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...

We show that the growth of plane tessellations and their edge graphs may be controlled from below by upper bounds for the combinatorial curvature. Under the assumption that every geodesic path may be extended to infinity we provide explicit estimates of the growth rate and isoperimetric constant of distance balls in negatively curved tessellations. We show that the assumption about geodesics holds for all tessellations with at least p faces meeting in each vertex and at least q edges bounding each face, where ( p, q) ∈ {(3, 6), (4, 4), (6, 3)}.

We introduce a natural combinatorial curvature function on the corners of plane tessellations and relate it to the global metric geometry of their corresponding edge and dual graphs. If the combinatorial curvature in the corners is non-positive then we prove that any geodesic path in such a graph may be extended to infinity. Moreover, if the combinatorial curvature is negative we show that every pair of geodesic segments with the same end points does not enclose any vertices. We apply these results to establish an estimate for the growth of distance balls, Gromov hyperbolicity, and four-colourability of certain classes of plane tessellations.

We study local properties of the Bakry-Émery curvature function K G,x : (0, ∞] → R at a vertex x of a graph G systematically. Here K G,x (N ) is defined as the optimal curvature lower bound K in the Bakry-Émery curvature-dimension inequality CD(K, 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, strongly regular 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, ∞) but are not Cayley 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.