Bakry–Émery Curvature Functions on Graphs

Cushing, D. H.; Liu, Shiping; Peyerimhoff, Norbert

doi:10.4153/cjm-2018-015-4

Cited by 34 publications

(91 citation statements)

References 41 publications

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“…Several adaptations of Ricci curvature such as Bakry-Émery curvature (see e.g. [19,17,7]), Ollivier-Ricci curvature [31], Entropic curvature introduced by Erbar and Maas [10], and Forman curvature [13,35], have emerged on graphs in recent years, and there is very active research on these notions. We refer to [29] and the references therein for this vibrant research field.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…The following reformulation can also be found in [7] and [20]. Then we have the following decomposition of the 2-ball B 2 (x):…”

Section: Bakry-émery Curvature As a Semidefinite Programming Problemmentioning

confidence: 99%

“…For each of the above options there are "Normalised" analogues in which the above procedure is carried out but with respect to the Normalised Laplacian. See [7] for further details.…”

Section: Bakry-émery Curvature As a Semidefinite Programming Problemmentioning

confidence: 99%

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The Graph Curvature Calculator and the Curvatures of Cubic Graphs

Cushing

Kangaslampi

Lipiäinen

et al. 2019

Experimental Mathematics

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We classify all cubic graphs with either non-negative Ollivier-Ricci curvature or non-negative Bakry-Émery curvature everywhere. We show in both curvature notions that the non-negatively curved graphs are the prism graphs and the Möbius ladders. We also highlight an online tool for calculating the curvature of graphs under several variants of the curvature notions that we use in the classification. As a consequence of the classification result we show that non-negatively curved cubic expanders do not exist.arXiv:1712.03033v2 [math.CO]

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…The following reformulation can also be found in [7] and [20]. Then we have the following decomposition of the 2-ball B 2 (x):…”

Section: Bakry-émery Curvature As a Semidefinite Programming Problemmentioning

confidence: 99%

See 1 more Smart Citation

The Graph Curvature Calculator and the Curvatures of Cubic Graphs

Cushing

Kangaslampi

Lipiäinen

et al. 2019

Experimental Mathematics

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“…Recently, there has been a tremendous interest in various notions of curvature, and especially of Ricci curvature, for graphs, see, for example, [4,8,10,15,17,24,[26][27][28][33][34][35]42,50,51,60,[63][64][65][66][67][68][69]71,[73][74][75][76][77][78][79][83][84][85]87,90,97]. In particular, two notions have been most prominently explored for finding analogues to results in the setting of Riemannian manifolds for graphs: the Bakry-Émery approach having its origins in [2] and the coarse Ollivier-Ricci curvature originating in the work [78].…”

Section: 3mentioning

confidence: 99%

Coverings and the heat equation on graphs: Stochastic incompleteness, the Feller property, and uniform transience

Hua

Münch

Wojciechowski

2019

Trans. Amer. Math. Soc.

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We study regular coverings of graphs and manifolds with a focus on properties of the heat equation. In particular, we look at stochastic incompleteness, the Feller property and uniform transience; and investigate the connection between the validity of these properties on the base space and its covering. For both graphs and manifolds, we prove the equivalence of stochastic incompleteness of the base and that of its cover. Along the way we also give some new conditions for the Feller property to hold on graphs.

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“…It is conceivable that the results in [GW18] may also have other applications, for example with regards to the following conjecture about expander graph families (Conjecture 9.11 in [CLP17]):…”

Section: Introductionmentioning

confidence: 99%

Quartic graphs which are Bakry-Émery curvature sharp

Cushing

Kamtue

Peyerimhoff

et al. 2020

Discrete Mathematics

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We give a classification of all connected quartic graphs which are (infinity) curvature sharp in all vertices with respect to Bakry-Émery curvature. The result is based on a computer classification by F. Gurr and L. Watson May and a combinatorial case by case investigation. 2 CUSHING, KAMTUE, PEYERIMHOFF, AND WATSON MAY equality. The main result in the paper is a complete classification of all 4-regular (quartic) curvature sharp graphs: Theorem 1.1. Let G = (V, E) be a connected quartic graph which is Bakry-Émery curvature sharp in all vertices. Then G is one of the following: (i) The complete graph K 5 with |V | = 5, K ∞ = 3.5, diam G = 1; (ii) The octahedral graph O with |V | = 6, K ∞ = 3, diam G = 2; (iii) The Cartesian product K 3 × K 3 of two copies of the complete graph K 3 with |V | = 9, K ∞ = 2.5, diam G = 2; (iv) The complete bipartite graph K 4,4 with |V | = 8, K ∞ = 2, diam G = 2; (v) The crown graph C(10) with |V | = 10, K ∞ = 2, diam G = 3; (vi) The Cayley graph Cay(D 12 , S) of the dihedral group D 12 of order 12 with generators S = {r 3 , s, sr 2 , sr 4 } with |V | = 12, K ∞ = 2, diam G = 3; (vii) The Cayley graph Cay(D 14 , S) of the dihedral group D 14 of order 14 with generators S = {s, sr, sr 4 , sr 6 } with |V | = 14, K ∞ = 2, diam G = 3; (viii) The 4-dimensional hypercube Q 4 with |V | = 16, K ∞ = 2, diam G = 4.

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Bakry–Émery Curvature Functions on Graphs

Cited by 34 publications

References 41 publications

The Graph Curvature Calculator and the Curvatures of Cubic Graphs

The Graph Curvature Calculator and the Curvatures of Cubic Graphs

Coverings and the heat equation on graphs: Stochastic incompleteness, the Feller property, and uniform transience

Quartic graphs which are Bakry-Émery curvature sharp

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