A graph-theoretical model for periodic discrete structures☆

Abstract A class of infinite graphs called periodic arrays is defined. The study of these graphs was motivated by the use of wiring diagrams which possess to a high degree a periodic structure. A characterization is obtained for those periodic arrays which are connected, planar, and point-symmetric; namely, it is shown that they are isomorphic to the eleven regular and semi-regular tessellations of the plane. Also a characterization is obtained for those directed periodic arrays which are point-symmetric and which, as undirected periodic arrays, are isomorphic to the quadratic lattice. Precisely four such arrays are possible.