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Norbert Peyerimhoff

Publications and source records attributed to Norbert Peyerimhoff.

At least 37 records · Page 2Linked to original sources
arXiv2021

Parameterized (Modular) Counting and Cayley Graph Expanders

We study the problem $\#\mathrm{EdgeSub}(Φ)$ of counting $k$-edge subgraphs satisfying a given graph property $Φ$ in a large host graph $G$. Building upon the breakthrough result of Curticapean, Dell and Marx (STOC 17), we express the number of such subgraphs as a finite linear combination of graph homomorphism counts and derive the complexity of computing this number by studying its coefficients. Our approach relies on novel constructions of low-degree Cayley graph expanders of $p$-groups, which might be of independent interest. The properties of those expanders allow us to analyse the coefficients in the aforementioned linear combinations over the field $\mathbb{F}_p$ which gives us significantly more control over the cancellation behaviour of the coefficients. Our main result is an exhaustive and fine-grained complexity classification of $\#\mathrm{EdgeSub}(Φ)$ for minor-closed properties $Φ$, closing the missing gap in previous work by Roth, Schmitt and Wellnitz (ICALP 21). Additionally, we observe that our methods also apply to modular counting. Among others, we investigate the problems of modular counting of paths, cycles, forests and matroid bases. In the course of our investigations we also provide an exhaustive parameterized complexity classification for the problem of counting graph homomorphisms modulo a prime $p$.

cs.CC↗
arXiv2020

A note on eigenvalue bounds for non-compact manifolds

In this article we prove upper bounds for the Laplace eigenvalues $λ_k$ below the essential spectrum for strictly negatively curved Cartan-Hadamard manifolds. Our bound is given in terms of $k^2$ and specific geometric data of the manifold. This applies also to the particular case of non-compact manifolds whose sectional curvature tends to $-\infty$, where no essential spectrum is present due to a theorem of Donnelly/Li. The result stands in clear contrast to Laplacians on graphs where such a bound fails to be true in general.

math.DG↗
arXiv2020

Eigenfunctions and the Integrated Density of States on Archimedean Tilings

We study existence and absence of $\ell^2$-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 $(3.12^2)$ tiling) have $\ell^2$-eigenfunctions. These eigenfunctions are infinitely degenerate and are constituted of explicitly described eigenfunctions which are supported on a finite number of vertices of the underlying graph (namely on the hexagons and $12$-gons in the tilings, respectively). Furthermore, we provide an explicit expression for the Integrated Density of States (IDS) of the Laplacian on Archimedean tilings in terms of eigenvalues of Floquet matrices and deduce integral formulas for the IDS of the Laplacian on the $(4^4)$, $(3^6)$, $(6^3)$, $(3.6)^2$, and $(3.12^2)$ tilings. Our method of proof can be applied to other $\mathbb{Z}^d$-periodic graphs as well.

math-ph↗
arXiv2019

Curvatures, graph products and Ricci flatness

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 non-negativity of Ollivier Ricci curvature implies non-negativity of Bakry-Émery curvature under triangle-freeness and an additional in-degree condition. We also provide examples that both conditions of this result are necessary. We investigate relations to graph products and show that Ricci flatness is preserved under all natural products. While non-negativity of both curvatures are preserved under Cartesian products, we show that in the case of strong products, non-negativity of Ollivier Ricci curvature is only preserved for horizontal and vertical edges. We also prove that all distance-regular graphs of girth $4$ attain their maximal possible curvature values.

math.CO↗
arXiv2019

The Fourier transform on harmonic manifolds of purely exponential volume growth

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 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 $o \in X$, for $ξ\in \partial X$, denote by $B_ξ$ the Busemann function at $ξ$ such that $B_ξ(o) = 0$. then for $λ\in \C$ the function $e^{(iλ- ρ)B_ξ}$ 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 $$\tilde{f}(λ, ξ) := \int_X f(x) e^{(-iλ- ρ)B_ξ(x)} dvol(x)$$ for all $λ\in \C, ξ\in \partial X$ for which the integral converges. We prove a Fourier inversion formula $$f(x) = C_0 \int_{0}^{\infty} \int_{\partial X} \tilde{f}(λ, ξ) e^{(iλ- ρ)B_ξ(x)} dλ_o(ξ) |c(λ)|^{-2} dλ$$ for $f \in C^{\infty}_c(X)$, where $c$ is a certain function on $\mathbb{R} - \{0\}$, $λ_o$ is the visibility measure on $\partial X$ with respect to the basepoint $o \in X$ and $C_0 > 0$ is a constant. We also prove a Plancherel theorem, and a version of the Kunze-Stein phenomenon.

math.DG↗
arXiv2019

Minimal codimension one foliation of a symmetric space by Damek-Ricci spaces

In this article we consider solvable hypersurfaces of the form $N \exp(\R H)$ with induced metrics in the symmetric space $M = SL(3,\C)/SU(3)$, where $H$ a suitable unit length vector in the subgroup $A$ of the Iwasawa decomposition $SL(3,\C) = NAK$. Since $M$ is rank $2$, $A$ is $2$-dimensional and we can parametrize these hypersurfaces via an angle $α\in [0,π/2]$ determining the direction of $H$. We show that one of the hypersurfaces (corresponding to $α= 0$) is minimally embedded and isometric to the non-symmetric $7$-dimensional Damek-Ricci space. We also provide an explicit formula for the Ricci curvature of these hypersurfaces and show that all hypersurfaces for $α\in (0,\fracπ{2}]$ admit planes of both negative and positive sectional curvature. Moreover, the symmetric space $M$ admits a minimal foliation with all leaves isometric to the non-symmetric $7$-dimensional Damek-Ricci space.

