Bethe grid

This article covers a specific type of graph. In contrast, the Bethe approach is a solution method for integrable one-dimensional quantum systems or two-dimensional classical many-body systems.

Bethe grid for z = 3

A Bethe lattice (after Hans Bethe ), also called Cayley tree (after Arthur Cayley ), is a connected, circular graph in which every node is connected to z other nodes ( z is also called the coordination number). The lattice can be viewed as a tree that grows from an origin node and in which all higher nodes lie in shells around the origin node. The origin node can be referred to as the root or the origin. The number of nodes in the k th shell is given by

${\displaystyle N_{k}=z(z-1)^{k-1}\quad \mathrm {f{\ddot {u}}r} \quad k>0.}$

Sometimes the definition is changed to the effect that the origin node has z -1 neighbors. Due to the unique topological structure, the statistical mechanics of this grid model can often be solved exactly, similar to the Bethe approximation for such systems.

A related graph is the Cayley graph .