Curvatures, graph products and Ricci flatness

Cushing, D. H.; Kamtue, Supanat; Kangaslampi, Riikka; Liu, Shiping; Peyerimhoff, Norbert

doi:10.1002/jgt.22630

Cited by 10 publications

(31 citation statements)

References 13 publications

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“…This formula has been established for the particular cases of strongly regular graphs with girth 4 and distance regular graphs with girth 4 in [11] and [13] respectively. Observe that the Lin-Lu-Yau curvature of a given edge only involves number of common neighbors of vertices with distance at most 2.…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

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Lin-Lu-Yau curvature and diameter of amply regular graphs

Li,

Liu

2021

Preprint

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By Hall's marriage theorem, we study lower bounds of the Lin-Lu-Yau curvature of amply regular graphs with girth 3 or 4 under different parameter restrictions. As a consequence, we derive that the Lin-Lu-Yau curvature of any strongly regular conference graph with parameterIn general, it is still open whether the Lin-Lu-Yau curvature of an amply regular graph of parameter (n, d, α, β) with girth 3 (i.e., α ≥ 1) and β ≥ 2 is always nonnegative or not. Our approach also provides a geometric proof of a classical diameter estimates of amply regular graphs in the case of girth 4 and some special cases of girth 3.

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Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

“…Observe that the Lin-Lu-Yau curvature of a given edge only involves number of common neighbors of vertices with distance at most 2. Therefore, the proofs of [11] and [13] apply directly to amply regular graphs with girth 4.…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

Lin-Lu-Yau curvature and diameter of amply regular graphs

Li,

Liu

2021

Preprint

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“…The case of (R)-Ricci-flat graphs will be discussed seperatly below. For the reader's convenience we recall the definition of (R)-Ricci flat graphs, which originates from the work of [7]. Definition 2.9.…”

Section: Definition 21 a Continuous Functionmentioning

confidence: 99%

“…(R)-Ricci-flat graphs constitute a subclass of Ricci-flat graphs with Bakry-Émery condition CD(2, ∞), see [7]. Important examples are given by complete bipartite graphs and, since (R)-Ricci-flat graphs are invariant under tensorization, by the hypercube, cf.…”

Section: Definition 21 a Continuous Functionmentioning

confidence: 99%

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Entropy-information inequalities under curvature-dimension conditions for continuous-time Markov chains

Weber

2021

Electron. J. Probab.

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In the setting of reversible continuous-time Markov chains, the CDΥ condition has been shown recently to be a consistent analogue to the Bakry-Émery condition in the diffusive setting in terms of proving Li-Yau inequalities under a finite dimension term and proving the modified logarithmic Sobolev inequality under a positive curvature bound. In this article we examine the case where both is given, a finite dimension term and a positive curvature bound. For this purpose we introduce the CDΥ(κ, F ) condition, where the dimension term is expressed by a so called CD-function F . We derive functional inequalities relating the entropy to the Fisher information, which we will call entropy-information inequalities. Further, we deduce applications of entropyinformation inequalities such as ultracontractivity bounds, exponential integrability of Lipschitz functions, finite diameter bounds and a modified version of the celebrated Nash inequality.

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Bakry-Émery curvature on graphs as an eigenvalue problem

et al. 2022

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In this paper, we reformulate the Bakry-Émery curvature on a weighted graph in terms of the smallest eigenvalue of a rank one perturbation of the so-called curvature matrix using Schur complement. This new viewpoint allows us to show various curvature function properties in a very conceptual way. We show that the curvature, as a function of the dimension parameter, is analytic, strictly monotone increasing and strictly concave until a certain threshold after which the function is constant. Furthermore, we derive the curvature of the Cartesian product using the crucial observation that the curvature matrix of the product is the direct sum of each component. Our approach of the curvature functions of graphs can be employed to establish analogous results for the curvature functions of weighted Riemannian manifolds. Moreover, as an application, we confirm a conjecture (in a general weighted case) of the fact that the curvature does not decrease under certain graph modifications.

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Curvatures, graph products and Ricci flatness

Cited by 10 publications

References 13 publications

Lin-Lu-Yau curvature and diameter of amply regular graphs

Lin-Lu-Yau curvature and diameter of amply regular graphs

Entropy-information inequalities under curvature-dimension conditions for continuous-time Markov chains

Bakry-Émery curvature on graphs as an eigenvalue problem

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