Two theorems concerning the Bannai-Ito conjecture

  • Authors:

  • S. Bang;J. H. Koolen;V. Moulton

  • Affiliations:

  • Com2MaC, Pohang University of Science and Technology, Hyoja-dong, Namgu, Pohang 790-784, Republic of Korea;Department of Mathematics, Pohang University of Science and Technology, Hyoja-dong, Namgu, Pohang 790-784, Republic of Korea;School of Computing Sciences, University of East Anglia, Norwich NR4 7TJ, UK

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2007

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Abstract

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.