Eigenvalues and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mn>1</mml:mn><mml:mtext>,</mml:mtext><mml:mi>n</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math>-odd factors

Lu, Hongliang; Wu, Zefang; Xu, Yang

doi:10.1016/j.laa.2010.04.002

Cited by 7 publications

(2 citation statements)

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“…Cui and Kano [18] showed a neighborhood condition for a graph having an odd [1, 𝑏]-factor. Lu et al [19] characterized a graph with an odd [1, 𝑏]-factor in terms of its Laplacian eigenvalues and adjacency eigenvalues, respectively. Kim et al [20] established an upper bound for the third largest adjacency eigenvalue in a regular graph to ensure the existence of an odd [1, 𝑏]-factor.…”

Section: Introductionmentioning

confidence: 99%

Characterizing an odd [1, b]-factor on the distance signless Laplacian spectral radius

Zhou¹,

Liu²

2023

RAIRO-Oper. Res.

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Let G be a connected graph of even order n. An odd [1,b]-factor of G is a spanning subgraph F of G such that dF(v) ∈ {1,3,5,··· ,b} for any v ∈ V (G), where b is positive odd integer. The distance matrix D(G) of G is a symmetric real matrix with (i,j)-entry being the distance between the vertices vi and vj. The distance signless Laplacian matrix Q(G) of G is defined by Q(G) = Tr(G) + D(G), where Tr(G) is the diagonal matrix of the vertex transmissions in G. The largest eigenvalue η1(G) of Q(G) is called the distance signless Laplacian spectral radius of G. In this paper, we verify sharp upper bounds on the distance signless Laplacian spectral radius to guarantee the existence of an odd [1,b]-factor in a graph; we provide some graphs to show that the bounds are optimal.

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Section: Introductionmentioning

confidence: 99%

Characterizing an odd [1, b]-factor on the distance signless Laplacian spectral radius

Zhou¹,

Liu²

2023

RAIRO-Oper. Res.

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“…In recent years, some sufficient conditions in terms of the eigenvalues are also established for the existence of an (a, b)-parity factor in graphs. For positive integers r ≥ 3 and even n, Lu, Wu, and Yang [14] proved a lower bound for λ 3 (G) in an n-vertex r-regular graph G to guarantee the existence of a (1, b)-parity factor in G, where b is a positive odd integer. In [11], Kim, O, Park and Ree improved this bound, and proved the following result.…”

Section: Introductionmentioning

confidence: 99%

School Mathematics Textbooks in China

Wang¹

2015

East China Normal University Scientific Reports

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Let µ and ν be two probability measures on R d , where µ(dx) = e −V (x) dx R d e −V (x) dx for some V ∈ C 1 (R d ). Explicit sufficient conditions on V and ν are presented such that µ * ν satisfies the log-Sobolev, Poincaré and super Poincaré inequalities. In particular, if V (x) = λ|x| 2 for some λ > 0 and ν(e λθ|•| 2 ) < ∞ for some θ > 1, then µ * ν satisfies the log-Sobolev inequality. This improves and extends the recent results on the log-Sobolev inequality derived in [20] for convolutions of the Gaussian measure and compactly supported probability measures. On the other hand, it is well known that the log-Sobolev inequality for µ * ν implies ν(e ε|•| 2 ) < ∞ for some ε > 0. RésuméDes conditions explicites suffisantes sur V et ν sont présentées telles que µ * ν satisfait des inégalités de Sobolev logarithmique, de Poincaré et de super-Poincaré. En particulier, si V (x) = λ|x| 2 pour quelque λ > 0 et ν(e λθ|•| 2 ) < ∞ avec θ > 1, alors µ * ν satisfait l'inégalité de Sobolev logarithmique. Cela améliore et étend des résultats récents sur l'inégalité de Sobolev logarithmique obtenus dans [20] pour des convolutions de la mesure de Gauss et des mesures de probabilité à support compact. D'autre part, il est bien connu que l'inégalité de Sobolev logarithmique pour µ * ν implique ν(e ε|•| 2 ) < ∞ pour quelque ε > 0.

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Eigenvalues and [a,b]‐factors in regular graphs

Suil

2021

Journal of Graph Theory

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For positive integers, r ≥ 3, h ≥ 1, and k ≥ 1, Bollobás, Saito, and Wormald proved some sufficient conditions for an h‐edge‐connected r‐regular graph to have a k‐factor in 1985. Lu gave an upper bound for the third largest eigenvalue in a connected r‐regular graph to have a k‐factor in 2010. Gu found an upper bound for certain eigenvalues in an h‐edge‐connected r‐regular graph to have a k‐factor in 2014. For positive integers a ≤ b, an even (or odd) [ a , b ]‐factor of a graph G is a spanning subgraph H such that for each vertex v ∈ V ( G ), d H ( v ) is even (or odd) and a ≤ d H ( v ) ≤ b. In this paper, we prove upper bounds (in terms of a , b , and r) for certain eigenvalues (in terms of a , b , r , and h) in an h‐edge‐connected r‐regular graph G to guarantee the existence of an even [ a , b ]‐factor or an odd [ a , b ]‐factor. This result extends the one of Bollbás, Saito, and Wormald, the one of Lu, and the one of Gu.

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Eigenvalues and [1,n]-odd factors

Cited by 7 publications

References 5 publications

Characterizing an odd [1, b]-factor on the distance signless Laplacian spectral radius

Characterizing an odd [1, b]-factor on the distance signless Laplacian spectral radius

School Mathematics Textbooks in China

Eigenvalues and [a,b]‐factors in regular graphs

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