Li-Yau inequality on graphs

Bauer, Frank; Horn, Paul; Lin, Yong; Lippner, Gábor; Mangoubi, Dan; Yau, Shing–Tung

doi:10.4310/jdg/1424880980

Cited by 144 publications

(227 citation statements)

References 25 publications

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“…Then for any 1 ≤ p < q ≤ ∞, ℓ p m ֒→ ℓ q m . Now we introduce the gradient forms associated to the Laplacian and curvature dimension conditions on graphs following [21,4]. Definition 1.1.…”

Section: Settingmentioning

confidence: 99%

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Li-Yau inequality for unbounded Laplacian on graphs

Gong

Lin

Liu

et al. 2019

Advances in Mathematics

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In this paper, we derive Li-Yau inequality for unbounded Laplacian on complete weighted graphs with the assumption of the curvature-dimension inequality CDE ′ (n, K), which can be regarded as a notion of curvature on graphs. Furthermore, we obtain some applications of Li-Yau inequality, including Harnack inequality, heat kernel bounds and Cheng's eigenvalue estimate. These are first kind of results on this direction for unbounded Laplacian on graphs.

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Section: Settingmentioning

confidence: 99%

“…CDE ′ (n, K) implies CD(n, K) on graphs but visa versa is not true(see [23]). For diffusion Laplace operator, for example the Laplace-Beltrami operator on Riemannian manifolds, the CDE ′ (n, K) is equivalent to CD(n, K)(see [4]).…”

Section: Settingmentioning

confidence: 99%

Li-Yau inequality for unbounded Laplacian on graphs

Gong

Lin

Liu

et al. 2019

Advances in Mathematics

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“…These successes have motivated the problem of defining the discrete Ricci curvature. There have so far been several proposed definitions of discrete Ricci curvature [47,37,43,7,42,24,23,10,15]. It is generally unclear whether or not any of these notions of curvature are equivalent, and in some instances examples illustrate that they are not equivalent.…”

Section: Introductionmentioning

confidence: 99%

“…Under many notions of discrete curvature it is unclear whether such a volume growth bound exists. In this work we will present a volume growth that is interesting for regular graphs with a negative lower bound on Ollivier curvature.We will also briefly discuss the CDE ′ curvature, which was created by Bauer, Jost, and Liu [10]. The CDE ′ inequality is a modification of the CD inequality of Bakry-Émery, which is a discrete generalization of the Bochner formula from Riemannian geometry.…”

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Volume Growth, Curvature, and Buser-Type Inequalities in Graphs

Benson

Ralli

Tetali

2019

International Mathematics Research Notices

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We study the volume growth of metric balls as a function of the radius in discrete spaces, and focus on the relationship between volume growth and discrete curvature. We improve volume growth bounds under a lower bound on the so-called Ollivier curvature, and discuss similar results under other types of discrete Ricci curvature.Following recent work in the continuous setting of Riemannian manifolds (by the first author), we then bound the eigenvalues of the Laplacian of a graph under bounds on the volume growth. In particular, λ 2 of the graph can be bounded using a weighted discrete Hardy inequality and the higher eigenvalues of the graph can be bounded by the eigenvalues of a tridiagonal matrix times a multiplicative factor, both of which only depend on the volume growth of the graph. As a direct application, we relate the eigenvalues to the Cheeger isoperimetric constant. Using these methods, we describe classes of graphs for which the Cheeger inequality is tight on the second eigenvalue. We also describe a method for proving Buser's Inequality in graphs, particularly under a lower bound assumption on curvature.1 on average, the distances between points in B(x, r) and B(y, r) will be closer than their counterparts under the parallel transport. Ollivier observed that the average distance can be replaced by the L 1 -Wasserstein distance between uniform distributions on B(x, r) and B(y, r), and this metric is used in definition of the so-called Ollivier curvature, which can be used to recover the manifold's Ricci curvature (up to a factor) [42].Ollivier used this concept to help define the discrete Ricci curvature [42]. The metric balls B(x, r) and B(y, r) can also be defined on a graph where r is a non-negative integer and x and y are vertices of the graph. Then the L 1 -Wasserstein distance between the balls B(x, r) and B(y, r) determines a notion of curvature on the graph.While definitions of Ollivier curvature can be applied to any metric measure space, arguably its most fruitful use has been to define curvature in graphs with the graph distance and counting measure, for example [9,16,29]. That will also be our focus in this work: A well-known fact due to Bishop is that a Riemannian manifold with a lower bound on its Ricci curvature will have the volume growth of its metric balls controlled by this lower bound [12]. Under many notions of discrete curvature it is unclear whether such a volume growth bound exists. In this work we will present a volume growth that is interesting for regular graphs with a negative lower bound on Ollivier curvature.We will also briefly discuss the CDE ′ curvature, which was created by Bauer, Jost, and Liu [10]. The CDE ′ inequality is a modification of the CD inequality of Bakry-Émery, which is a discrete generalization of the Bochner formula from Riemannian geometry. Those authors demonstrated a version of the Li-Yau gradient estimate for graphs under the CDE ′ curvature. This is a result that does not have any known analogue in the setting of Ollivier curvature.Volume growth est...

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Graph curvature and local discrepancy

Horn,

Purcilly,

Stevens

2024

Journal of Graph Theory

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In recent years, discrete notions of curvature have been defined and exploited to understand various geometric properties of graphs; especially regarding heat flow, and spectral properties. In this paper, we study various combinatorial properties implied by satisfying the Bakry–Émery curvature dimension inequality . In particular we derive a local discrepancy inequality, similar in spirit to the expander mixing lemma from spectral graph theory, which certifies a type of “local pseudo‐randomness” of the edge set of the graph, for graphs satisfying a curvature lower bound. In addition, several other consequences are derived regarding graph connectivity and cycle statistics of the graph.

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Li-Yau inequality on graphs

Cited by 144 publications

References 25 publications

Li-Yau inequality for unbounded Laplacian on graphs

Li-Yau inequality for unbounded Laplacian on graphs

Volume Growth, Curvature, and Buser-Type Inequalities in Graphs

Graph curvature and local discrepancy

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