuples of integers. We have found a necessary and sufficient condition for an arbitrary matrix to be canonical. This condition could be the base for realizing recursive algorithm for finding all n x m canonical binary matrices and consequently for finding all bipartite graphs, up to isomorphism with cardinality of each part equal to n and m. (C) 2017 Elsevier Inc. All rights reserved.

关键词： Ordered n-tuple of integers canonical binary matrix semi-canonical binary matrix Bipartite graph

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Calculating the cardinality of some classes of binary matrices using bitwise operations - A polynomial algorithm

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DISCRETE MATHEMATICS ALGORITHMS AND APPLICATIONS 2019年第3期11卷

作者： Yordzhev, Krasimir South West Univ Neofit Rilski Fac Math & Nat Sci Blagoevgrad Bulgaria

Let Lambda(k)(n) be the set of all n x n binary matrices with exactly k units in each row and each column, 1 <= k <= n. A matrix A is an element of Lambda(k)(n) will be called primitive, if there is no l x l submatrix of A that belongs to the set Lambda(k)(l), k <= l < n .The article describes a polynomial algorithm, which works in time O(n(2)) for verifying whether a Lambda(k)(n)-matrix is primitive. The work applies this algorithm for finding all primitive Lambda(k)(n)-matrices which rows and columns are in lexicographically nondecreasing order (semi-canonical binary matrices) for some integers n and k.

关键词： binary matrix primitive binary matrix semi-canonical binary matrix n-tuples of integers bitwise operations polynomial algorithm

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