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What are distance regular graphs

A connected graph Γ with diameter D is called distance-regular if there are constants ci,ai,bi — the so-called intersection numbers — such that for all i = 0,1,...,D, and all vertices x and y at distance i = d(x,y), among the neighbors ofy, there are ci at distance i−1 from x, ai at distance i, and bi at distance i+1. It follows that Γ is a regular graph with valency k = b0, and that ci +ai +bi = kfor all i = 0, 1, . . . , D. By these equations, the intersection numbers ai can be expressed in terms of the others, and it is standard to put these others in the so-called intersection array