regulargraph

A regular graph is a type of graph in graph theory where each vertex has the same degree, meaning each vertex is connected to exactly the same number of other vertices. The common notation for a regular graph is r-regular, where r indicates the degree of each vertex. For example, a 3-regular graph (also called cubic graph) is one where every vertex has three neighbors.

Regular graphs can be simple, with no loops or multiple edges between the same pair of vertices, or they can include such features. They are fundamental in the study of symmetric structures and network designs because their uniformity simplifies many theoretical analyses.

Some key properties of regular graphs include each vertex having degree r, and the sum of the degrees of all vertices being twice the number of edges, as per the Handshaking Lemma. The number of vertices in a regular graph with degree r must be at least r+1, especially in simple graphs.

Regular graphs are closely related to other classes of symmetric graphs, such as strongly regular graphs and vertex- or edge-transitive graphs. They are used not only in pure mathematics but also in designing efficient communication networks, coding theory, and the study of molecule structures in chemistry.

Existence conditions for regular graphs depend on parameters like the number of vertices and the degree r. For instance, a simple regular graph of degree r on n vertices exists only if n ≥ r+1 and the product n*r is even, satisfying the necessary conditions for the degree sum and edge count.