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Periodic graph (geometry)
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<p>A <a href="/facts/Geometric_graph_theory/4vIAySX3">Euclidean graph</a> (a graph embedded in some <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>) is periodic if there exists a <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a> of that Euclidean space whose corresponding <a href="/facts/Translation_(geometry)/KlpWmnfX">translations</a> induce <a href="/facts/Symmetry_groups/CNqeseMp">symmetries</a> of that graph (i.e., application of any such translation to the graph embedded in the Euclidean space leaves the graph unchanged). Equivalently, a periodic Euclidean graph is a periodic realization of an abelian covering graph over a finite graph. A Euclidean graph is <a href="/facts/Discrete_space/hpvXA23u">uniformly discrete</a> if there is a minimal distance between any two vertices. Periodic graphs are closely related to <a href="/facts/Tessellation_of_space/seyRBcel">tessellations of space</a> (or honeycombs) and the geometry of their <a href="/facts/Symmetry_groups/CNqeseMp">symmetry groups</a>, hence to <a href="/facts/Geometric_group_theory/AJhxpFUe">geometric group theory</a>, as well as to <a href="/facts/Discrete_geometry/Ixnl3GBc">discrete geometry</a> and the theory of <a href="/facts/Polytope/a76nXrL0">polytopes</a>, and similar areas. </p><p>Much of the effort in periodic graphs is motivated by applications to natural science and engineering, particularly of <a href="/facts/Three-dimensional/KAqKWUbS">three-dimensional</a> <a href="/facts/Crystal_net/fhYsq5z4">crystal nets</a> to <a href="/facts/Crystal_engineering/aj0XSqIQ">crystal engineering</a>, <a href="/facts/Crystal_structure_prediction/zRCvlXa9">crystal prediction (design)</a>, and modeling crystal behavior. Periodic graphs have also been studied in modeling <a href="/facts/Vlsi/l1Wgqr27">very-large-scale integration (VLSI)</a> circuits. </p>