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Journal of Statistical Theory and Practice

, Volume 6, Issue 1, pp 162–168 | Cite as

A Note on the E-Optimality of Regular Line Graph Designs

Article

Abstract

The recent work of Cameron (2009) on real symmetric matrices with zero diagonals, −1, 0, 1 entries, constant row sums, and the smallest eigenvalues greater than −2 provides a useful tool for studying E-optimal block designs. We use a result that restricts the possible values of the common row sum to show the E-optimality of some regular graph designs whose associated graphs are line graphs. In particular, L2-type and triangular partially balanced incomplete block designs with λ2 = λ1 + 1 are shown to be E-optimal.

AMS Subject Classification

62K05 62K10 

Keywords

L2-type partially balanced incomplete block designs Regular graph designs Root systems Triangular partially balanced incomplete block designs 

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References

  1. Cameron, P. J. 2009. Root systems and optimal block designs. Michigan Math. J., 58, 181–194.MathSciNetCrossRefGoogle Scholar
  2. Cameron, P. J., J. M., Goethals, J. J., Seidel, and E. E. Shult. 1976. Line graphs, root systems, and elliptic geometry. J. Algebra, 43, 305–327.MathSciNetCrossRefGoogle Scholar
  3. Cheng, C. S. 1980. On the E-optimality of some block designs. J. R. Stat. Soc. B, 42, 199–204.MathSciNetMATHGoogle Scholar
  4. Cheng, C. S., and G. M. Constantine. 1986. On the efficiency of regular generalized line graph designs. J. Stat. Plan. Inference, 15, 1–10.MathSciNetCrossRefGoogle Scholar
  5. Constantine, G. M. 1986. On the optimality of block designs. Ann. Inst. Stat. Math., 38, 161–174.MathSciNetCrossRefGoogle Scholar
  6. Doob, M., and D. Cvetković. 1979. On spectral characterizations and embeddings of graphs. Linear Algebra Appl., 27, 17–26.MathSciNetCrossRefGoogle Scholar
  7. Jacroux, M. 1980. On the E-optimality of regular graph designs. J. R. Stat. Soc. B, 42, 205–209.MathSciNetMATHGoogle Scholar
  8. John, J. A., and T. J. Mitchell. 1977. Optimal incomplete block designs. J. R. Stat. Soc. B, 39, 39–43.MathSciNetMATHGoogle Scholar
  9. Kiefer, J. 1958. On the nonrandomized optimality and randomized nonoptimality of symmetrical designs. Ann. Math. Stat., 29, 675–699.MathSciNetCrossRefGoogle Scholar
  10. Masuyama, M. 1957. On the optimality of balanced incomplete block designs. Rep. Stat. Appl. Res. Un. Japan Sci. Eng., 5, 4–8.MathSciNetGoogle Scholar
  11. Shah, K. R., and B. K. Sinha. 1989. Theory of optimal designs. New York, Springer-Verlag.CrossRefGoogle Scholar
  12. Takeuchi, K. 1961. On the optimality of certain type of PBIB designs. Rep. Stat. Appl. Res. Un. Japan Sci. Eng., 8, 140–145.MathSciNetMATHGoogle Scholar
  13. Takeuchi, K. 1963. A remark added to ‘On the optimality of certain type of PBIB designs’. Rep. Stat. Appl. Res. Un. Japan Sci. Eng., 10, 47.MathSciNetMATHGoogle Scholar

Copyright information

© Grace Scientific Publishing 2012

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of CaliforniaBerkeleyUSA

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