In graph theory, a circulant graph is an undirected graph acted on by a cyclic group of symmetries which takes any vertex to any other vertex. It is sometimes called a cyclic graph, but this term has other meanings. Equivalent definitions Circulant graphs can be described in several equivalent ways: The automorphism group of the graph includes a cyclic subgroup that acts transitively on the graph's vertices. In other words, the graph has a graph automorphism, which is a cyclic permutation of its vertices. The graph has an adjacency matrix that is a circulant matrix. The vertices of the graph can be numbered from 0 to in such a way that, if some two vertices numbered and are adjacent, then every two vertices numbered and are adjacent. The graph can be drawn (possibly with crossings) so that its vertices lie on the corners of a regular polygon, and every rotational symmetry of the polygon is also a symmetry of the drawing. The graph is a Cayley graph of a cyclic group. Examples Every cycle graph is a circulant graph, as is every crown graph with vertices. The Paley graphs of order (where is a prime number congruent to ) is a graph in which the vertices are the numbers from 0 to and two vertices are adjacent if their difference is a quadratic residue modulo . Since the presence or absence of an edge depends only on the difference modulo of two vertex numbers, any Paley graph is a circulant graph. Every Möbius ladder is a circulant graph, as is every complete graph. A complete bipartite graph is a circulant graph if it has the same number of vertices on both sides of its bipartition. If two numbers and are relatively prime, then the rook's graph (a graph that has a vertex for each square of an chessboard and an edge for each two squares that a chess rook can move between in a single move) is a circulant graph. This is because its symmetries include as a subgroup the cyclic group Cmn Cm×Cn. More generally, in this case, the tensor product of graphs between any - and -vertex circulants is itself a circulant. Many of the known lower bounds on Ramsey numbers come from examples of circulant graphs that have small maximum cliques and small maximum independent sets. A specific example The circulant graph with jumps is defined as