Graphs whose adjacency matrices have rank equal to the number of distinct nonzero rows

Huang, Li; Tam, Bit-Shun; Wu, Shuhui

doi:10.1016/j.laa.2012.06.027

Cited by 14 publications

(9 citation statements)

References 9 publications

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“…Next, consider the case when n is odd (n ≥ 7). Consider the n × n matrix 2) . Then Q is an upper triangular, {0, 1}, group invertible, singular matrix.…”

Section: Resultsmentioning

confidence: 99%

“…Let S(X) denote the sum of the entries of a matrix X. Huang, Tam and Wu [2] showed that a number s is equal to S(A −1 ) for a symmetric (0, 1) matrix A with trace zero if and only if s is rational. Motivated by this work, Farber and Berman [3] presented an interesting relationship between Fibonacci numbers and matrix theory, thereby providing a partial answer to the question "what can be said about the sum of the entries of the inverse of a (0, 1) matrix?".…”

Section: Introductionmentioning

confidence: 99%

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Group inverses of $\{0,1\}$-triangular matrices and Fibonacci numbers}\\

Chatterjee,

Sivakumar

2020

Preprint

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“…Next, consider the case when n is odd (n ≥ 7). Consider the n × n matrix 2) . Then Q is an upper triangular, {0, 1}, group invertible, singular matrix.…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Group inverses of $\{0,1\}$-triangular matrices and Fibonacci numbers}\\

Chatterjee,

Sivakumar

2020

Preprint

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“…The conjecture was proved by Royle [27]. Since then alternative proofs and extensions of this result have appeared [2,5,15,28]. Furthermore, in [19] an algorithm is introduced for locating eigenvalues of cographs in a given interval.…”

Section: Introductionmentioning

confidence: 99%

Cographs: Eigenvalues and Dilworth number

Ghorbani

2019

Discrete Mathematics

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A cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. The vicinal preorder on the vertex set of a graph is defined in terms of inclusions among the neighborhoods of vertices. The minimum number of chains with respect to the vicinal preorder required to cover the vertex set of a graph G is called the Dilworth number of G. We prove that for any cograph G, the multiplicity of any eigenvalue λ = 0, −1, does not exceed the Dilworth number of G and show that this bound is tight. G. F. Royle [The rank of a cograph, Electron. J. Combin. 10 (2003), Note 11] proved that if a cograph G has no pair of vertices with the same neighborhood, then G has no 0 eigenvalue, and asked if beside cographs, there are any other natural classes of graphs for which this property holds. We give a partial answer to this question by showing that an H-free family of graphs has this property if and only if it is a subclass of the family of cographs. A similar result is also shown to hold for the −1 eigenvalue.

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“…For more examples, see [4,6,8,11,15,20]. A zero-sum k-flow for a graph G is a vector in the null space of the 0,1-incidence matrix of G such that its entries belong to {±1, · · · , ±(k − 1)}.…”

Section: Introductionmentioning

confidence: 99%