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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.

Let X be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of non-compact harmonic manifolds except for the flat spaces. Denote by h > 0 the mean curvature of horospheres in X , and set ρ = h / 2 . Fixing a basepoint o ∈ X , for ξ ∈ ∂ X , denote by B ξ the Busemann function at ξ such that B ξ ( o ) = 0 . Then for λ ∈ C the function e ( i λ - ρ ) B ξ is an eigenfunction of the Laplace–Beltrami operator with eigenvalue - ( λ 2 + ρ 2 ) . For a function f on X , we define the Fourier transform of f by f ~ ( λ , ξ ) : = ∫ X f ( x ) e ( - i λ - ρ ) B ξ ( x ) d v o l ( x ) for all λ ∈ C , ξ ∈ ∂ X for which the integral converges. We prove a Fourier inversion formula f ( x ) = C 0 ∫ 0 ∞ ∫ ∂ X f ~ ( λ , ξ ) e ( i λ - ρ ) B ξ ( x ) d λ o ( ξ ) | c ( λ ) | - 2 d λ for f ∈ C c ∞ ( X ) , where c is a certain function on R - { 0 } , λ o is the visibility measure on ∂ X with respect to the basepoint o ∈ X and C 0 > 0 is a constant. We also prove a Plancherel theorem, and a version of the Kunze–Stein phenomenon.

Correcting for anomalous dispersion is part of any refinement of an X-ray diffraction crystal structure determination. The procedure takes the inelastic scattering in the diffraction experiment into account. This X-ray absorption effect is specific to each chemical compound and is particularly sensitive to radiation energies in the region of the absorption edges of the elements in the compound. Therefore, the widely used tabulated values for these corrections can only be approximations as they are based on calculations for isolated atoms. Features of the unique spatial and electronic environment that are directly related to the anomalous dispersion are ignored, although these can be observed spectroscopically. This significantly affects the fit between the crystallographic model and the measured intensities when the excitation wavelength in an X-ray diffraction experiment is close to an element's absorption edge. Herein, we report on synchrotron multi-wavelength single-crystal X-ray diffraction, as well as X-ray absorption spectroscopy experiments which we performed on the molecular compound Mo(CO) 6 at energies around the molybdenum K edge. The dispersive ( f ′) and absorptive ( f ′′) terms of the anomalous dispersion can be refined as independent parameters in the full-matrix least-squares refinement. This procedure has been implemented as a new feature in the well-established OLEX2 software suite. These refined parameters are in good agreement with the independently recorded X-ray absorption spectrum. The resulting crystallographic models show significant improvement compared to those employing tabulated values.

In this paper, we discuss the implementation of a curvature flow on weighted graphs based on the Bakry–Émery calculus. This flow can be adapted to preserve the Markovian property and its limits as time goes to infinity turn out to be curvature sharp weighted graphs. After reviewing some of the main results of the corresponding paper concerned with the theoretical aspects, we present various examples (random graphs, paths, cycles, complete graphs, wedge sums and Cartesian products of complete graphs, and hypercubes) and exhibit various properties of this flow. One particular aspect of our investigations is asymptotic stability and instability of curvature flow equilibria. The paper ends with a description of the Python functions and routines freely available in an ancillary file on arXiv or via github. We hope that the explanations of the Python implementation via examples will help users to carry out their own curvature flow experiments.

In this article, we study relations between the local geometry of planar graphs (combinatorial curvature) and global geometric invariants , namely the Cheeger constants and the exponential growth. We also discuss spectral applications.

We show various sharp Hardy-type inequalities for the linear and quasi-linear Laplacian on non-compact harmonic manifolds with a particular focus on the case of Damek–Ricci spaces. Our methods make use of the optimality theory developed by Devyver, Fraas and Pinchover and Devyver and Pinchover and are motivated by corresponding results for hyperbolic spaces by Berchio, Ganguly and Grillo, and Berchio, Ganguly, Grillo and Pinchover.

We prove distance bounds for graphs possessing positive Bakry-Émery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-positively curved vertices is finite, then the graph admits an explicit upper bound for the diameter. Otherwise, the graph is a subset of the tubular neighborhood with an explicit radius around the non-positively curved vertices. Those results seem to be the first assuming non-constant Bakry-Émery curvature assumptions on graphs.

We derive an optimal eigenvalue ratio estimate for finite weighted graphs satisfying the curvature-dimension inequality CD(0, ∞). This estimate is independent of the size of the graph and provides a general method to obtain higher-order spectral estimates. The operation of taking Cartesian products is shown to be an efficient way for constructing new weighted graphs satisfying CD(0, ∞). We also discuss a higher-order Cheeger constant-ratio estimate and related topics about expanders.

We study local properties of the Bakry–Émery curvature function ${\\mathcal{K}}_{G,x}:(0,\\infty ]\\rightarrow \\mathbb{R}$ at a vertex $x$ of a graph $G$ systematically. Here ${\\mathcal{K}}_{G,x}({\\mathcal{N}})$ is defined as the optimal curvature lower bound ${\\mathcal{K}}$ in the Bakry–Émery curvature-dimension inequality $CD({\\mathcal{K}},{\\mathcal{N}})$ that $x$ satisfies. We provide upper and lower bounds for the curvature functions, introduce fundamental concepts like curvature sharpness and $S^{1}$ -out regularity, and relate the curvature functions of $G$ with various spectral properties of (weighted) graphs constructed from local structures of $G$ . We prove that the curvature functions of the Cartesian product of two graphs $G_{1},G_{2}$ are equal to an abstract product of curvature functions of $G_{1},G_{2}$ . We explore the curvature functions of Cayley graphs and many particular (families of) examples. We present various conjectures and construct an infinite increasing family of 6-regular graphs which satisfy $CD(0,\\infty )$ but are not Cayley graphs.

We prove finiteness and diameter bounds for graphs having a positive Ricci-curvature bound in the Bakry–Émery sense. Our first result using only curvature and maximal vertex degree is sharp in the case of hypercubes. The second result depends on an additional dimension bound, but is independent of the vertex degree. In particular, the second result is the first Bonnet–Myers type theorem for unbounded graph Laplacians. Moreover, our results improve diameter bounds from Fathi and Shu (Bernoulli 24(1):672–698, 2018) and Horn et al. (J für die reine und angewandte Mathematik (Crelle’s J), 2017, https://doi.org/10.1515/crelle-2017-0038) and solve a conjecture from Cushing et al. (Bakry–Émery curvature functions of graphs, 2016).