On the b-chromatic number of cartesian products

Guo, Chuan; Newman, Mike

doi:10.1016/j.dam.2017.12.025

Cited by 5 publications

(4 citation statements)

References 10 publications

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“…According to Lemmas 2.1 and 2.2, A is a discrete Riesz spectral operator if and only if both K and L are invertible. In order to establish (10) and (11), let us focus on the following eigenvalue problem, where we seek for functions φ in D(A) and scalars µ ∈ C such that Aφ = µφ .…”

Section: Spectral Analysismentioning

confidence: 99%

“…Therein, it is specified that for our class of systems, the spectrum determined growth assumption (SDGA) is satisfied when A is a discrete Riesz spectral operator, that is, ω 0 = s(A), where s(A) := sup s∈σ (A) Re(s) is the spectral bound of A. According to (10) and to the SDGA, it is easy to obtain (13).…”

Section: Spectral Analysismentioning

confidence: 99%

“…Few years later, discrete spectral operators were discussed in [3] while Riesz-spectral operators came in [4], in which the eigenvalues are assumed to be simple. A generalization to the multiple case may be found in [10]. Many different works investigating the Riesz-spectral property of linear operators describing hyperbolic PDEs may be found in the literature.…”

Section: Introductionmentioning

confidence: 99%

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Well-posedness of infinite-dimensional linear systems with nonlinear feedback

Hastir

Califano

Zwart

2019

Systems & Control Letters

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We study existence of solutions, and in particular well-posedness, for a class of inhomogeneous, nonlinear partial differential equations (PDE's). The main idea is to use system theory to write the nonlinear PDE as a well-posed infinitedimensional linear system interconnected with a static nonlinearity. By a simple example, it is shown that in general well-posedness of the closed-loop system is not guaranteed. We show that well-posedness of the closed-loop system is guaranteed for linear systems whose input to output map is coercive for small times interconnected to monotone nonlinearities. This work generalizes the results presented in (Tucsnak and Weiss, 2014), where only globally Lipschitz continuous nonlinearities were considered. Furthermore, it is shown that a general class of linear port-Hamiltonian systems satisfies the conditions asked on the open-loop system. The result is applied to show well-posedness of a system consisting of a vibrating string with nonlinear damping at the boundary.

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Section: Spectral Analysismentioning

confidence: 99%

Section: Spectral Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Well-posedness of infinite-dimensional linear systems with nonlinear feedback

Hastir

Califano

Zwart

2019

Systems & Control Letters

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“…The hamming graphs have been investigated in the past. See for instance [6,7,11,12,16]. The p th power of a graph G denoted by G p is a graph whose vertex set V (G p ) = V (G) and edge set E(G p ) = {xy : d G (x, y) ≤ p}, where d G (x, y) denotes the distance between x and y in G.…”

Section: Introductionmentioning

confidence: 99%

Bounds for the b-chromatic number of powers of hypercubes

Francis,

Raj,

Gokulnath

2021

Preprint

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The b-chromatic number b(G) of a graph G is the maximum k for which G has a proper vertex coloring using k colors such that each color class contains at least one vertex adjacent to a vertex of every other color class. In this paper, we mainly investigate on one of the open problems given in [5]. As a consequence, we have obtained an upper bound for the b-chromatic number of some powers of hypercubes. This turns out to be an improvement of the already existing bound in [5]. Further, we have determined a lower bound for the b-chromatic number of some powers of the Hamming graph, a generalization of the hypercube.

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