On bounding the eigenvalues of matrices with constant row-sums

Marsli, Rachid; Hall, Frank J.

doi:10.1080/03081087.2018.1430736

Cited by 9 publications

(8 citation statements)

References 8 publications

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“…In [13], the authors obtained results similar to the above theorem for the real matrices. In this paper we extend this result for any complex square matrix A with constant row sum.…”

Section: Introductionsupporting

confidence: 61%

Eigenvalue bounds for some classes of matrices associated with graphs

Mehatari¹,

Kannan²

2018

Preprint

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For a given complex square matrix A with constant row sum, we establish two new eigenvalue inclusion sets. Using these bounds, first we derive bounds for the second largest and smallest eigenvalues of adjacency matrices of k-regular graphs. Then, we establish some bounds for the second largest and the smallest eigenvalues of the normalized adjacency matrices of graphs and the second smallest eigenvalue and the largest eigenvalue of the Laplacian matrices of graphs. Sharpness of these bounds are verified by examples.

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“…In [13], the authors obtained results similar to the above theorem for the real matrices. In this paper we extend this result for any complex square matrix A with constant row sum.…”

Section: Introductionsupporting

confidence: 61%

Eigenvalue bounds for some classes of matrices associated with graphs

Mehatari¹,

Kannan²

2018

Preprint

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“…In our paper [4], published in 2019, without being aware of the explicit form of τ ∞ (Theorem 1.6), we obtained in a totally different manner the same result implied by the combination of Theorems 1.2 and 1.6, that if λ = 1 is an eigenvalue of the stochastic matrix S, then |λ| ≤ ρ(S). Our approach was based on the use of the following known optimization theorem, a form of which was introduced in the study of the location of eigenvalues of real matrices, for the first time (to the best of our knowledge), by Barany and Solymosi in their 2017 paper [6].…”

Section: Introductionmentioning

confidence: 84%

“…Even more important is the following enhancement of Theorem 1.9; this allows for better bounds on the eigenvalues, as was shown in [4].…”

Section: This Minimum Is Reached Whenmentioning

confidence: 99%

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Some properties of ergodicity coefficients with applications in spectral graph theory

Marsli¹,

Hall²

2019

Preprint

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The main result is Corollary 2.9 which provides upper bounds on, and even better, approximates the largest non-trivial eigenvalue in absolute value of real constant row-sum matrices by the use of vector norm based ergodicity coefficients {τ p }. If the constant row-sum matrix is nonsingular, then it is also shown how its smallest non-trivial eigenvalue in absolute value can be bounded by using {τ p }. In the last section, these two results are applied to bound the spectral radius of the Laplacian matrix as well as the algebraic connectivity of its associated graph. Many other results are obtained. In particular, Theorem 2.15 is a convergence theorem for τ p and Theorem 4.7 says that τ 1 is less than or equal to τ ∞ for the Laplacian matrix of every simple graph. Other discussions, open questions and examples are provided.

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“…If A is any real matrix having an eigenvector v with no zero component and S = diag(v), then every bound M that applies to the general case of real constant row-sum matrices applies also to the matrix A, since the matrix B = S − AS is constant row-sum and has the same spectrum as A. Such upper bounds are discussed in our recent articles [9] and [10]. If the eigenvector v of A has some components equal to zero, then Theorem 4.1 can be used.…”

Section: Bounding the Eigenvalues Of Real Matrices By Using A Classmentioning

confidence: 99%

Inclusion regions and bounds for the eigenvalues of matrices with a known eigenpair

Marsli

Hall

2020

Special Matrices

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Let (λ, v) be a known real eigenpair of an n×n real matrix A. In this paper it is shown how to locate the other eigenvalues of A in terms of the components of v. The obtained region is a union of Gershgorin discs of the second type recently introduced by the authors in a previous paper. Two cases are considered depending on whether or not some of the components of v are equal to zero. Upper bounds are obtained, in two different ways, for the largest eigenvalue in absolute value of A other than. Detailed examples are provided. Although nonnegative irreducible matrices are somewhat emphasized, the main results in this paper are valid for any n × n real matrix with n≥3.

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On bounding the eigenvalues of matrices with constant row-sums

Cited by 9 publications

References 8 publications

Eigenvalue bounds for some classes of matrices associated with graphs

Eigenvalue bounds for some classes of matrices associated with graphs

Some properties of ergodicity coefficients with applications in spectral graph theory

Inclusion regions and bounds for the eigenvalues of matrices with a known eigenpair

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