Infinite geodesic paths and fibers, new topological invariants in periodic graphs

Rings are well known invariants of nets. In this work, a generalization of the concepts of cycles and rings is introduced. Infinite paths in periodic graphs are defined as connected, acyclic, regular subgraphs of degree two; geodesics are defined as infinite paths such that the unique path between any pair of vertices is a geodesic path in the whole graph. An infinite path can be thought of as an infinite cycle and a geodesic as an infinite ring. In a further step, a geodesic fiber is defined as a minimal 1-periodic subgraph that contains all geodesic paths between any pair of its vertices. Geodesic fibers are topological invariants of periodic graphs whose labeled quotient graphs are subgraphs of the labeled quotient graph of the whole graph; the paper describes applications of geodesic fibers to the analysis of the automorphisms of minimal nets, crystallographic and non-crystallographic nets.