A connected graph G is distance-regular if for any vertices x and y of G and any integers i, j = 0, 1, ...d (where d is the graph diameter), the number of vertices at distance i from x and distance j from y depends only on i, j, and the graph distance between x and y, independently of the choice of x and y. In particular, a distance-regular graph is a graph for which there exist integers b_i, c_i, i = 0, ..., d such that for any two vertices x, y element G and distance i = d(x, y), there are exactly c_i neighbors of y element G_(i - 1)(x) and b_i neighbors of y element G_(i + 1)(x), where G_i(x) is the set of vertices y of G with d(x, y) = i . The array of integers characterizing a distance-regular graph is known as its intersection array.

automorphic graph | Biggs-Smith graph | Coxeter graph | cubical graph | cubic symmetric graph | Desargues graph | distance-transitive graph | dodecahedral graph | Foster graph | global parameters | Heawood graph | intersection array | Moore graph | Pappus graph | Petersen graph | regular graph | Shrikhande graph | Sylvester graph | Taylor graph | Wells graph