Annals of Combinatorics 2 (1998) 131-135


An Improvement of the Ivanov Bound

Akira Hiraki and Jack Koolen

Division of Mathematical Sciences, Osaka Kyoiku University, Kashiwara, Osaka 582, Japan
hiraki@cc.osaka-kyoiku.ac.jp

Graduate School of Mathematics, Kyushu University, Fukuoka 812, Japan
jack@math.kyushu-u.ac.jp

Received October 24, 1997

AMS Subject Classification: 05E30

Abstract. Let Γ be a distance-regular graph of diameter d, valency k and r=max{i|(ci, bi)=(c1, b1)}. In this paper, we prove that

d < (1/2) k3r.

Keywords: distance-regular graph, diameter bound


References

1.  E. Bannai and T. Ito, Algebraic Combinatorics I, Benjamin-Cummings, California, 1984.

2.  A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer Verlag, Berlin, Heidelberg, 1989.

3.  A. Boshier and K. Nomura, A remark on the intersection arrays of distance-regular graphs, J. Combin. Theory Ser. B 44 (1988) 14–153.

4.  A. Hiraki, Distance-regular subgraphs in a distance-regular graph, I–II, Europ. J. Combin. 16 (1995) 589–602, 603–615.

5.  A.A. Ivanov, Bounding the diameter of a distance-regular graph, Soviet Math. Doklady 28 (1983) 149–153.

6.  A.V. Ivanov, Problem, In: Algebraic, Extremal and Metrics Combinatorics, 1986, London Math. Soc. Lecture Note Series 131, M-M. Deza, P. Frankl, and I.G. Rosenberg, Eds., Cambridge Univ. Press, Cambridge, 1988, pp. 240–241.

7.  J.H. Koolen, On subgraphs in distance-regular graphs, J. Algebraic Combin. 1 (1992) 353–362.


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