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In the mathematical field of graph theory, a distance-regular graph is a regular graph such that for any two vertices v and w, the number of vertices at distance j from v and at distance k from w depends only upon j, k, and the distance between v and w. Some authors exclude the complete graphs and disconnected graphs from this definition.
A metric space defined over a set of points in terms of distances in a graph defined over the set is called a graph metric. The vertex set (of an undirected graph) and the distance function form a metric space, if and only if the graph is connected. The eccentricity ϵ(v) of a vertex v is the greatest distance between v and any other vertex; in ...
Regular graphs of degree at most 2 are easy to classify: a 0-regular graph consists of disconnected vertices, a 1-regular graph consists of disconnected edges, and a 2-regular graph consists of a disjoint union of cycles and infinite chains. A 3-regular graph is known as a cubic graph.
In mathematics, the Perkel graph, named after Manley Perkel, is a 6-regular graph with 57 vertices and 171 edges. It is the unique distance-regular graph with intersection array (6, 5, 2; 1, 1, 3). [1] The Perkel graph is also distance-transitive. It is also the skeleton of an abstract regular polytope, the 57-cell.
In 1969, before publication of the Biggs–Smith definition, a Russian group led by Georgy Adelson-Velsky showed that there exist graphs that are distance-regular but not distance-transitive. The smallest distance-regular graph that is not distance-transitive is the Shrikhande graph, with 16 vertices and degree 6.
Such a strongly regular graph is denoted by srg(v, k, λ, μ); its "parameters" are the numbers in (v, k, λ, μ). Its complement graph is also strongly regular: it is an srg(v, v − k − 1, v − 2 − 2k + μ, v − 2k + λ). A strongly regular graph is a distance-regular graph with diameter 2 whenever μ is non
In some cases, Hamming graphs may be considered more generally as the Cartesian products of complete graphs that may be of varying sizes. [3] Unlike the Hamming graphs H ( d , q ) , the graphs in this more general class are not necessarily distance-regular , but they continue to be regular and vertex-transitive .
Generalizing the triangle graph, every cycle graph is a unit distance graph, realized by a regular polygon. [4] Two finite unit distance graphs, connected at a single shared vertex, yield another unit distance graph, as one can be rotated with respect to the other to avoid undesired additional unit distances. [6]