math.DG↗
arXiv2019

Distance bounds for graphs with some negative Bakry-Émery curvature

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.

math.DG↗
arXiv2019

Quartic graphs which are Bakry-Émery curvature sharp

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.

math.CO↗
arXiv2018

Trivalent expanders, $(Δ-Y)$-transformation, and hyperbolic surfaces

We construct a new family of trivalent expanders tessellating hyperbolic surfaces with large isometry groups. These graphs are obtained from a family of Cayley graphs of nilpotent groups via $(Δ-Y)$-transformations. We compare this family with Platonic graphs and their associated hyperbolic surfaces and see that they are generally very different with only one hyperbolic surface in the intersection. Moreover, we study combinatorial, topological and spectral properties of our trivalent graphs and their associated hyperbolic surfaces.

math.CO↗
arXiv2018

Rigidity of the Bonnet-Myers inequality for graphs with respect to Ollivier Ricci curvature

We introduce the notion of Bonnet-Myers and Lichnerowicz sharpness in the Ollivier Ricci curvature sense. Our main result is a classification of all self-centered Bonnet-Myers sharp graphs (hypercubes, cocktail party graphs, even-dimensional demi-cubes, Johnson graphs $J(2n,n)$, the Gosset graph and suitable Cartesian products). We also present a purely combinatorial reformulation of this result. We show that Bonnet-Myers sharpness implies Lichnerowicz sharpness. We also relate Bonnet-Myers sharpness to an upper bound of Bakry-Émery $\infty$-curvature, which motivates a generalconjecture about Bakry-Émery $\infty$-curvature.

math.CO↗
arXiv2018

Eigenvalue ratios of nonnegatively curved graphs

We derive an optimal eigenvalue ratio estimate for finite weighted graphs satisfying the curvature-dimension inequality $CD(0,\infty)$. 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 $CD(0,\infty)$. We also discuss a higher order Cheeger constant ratio estimate and related topics about expanders.

math.SP↗
arXiv2018

Curvature calculations for antitrees

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 curvature case. We also show that these graphs do not have global positive lower curvature bounds, which one would expect in view of discrete analogues of the Bonnet-Myers theorem. The proofs in the different settings require different techniques.

math.CO↗
arXiv2018

A support theorem for the X-ray transform on manifolds with plane covers

This article is concerned with support theorems of the X-ray transform on non-compact manifolds with conjugate points. In particular, we prove that all simply connected 2-step nilpotent Lie groups have a support theorem. Important ingredients of the proof are the concept of plane covers and a support theorem for simple manifolds by Krishnan.We also provide examples of non-homogeneous 3-dimensional simply connected manifolds with conjugate points which have support theorems.

math.DG↗
arXiv2017

Bakry-Émery curvature functions of graphs

We study the Bakry-Émery curvature function $\mathcal{K}_{G,x}:(0,\infty]\to \mathbb{R}$ of a vertex $x$ in a locally finite 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 prove the curvature functions of the Cartesian product of two graphs $G_1,G_2$ equal an abstract product of curvature functions of $G_1,G_2$. We relate the curvature functions of $G$ with various spectral properties of (weighted) graphs constructed from local structures of $G$. We explore the curvature functions of Cayley graphs, strongly regular graphs, and many particular (families of) examples including Johnson graphs and complete bipartite graphs. We construct an infinite family of $6$-regular graphs which satisfy $CD(0,\infty)$ but are not Cayley graphs.

math.CO↗
arXiv2017

Ollivier-Ricci idleness functions of graphs

We study the Ollivier-Ricci curvature of graphs as a function of the chosen idleness. We show that this idleness function is concave and piecewise linear with at most $3$ linear parts, with at most $2$ linear parts in the case of a regular graph. We then apply our result to show that the idleness function of the Cartesian product of two regular graphs is completely determined by the idleness functions of the factors.

math.CO↗
arXiv2017

Rigidity properties of the hypercube via Bakry-Emery curvature

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 direct methods which translate curvature to combinatorial properties. Our results can be seen as first known discrete analogues of Cheng's and Obata's rigidity theorems.

math.DG↗
arXiv2017

Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space

For the analysis of the Schrödinger and related equations it is of central importance whether a unique continuation principle (UCP) holds or not. In continuum Euclidean space quantitative forms of unique continuation imply Wegner estimates and regularity properties of the integrated density of states (IDS) of Schrödinger operators with random potentials. For discrete Schrödinger equations on the lattice only a weak analog of the UCP holds, but it is sufficient to guarantee the continuity of the IDS. For other combinatorial graphs this is no longer true. Similarly, for quantum graphs the UCP does not hold in general and consequently, the IDS does not need to be continuous.

math-ph↗