Department of Mathematics, Pusan National University, Pusan
In their 1984 book “Algebraic Combinatorics I: Association Schemes”, E. Bannai and T. Ito conjectured that there are only finitely many distance-regular graphs with fixed valency k≥3.
In the series of papers, they showed that their conjecture holds for k=3, 4, and for the class of bipartite distance-regular graphs. J. H. Koolen and V. Moulton also show that there are only finitely many distance-regular graphs with k=5, 6, or 7, and there are only finitely many triangle-free distance-regular graphs with k=8, 9 or 10. In this talk, we show that the Bannai-Ito conjecture holds for any integer k>2 (i.e., for fixed integer k>2, there are only finitely many distance-regular graphs with valency k).
This is a joint work with A. Dubickas, J. H. Koolen and V. Moulton.