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Professor Norbert Peyerimhoff's Outputs (70)

NP-completeness of the combinatorial distance matrix realisation problem (2025)
Presentation / Conference Contribution
Fairbairn, D., Mertzios, G., & Peyerimhoff, N. (2025, December). NP-completeness of the combinatorial distance matrix realisation problem. Presented at 14th International Symposium on Algorithms and Complexity (CIAC 2025), Rome, Italy

The k-CombDMR problem is that of determining whether an n×n distance matrix can be realised by n vertices in some undirected graph with n+k vertices. This problem has a simple solution in the case k=0. In this paper we show that this problem is polyn... Read More about NP-completeness of the combinatorial distance matrix realisation problem.

Bakry-Émery curvature sharpness and curvature flow in finite weighted graphs: theory (2025)
Journal Article
Cushing, D., Kamtue, S., Liu, S., Münch, F., Peyerimhoff, N., & Snodgrass, B. (2025). Bakry-Émery curvature sharpness and curvature flow in finite weighted graphs: theory. manuscripta mathematica, 176(1), Article 11. https://doi.org/10.1007/s00229-024-01606-7

In this sequence of two papers, we introduce a curvature flow on (mixed) weighted graphs which is based on the Bakry-Émery calculus. The flow is described via a time-continuous evolution through the weighting schemes. By adapting this flow to preserv... Read More about Bakry-Émery curvature sharpness and curvature flow in finite weighted graphs: theory.

Refinement of X‐ray and electron diffraction crystal structures using analytical Fourier transforms of Slater‐type atomic wavefunctions in Olex2 (2024)
Journal Article
Kleemiss, F., Peyerimhoff, N., & Bodensteiner, M. (2024). Refinement of X‐ray and electron diffraction crystal structures using analytical Fourier transforms of Slater‐type atomic wavefunctions in Olex2. Journal of Applied Crystallography, 57, 161-174. https://doi.org/10.1107/s1600576723010981

An implementation of Slater‐type spherical scattering factors for X‐ray and electron diffraction for elements in the range Z = 1–103 is presented within the software Olex2. Both high‐ and low‐angle Fourier behaviour of atomic electron density and ele... Read More about Refinement of X‐ray and electron diffraction crystal structures using analytical Fourier transforms of Slater‐type atomic wavefunctions in Olex2.

Eigenvalue estimates for the magnetic Hodge Laplacian on differential forms (2023)
Journal Article
Egidi, M., Gittins, K., Habib, G., & Peyerimhoff, N. (2023). Eigenvalue estimates for the magnetic Hodge Laplacian on differential forms. Journal of Spectral Theory, 13(4), 1297-1343. https://doi.org/10.4171/JST/480

In this paper we introduce the magnetic Hodge Laplacian, which is a generalization of the magnetic Laplacian on functions to differential forms. We consider various spectral results, which are known for the magnetic Laplacian on functions or for the... Read More about Eigenvalue estimates for the magnetic Hodge Laplacian on differential forms.

Bakry–Émery Curvature Sharpness and Curvature Flow in Finite Weighted Graphs. Implementation (2023)
Journal Article
Cushing, D., Kamtue, S., Liu, S., Münch, F., Peyerimhoff, N., & Snodgrass, B. (2023). Bakry–Émery Curvature Sharpness and Curvature Flow in Finite Weighted Graphs. Implementation. Axioms, 12(6), Article 577. https://doi.org/10.3390/axioms12060577

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... Read More about Bakry–Émery Curvature Sharpness and Curvature Flow in Finite Weighted Graphs. Implementation.

Going round in circles: Geometry in the early years (2023)
Journal Article
Oughton, R. H., Wheadon, D. M., Bolden, D. S., Nichols, K., Fearn, S., Darwin, S., Dixon-Jones, S., Mistry, M., Peyerimhoff, N., & Townsend, A. (2023). Going round in circles: Geometry in the early years. Mathematics teaching, 286, 29-34

The research described here came from a collaboration between university-based mathematicians and early years (EY) educators. The project emerged naturally, driven by the felt need of the EY educators to develop a broader understanding and appreciati... Read More about Going round in circles: Geometry in the early years.

