Bethe lattice

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A Bethe lattice with coordination number z = 3

A Bethe lattice or Cayley tree, introduced by Hans Bethe in 1935, is a connected cycle-free graph where each node is connected to z neighbours, where z is called the coordination number. It can be seen as a tree-like structure emanating from a central node, with all the nodes arranged in shells around the central one. The central node may be called the root or origin of the lattice. The number of nodes in the kth shell is given by

$\, N_k=z(z-1)^{k-1}\text{ for }k > 0.$

In some situations the definition is modified to specify that the root node has z − 1 neighbours.

Due to its distinctive topological structure, the statistical mechanics of lattice models on this graph are often exactly solvable. The solutions are related to the often used Bethe approximation for these systems.

Note that the Bethe lattice visually resembles the Cayley graph used to visualize the structure of groups. Bethe lattices also occur as the discrete group subgroups of certain hyperbolic Lie groups, such as the Fuchsian groups. As such, they are also lattices in the sense of a lattice in a Lie group.