The Kagome lattice is the infinite periodic planar graph formed by the vertices and edges of the trihexagonal tiling, with vertex configuration
. Equivalently, it is the line graph of the honeycomb
graph: place a vertex at the midpoint of every honeycomb edge and join two new vertices
when the corresponding edges have a common endpoint. It is therefore a 4-regular
graph composed of corner-sharing triangles surrounding
hexagons (Syôzi 1951, Kotani and Sunada 2000).
The translation group of the Kagome lattice is isomorphic to 
. A combinatorial fundamental region contains three vertices
and six edges. Its quotient on the torus has two triangular
faces and one hexagonal face, in agreement with 
. More generally, quotienting by a translation sublattice
of index 
gives a periodic toroidal graph
with 
vertices, 
edges, and 
faces, provided the quotient introduces no loops or multiple edges. It can equivalently
be constructed as the line graph of the corresponding
honeycomb toroidal quotient (Kotani and Sunada 2000).
The name comes from the Japanese word kagome for a woven bamboo-basket pattern. It was introduced in the physics literature by Syôzi (1951) for an Ising model on this graph.