Parameterized Counting and Cayley Graph Expanders (2023)
Journal Article
Peyerimhoff, N., Roth, M., Schmitt, J., Stix, J., Vdovina, A., & Wellnitz, P. (2023). Parameterized Counting and Cayley Graph Expanders. SIAM Journal on Discrete Mathematics, 37(2), 405-486. https://doi.org/10.1137/22m1479804

Given a graph property \Phi , we consider the problem \# EdgeSub(\Phi ), where the input is a pair of a graph G and a positive integer k, and the task is to compute the number of k-edge subgraphs in G that satisfy \Phi . Specifically, we study the pa... Read More about Parameterized Counting and Cayley Graph Expanders.

Rigidity properties of the hypercube via Bakry–Émery curvature (2022)
Journal Article
Liu, S., Münch, F., & Peyerimhoff, N. (2024). Rigidity properties of the hypercube via Bakry–Émery curvature. Mathematische Annalen, 388(2), 1225-1259. https://doi.org/10.1007/s00208-022-02537-y

We give rigidity results for the discrete Bonnet–Myers diameter bound and the Lichnerowicz eigenvalue estimate. Both inequalities are sharp if and only if the underlying graph is a hypercube. The proofs use well-known semigroup methods as well as new... Read More about Rigidity properties of the hypercube via Bakry–Émery curvature.

Developing ‘deep mathematical thinking’ in geometry with 3- and 4-year-olds: A collaborative study between early years teachers and University-based mathematicians (2022)
Journal Article
Oughton, R., Nichols, K., Bolden, D. S., Dixon-Jones, S., Fearn, S., Darwin, S., Mistry, M., Peyerimhoff, N., & Townsend, A. (2024). Developing ‘deep mathematical thinking’ in geometry with 3- and 4-year-olds: A collaborative study between early years teachers and University-based mathematicians. Mathematical Thinking and Learning, 26(3), 306-325. https://doi.org/10.1080/10986065.2022.2119497

Mathematics in early years settings is often restricted to learning to count and identifying simple shapes. This is partly due to the narrow scope of many early years curricula and insufficient teacher training for exploring deeper mathematical conce... Read More about Developing ‘deep mathematical thinking’ in geometry with 3- and 4-year-olds: A collaborative study between early years teachers and University-based mathematicians.

Refinement of anomalous dispersion correction parameters in single-crystal structure determinations (2022)
Journal Article
Meurer, F., Dolomanov, O. V., Hennig, C., Peyerimhoff, N., Kleemiss, F., Puschmann, H., & Bodensteiner, M. (2022). Refinement of anomalous dispersion correction parameters in single-crystal structure determinations. IUCrJ, 9(5), https://doi.org/10.1107/s2052252522006844

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... Read More about Refinement of anomalous dispersion correction parameters in single-crystal structure determinations.

Bakry-Émery curvature on graphs as an eigenvalue problem (2022)
Journal Article
Cushing, D., Kamtue, S., Liu, S., & Peyerimhoff, N. (2022). Bakry-Émery curvature on graphs as an eigenvalue problem. Calculus of Variations and Partial Differential Equations, 61, Article 62. https://doi.org/10.1007/s00526-021-02179-z

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 curvatur... Read More about Bakry-Émery curvature on graphs as an eigenvalue problem.

Vanishing of the atomic form factor derivatives in non-spherical structural refinement – a key approximation scrutinized in the case of Hirshfeld atom refinement (2021)
Journal Article
Midgley, L., Bourhis, L. J., Dolomanov, O. V., Grabowsky, S., Kleemiss, F., Puschmann, H., & Peyerimhoff, N. (2021). Vanishing of the atomic form factor derivatives in non-spherical structural refinement – a key approximation scrutinized in the case of Hirshfeld atom refinement. Acta Crystallographica Section A: Foundations and Advances, 77(6), 519-533. https://doi.org/10.1107/s2053273321009086

When calculating derivatives of structure factors, there is one particular term (the derivatives of the atomic form factors) that will always be zero in the case of tabulated spherical atomic form factors. What happens if the form factors are non-sph... Read More about Vanishing of the atomic form factor derivatives in non-spherical structural refinement – a key approximation scrutinized in the case of Hirshfeld atom refinement.

