In statistical mechanics and mathematics, the Bethe lattice (also called a regular tree) is an infinite connected cycle-free graph where all vertices have the same number of neighbors. The Bethe lattice was introduced into the physics literature by Hans Bethe in 1935. In such a graph, each node is connected to z neighbors; the number z is called either the coordination number or the degree, depending on the field. Due to its distinctive topological structure, the statistical mechanics of lattice models on this graph are often easier to solve than on other lattices. The solutions are related to the often used Bethe approximation for these systems. Basic Properties When working with the Bethe lattice, it is often convenient to mark a given vertex as the root, to be used as a reference point when considering local properties of the graph. Sizes of Layers Once a vertex is marked as the root, we can group the other vertices into layers based on their distance from the root. The number of vertices at a distance from the root is , as each vertex other than the root is adjacent to vertices at a distance one greater from the root, and the root is adjacent to vertices at a distance 1. In statistical mechanics The Bethe lattice is of interest in statistical mechanics mainly because lattice models on the Bethe lattice are often easier to solve than on other lattices, such as the two-dimensional square lattice. This is because the lack of cycles removes some of the more complicated interactions. While the Bethe lattice does not as closely approximate the interactions in physical materials as other lattices, it can still provide useful insight. Exact solutions to the Ising model The Ising model is a mathematical model of ferromagnetism, in which the magnetic properties of a material are represented by a "spin" at each node in the lattice, which is either +1 or -1. The model is also equipped with a constant representing the strength of the interaction between adjacent nodes, and a constant representing an external magnetic field. The Ising model on the Bethe lattice is defined by the partition function Magnetization In order to compute the local magnetization, we can break the lattice up into several identical parts by removing a vertex. This gives us a recurrence relation which allows us to compute the magnetization of a Cayley tree with n shells