In this paper we prove the Bannai-Ito conjecture, namely that there are only finitely many distance-regular graphs of fixed valency greater than two.

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A non-complete geometric distance-regular graph is the point graph of a partial linear space in which the set of lines is a set of Delsarte cliques. In this paper, we prove that for a fixed integer m 2, there are only finitely many non-geometric distance-regular graphs with smallest eigenvalue at least −m, diameter at least three and intersection number c 2 2.