In this article we study two discrete curvature notions, Bakry-\'Emery curvature and Ollivier Ricci curvature, on Cayley graphs. We introduce Right Angled Artin-Coxeter Hybrids (RAACHs) generalizing Right Angled Artin and Coxeter groups (RAAGs and RACGs) and derive the curvatures of Cayley graphs of certain RAACHs. Moreover, we show for general finitely presented groups $\Gamma = \langle S \, \mid\, R \rangle$ that addition of relators does not lead to a decrease the weighted curvatures of their Cayley graphs with adapted weighting schemes.

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 note, we provide Steinerberger curvature formulas for block graphs, discuss curvature relations between two graphs and the graph obtained by connecting them via a bridge, and show that self-centered Bonnet-Myers sharp graphs are precisely those which are antipodal. We also discuss similarities and differences between Steinerberger and Ollivier Ricci curvature results.

We describe some recent computer investigations with the `Constraint Logic Programming over Finite Domains' -- CLP(FD) -- library in the Prolog programming environment to search for new simple Lie algebras over the field $\GF(2)$ of $2$ elements. Motivated by a paper of Grishkov et. al., we specifically look for those with a `thin decomposition', and we settle one of their conjectures. We extrapolate from our results the existence of two new infinite families of simple Lie algebras, in addition to finding seven new sporadic examples in dimension $31$. We also better contextualise some previously discovered simple algebras, putting them into families which do not seem to have ever appeared in the literature, and give an updated table of those currently known.

Self-centred regular graphs which are Ollivier-Ricci Bonnet-Myers sharp have been completely classified. When the conditions of self-centeredness and regularity are removed it is an open problem on what the classification is. We present a complete classification of Bonnet-Myers sharp graphs with a diameter 2 and show that there exists Bonnet-Myers sharp graphs of diameter 3, 4 and 6 that belong to a family of graphs called symmetrical antitees.

We develop and deploy a set of constraints for the purpose of calculating minimal sizes of lottery designs. Specifically, we find the minimum number of tickets of size six which are needed to match at least two balls on any draw of size six, whenever there are at most 70 balls.

This paper develops and deploys a set of constraints for the purpose of calculating minimal sizes of lottery designs, with an application to the UK National Lottery: in case there are n = 59 balls, the minimum number of tickets of size six which are needed to match at least two balls on any draw of size six is precisely 27. In fact, we calculate this minimum value for all n at most 70.

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.

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 these 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.

In this paper, we compare Ollivier Ricci curvature and Bakry-\'Emery 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-\'Emery 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.

We classify all connected, simple, 3-regular graphs with girth at least 5 that are Ricci-flat. We use the definition of Ricci curvature on graphs given in Lin-Lu-Yau, Tohoku Math., 2011, which is a variation of Ollivier, J. Funct. Anal., 2009. A graph is Ricci-flat, if it has vanishing Ricci curvature on all edges. We show, that the only Ricci-flat cubic graphs with girth at least 5 are the Petersen graph, the Triplex and the dodecahedral graph. This will correct the classification in Lin-Lu-Yau, Comm. Anal. Geom., 2014, that misses the Triplex.

In Theorem 1 of [1], the classification of Ricci-flat graph with girth g(G) > 5 missed one graph ??? the Triplex graph, as discovered by three authors: Cushing, Kangaslampi, and Liu. Here is the correct theorem.

This erratum will correct the classification of Theorem 1 in Lin-Lu-Yau, Comm. Anal. Geom., 2014, that misses the Triplex graph.

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.

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.

Abstract 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 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 and classify all distance-regular Lichnerowicz sharp graphs under the additional condition θ1=b1−1. We also relate Bonnet-Myers sharpness to an upper bound of Bakry-Émery ∞-curvature, which motivates a general conjecture about Bakry-Émery ∞-curvature.

We study the long-scale Ollivier Ricci curvature of graphs as a function of the chosen idleness. Similarly to the previous work on the short-scale case, we show that this idleness function is concave and piecewise linear with at most 3 linear parts. We provide bounds on the length of the first and last linear pieces. We also study the long-scale curvature for the Cartesian product of two regular 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.