Coordination sequences of crystals are of quasi-polynomial type

Abstract

The coordination sequence of a graph measures how many vertices the graph has at each distance from a fixed vertex and is a generalization of the coordination number. Here it is proved that the coordination sequence of the graph obtained from a crystal is of quasi-polynomial type, as had been postulated by Grosse-Kunstleve et al. [Acta Cryst. (1996), A52, 879–889].

Coordination sequences of periodic structures are rational via automata theory

“…Coordination sequences have been found in specific cases by O'Keeffe (1991), Eon (2002), Shutov and Maleev (2019), Shutov and Maleev (2020). This conjecture has been recently proven by Nakamura et al (2021), however, we believe that our simple proof is still insightful, as it shows another strong connection between apparently unrelated fields of crystallography and automata theory. In particular, we believe that recent research in the computational aspects of Parikh images Kopczynski and To (2010); Chistikov and Haase (2016) may prove useful in crystallography.…”

An Application of Singular Traces to Crystals and Percolation

“…This is a simple observation by considering the underlying Bravais lattice of any crystal [6,Chapter 4]. A very recent advancement by Y. Nakamura, R. Sakamoto, T. Masea and J. Nakagawa [31] concerns exactly such (even possibly directed) graphs Γ with a free Z d action such that Γ/Z d is finite. Namely, these authors have proven that the coordination sequence {|S(x 0 , k)|} k∈N is then of quasi-polynomial type, by which they mean the following.…”