Acta Crystallographica Section A

Foundations of Crystallography

Volume 58, Part 1 (January 2002)

research papers

Acta Cryst. (2002). A58, 47-53 [ doi:10.1107/S0108767301016609 ]

Algebraic determination of generating functions for coordination sequences in crystal structures

J.-G. Eon

Abstract: Given a connected crystalline structure, the set of paths and cycles of the quotient graph of its bond net is embedded into a commutative ring structure. Multiplication combines walks to build up geodesics of the net whereas addition stands symbolically for enumerating a collection of walks. Topological criteria are used to define zero divisors which enable the development of an algebraic generator into a combination of geodesics that are in a one-to-one correspondence with the vertices of the net. A ring mapping gives the generating function of the coordination sequence in the net.

Keywords: geodesics; nets; coordination sequences.