Bethe lattice

A Bethe lattice or Cayley tree 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 number of nodes in the k'th shell is given by

<math>N_k=z(z-1)^{(k-1)}

<math> for k > 0

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.