Investigation of hydrogen bond patterns: A review of mathematical tools for the graph set approach

In this paper we review a graph-theoretical reformulation of the elements of graph set analysis for describing hydrogen bond patterns in crystal structures. We first collect a number of mathematical tools which are convenient for this purpose such as crystal structure graph, H-bond graph, G-labeling, significant labeling, walks, paths and periodic paths, qualitative descriptors for walks in a labeled graph, graph sets, and show how to use these tools for graph set analysis. We demonstrate by mathematical reasoning that traditional graph sets, namely selfs, rings, chains and discretes, are sufficient to describe H-bond patterns completely and we show that qualitative descriptors characterize graph sets uniquely up to crystallographic equivalence. The second part of the paper is restricted to the consideration of crystal structures in which all molecules are crystallographically equivalent and in a general position. For this particular case, it is demonstrated how one can determine algorithmically whether a given descriptor characterizes a graph set. With the aid of a particular example (the crystal structure of L-alanine) it is shown that the H-bond graph is determined by the symmetry operations corresponding to the different types of H-bonds observed in the crystal structure and by a finite set of rings. Further, we conclude that the H-bond graph is a Cayley graph, or may be found starting with a Cayley graph and replacing edges by multiple edges.