Fiveregular

Fiveregular is a term used in graph theory to describe a class of graphs in which every vertex has degree five. It is a specific instance of the broader concept of regular graphs, where all vertices share the same degree.

Formally, a fiveregular graph is a simple graph G = (V,E) such that for every vertex v in V, the degree deg(v) = 5. By the Handshaking Lemma, the sum of degrees equals 2|E|, so 5|V| is even and thus |V| must be even. Consequently, the number of edges is |E| = 5|V|/2. The smallest such graph is the complete graph on six vertices, K6, which is 5-regular.

Existence and construction: For every even n ≥ 6, there exists a five-regular simple graph on n vertices. One way to build examples is to take the complement of a 2-regular graph on n vertices, or to start with a K6 and apply standard regular-graph constructions (such as circulant graphs or other methods for generating regular graphs) to obtain larger fiveregular graphs.

Properties and variations: Five-regular graphs need not be connected; they can decompose into multiple components, each of which is five-regular on its vertex set. They are a special case of k-regular graphs with k = 5 and are studied in general graph theory for their symmetry and structural properties, including questions about existence, connectivity, and girth.

Applications: Five-regular graphs appear in network design, chemistry, and coding theory as models of symmetric adjacency relations. Practical uses depend on additional properties such as connectivity, girth, and symmetry.

See also: Regular graph, k-regular graphs.