Two theorems concerning the Bannai-Ito Conjecture

S. Bang, J. Koolen, V. Moulton

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5 Citations (Scopus)


In 1984 Bannai and Ito conjectured that there are finitely many distance-regular graphs with fixed valencies greater than two. In a series of papers, they showed that this is the case for valency 3 and 4, and also for the class of bipartite distance-regular graphs. To prove their result, they used a theorem concerning the intersection array of a triangle-free distance-regular graph, a theorem that was subsequently generalized in 1994 by Suzuki to distance-regular graphs whose intersection numbers satisfy a certain simple condition. More recently, Koolen and Moulton derived a more general version of Bannai and Ito’s theorem which they used to show that the Bannai–Ito conjecture holds for valencies 5, 6 and 7, and which they subsequently extended to triangle-free distance-regular graphs in order to show that the Bannai–Ito conjecture holds for such graphs with valencies 8, 9 and 10. In this paper, we extend the theorems of Bannai and Ito, and Koolen and Moulton to arbitrary distance-regular graphs.
Original languageEnglish
Pages (from-to)2026-2052
Number of pages27
JournalEuropean Journal of Combinatorics
Issue number7
Publication statusPublished - 2007

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