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mathworld.wolfram.com

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 in G and distance i=d(x,y), there are exactly c_i...

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