Eigenfunctions and the Integrated Density of States on Archimedean Tilings (2021)
Journal Article
Peyerimhoff, N., & Taeufer, M. (2021). Eigenfunctions and the Integrated Density of States on Archimedean Tilings. Journal of Spectral Theory, 11(2), 461-488. https://doi.org/10.4171/jst/347

We study existence and absence of ` 2 -eigenfunctions of the combinatorial Laplacian on the 11 Archimedean tilings of the Euclidean plane by regular convex polygons. We show that exactly two of these tilings (namely the .3:6/2 “kagome” tiling and the... Read More about Eigenfunctions and the Integrated Density of States on Archimedean Tilings.

Ricci curvature and eigenvalue estimates for the magnetic Laplacian on manifolds (2021)
Journal Article
Egidi, M., Liu, S., Muench, F., & Peyerimhoff, N. (2021). Ricci curvature and eigenvalue estimates for the magnetic Laplacian on manifolds. Communications in Analysis and Geometry, 29(5), 1127-1156. https://doi.org/10.4310/cag.2021.v29.n5.a4

In this paper, we present a Lichnerowicz type estimate and (higher order) Buser type estimates for the magnetic Laplacian on a closed Riemannian manifold with a magnetic potential. These results relate eigenvalues, magnetic fields, Ricci curvature, a... Read More about Ricci curvature and eigenvalue estimates for the magnetic Laplacian on manifolds.

Accurate Crystal Structures and Chemical Properties from NoSpherA2 (2020)
Journal Article
Kleemiss, F., Dolomanov, O. V., Bodensteiner, M., Peyerimhoff, N., Midgley, L., Bourhis, L. J., Genoni, A., Malaspina, L. A., Jayatilaka, D., Spencer, J. L., White, F., Grundkötter-Stock, B., Steinhauer, S., Lentz, D., Puschmann, H., & Grabowsky, S. (2021). Accurate Crystal Structures and Chemical Properties from NoSpherA2. Chemical Science, 12, 1675-1692. https://doi.org/10.1039/d0sc05526c

The relationship between the structure and the properties of a drug or material is a key concept of chemistry. Knowledge of the three-dimensional structure is considered to be of such importance that almost every report of a new chemical compound is... Read More about Accurate Crystal Structures and Chemical Properties from NoSpherA2.

Curvatures, Graph Products and Ricci Flatness (2020)
Journal Article
Cushing, D., Kamtue, S., Kangaslampi, R., Liu, S., & Peyerimhoff, N. (2021). Curvatures, Graph Products and Ricci Flatness. Journal of Graph Theory, 96(4), 522-553. https://doi.org/10.1002/jgt.22630

In this paper, we compare Ollivier–Ricci curvature and Bakry–Émery curvature notions on combinatorial graphs and discuss connections to various types of Ricci flatness. We show that nonnegativity of Ollivier–Ricci curvature implies the nonnegativity... Read More about Curvatures, Graph Products and Ricci Flatness.

Signatures, Lifts, and Eigenvalues of Graphs (2020)
Book Chapter
Liu, S., Peyerimhoff, N., & Vdovina, A. (2020). Signatures, Lifts, and Eigenvalues of Graphs. In F. M. Atay, P. B. Kurasov, & D. Mugnolo (Eds.), Discrete and continuous models in the theory of networks : operator theory : advances and applications (255-269). Birkhäuser Verlag. https://doi.org/10.1007/978-3-030-44097-8_13

We study the spectra of cyclic signatures of finite graphs and the corresponding cyclic lifts. Starting from a bipartite Ramanujan graph, we prove the existence of an infinite tower of 3-cyclic lifts, each of which is again Ramanujan.

Curvature Calculations for Antitrees (2020)
Book Chapter
Cushing, D., Liu, S., Muench, F., & Peyerimhoff, N. (2020). Curvature Calculations for Antitrees. In M. Keller, D. Lenz, & R. K. Wojciechowski (Eds.), Analysis and geometry on graphs and manifolds (21-54). Cambridge University Press. https://doi.org/10.1017/9781108615259.003

In this article we prove that antitrees with suitable growth properties are examples of infinite graphs exhibiting strictly positive curvature in various contexts: in the normalized and non-normalized Bakry-Émery setting as well in the Ollivier-Ricci... Read More about Curvature Calculations for Antitrees.