Matrices in Combinatorics and Graph Theory

Liu, Bolian; Lai, Hong‐Jian

doi:10.1007/978-1-4757-3165-1

Cited by 32 publications

(14 citation statements)

References 71 publications

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“…Therefore, matrix B is a doubly stochastic matrix, and it has a largest eigenvalue 1 and an eigenvector ½1 1 Á Á Á 1 T corresponding to the largest eigenvalue [26]. Since the associated graph of matrix M (also matrix B) is strongly connected (for each entry ði; jÞ in matrix M there exists an integer k), matrix B is an irreducible matrix.…”

Section: Proof Consider the Matrixmentioning

confidence: 98%

Some further development on the eigensystem approach for graph isomorphism detection

Zhang

2005

Journal of the Franklin Institute

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Section: Proof Consider the Matrixmentioning

confidence: 98%

Some further development on the eigensystem approach for graph isomorphism detection

Zhang

2005

Journal of the Franklin Institute

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“…In this paper, we follow the terminology and notation used in [3,7]. A Boolean matrix is a matrix over the binary Boolean algebra {0, 1}.…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

“…The smallest such l is called the exponent of D, denoted by exp(D). Exponents have been studied by several researchers [3,7,8,9,10].…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

The Competition Index of a Nearly Reducible Boolean Matrix

Cho¹,

Kim²

2013

Bulletin of the Korean Mathematical Society

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Abstract. Cho and Kim [4] have introduced the concept of the competition index of a digraph. Similarly, the competition index of an n × n Boolean matrix A is the smallest positive integer q such that A q+i (A T ) q+i = A q+r+i (A T ) q+r+i for some positive integer r and every nonnegative integer i, where A T denotes the transpose of A. In this paper, we study the upper bound of the competition index of a Boolean matrix. Using the concept of Boolean rank, we determine the upper bound of the competition index of a nearly reducible Boolean matrix. Preliminaries and notationsIn this paper, we follow the terminology and notation used in [3,7]. A Boolean matrix is a matrix over the binary Boolean algebra {0, 1}. For m × n Boolean matrices A = (a ij ) and B = (b ij ), we say that B is dominated by A (denoted by B ≤ A) if b ij ≤ a ij for all i and j. We denote the m × n all-ones Boolean matrix by J m,n (and by J n if m = n), the m × n all-zeros Boolean matrix by O m,n (and by O n if m = n), and the n × n identity Boolean matrix by I n . The subscripts m and n will be omitted whenever their values are clear from the context. Let D = (V, E) denote a digraph (directed graph) with vertex set

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“…However, for digraphs whose competition periods are not one, there are significant differences between their competition indices and ordinary indices. In the case of the ordinary index, it is well known that index(D) ≤ (n − 2) 2 + 2 for a reducible digraph D of order n, [7,9]. This implies that the index of a Wielandt digraph is the maximum possible index among indices of digraphs of order n. However, in the case of a competition index, the similarity between the properties does not hold.…”

Section: Closing Remarkmentioning

confidence: 99%

Competition Indices of Strongly Connected Digraphs

Cho¹,

Kim²

2011

Bulletin of the Korean Mathematical Society

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Abstract. Cho and Kim [4] and Kim [6] introduced the concept of the competition index of a digraph. Cho and Kim [4] and Akelbek and Kirkland [1] also studied the upper bound of competition indices of primitive digraphs. In this paper, we study the upper bound of competition indices of strongly connected digraphs. We also study the relation between competition index and ordinary index for a symmetric strongly connected digraph.

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Matrices in Combinatorics and Graph Theory

Cited by 32 publications

References 71 publications

Some further development on the eigensystem approach for graph isomorphism detection

Some further development on the eigensystem approach for graph isomorphism detection

The Competition Index of a Nearly Reducible Boolean Matrix

Competition Indices of Strongly Connected Digraphs

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