Graph-theoretical characterization of periodicity in crystallographic nets and other infinite graphs

Local automorphisms in infinite graphs are defined as those automorphisms for which the distance (in the graph-theoretical sense) between any vertex and its image possesses an upper bound. Abelian subgroups of direction-preserving local automorphisms without fixed point, so-called shift groups, are used to determine the quotient graph of infinite graphs. It is shown that the shift group, the closest topological analogue to a translation group in crystal structures, is isomorphic to the quotient group C/C(0) of the cycle space C of the quotient graph by some subgroup C(0), its kernel. As a main consequence, the isomorphism class of nets can be determined directly from their labeled quotient graph, without having recourse to any embedding. A general method is formulated and illustrated in the case of cristobalite and moganite structures. Application to carbon and other kinds of nanotubes is also described.