Bounding the diameter of distance-regular graphs
LetG be a connected distance-regular graph with valencyk>2 and diameterd, but not a complete multipartite graph. Suppose thatθ is an eigenvalue ofG with multiplicitym and thatθ≠±k. We prove that bothd andk are bounded by functions ofm. This implies that, ifm>1 is given, there are only finitely many connected, co-connected distance-regular graphs with an eigenvalue of multiplicitym.
AMS subject classification code (1980)05 C 99
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- E. Bannai and T. Ito,Algebraic Combinatorics I, Benjamin/Cummings, 1984.Google Scholar
- N. L. Biggs,Algebraic Graph Theory, Cambridge University Press, 1974.Google Scholar
- N. L. Biggs, A. G. Boshier andJ. Shawe-Taylor, Cubic distance-regular graphs, J. London Math. Soc.,33 (1986), 385–394.Google Scholar
- Ph. Delsarte, J. M. Goethals andJ. J. Seidel, Spherical codes and designs, Geom. Dedicata,6 (1977), 363–388.Google Scholar
- C. D. Godsil, Graphs, groups and polytopes, in: Combinatorial Mathematics, (ededited by D. A. Holton and Jannifer Seberry) pp. 157–164, Lecture Notes in Mathematics 686, Springer, Berlin 1978.Google Scholar
- P. Henrici, Applied and Computational Complex Analysis, Vol. 1, Wiley, New York, 1974.Google Scholar
- D. L. Powers, Eigenvectors of distance-regular graphs,submitted.Google Scholar
- D. E. Taylor andR. Levingston, Distance-regular graphs, in: Combinatorial Mathematics, (ededited by D. A. Holton and Jennifer Seberry) pp. 313–323, Lecture Notes in Mathematics 686, Springer, Berlin 1978.Google Scholar