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Hoffman’s Coclique Bound for Normal Regular Digraphs, and Nonsymmetric Association Schemes

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Mathematics Across Contemporary Sciences (AUS-ICMS 2015)

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Abstract

We extend Hoffman’s coclique bound for regular digraphs with the property that its adjacency matrix is normal, and discuss cocliques attaining the inequality. As a consequence, we characterize skew-Bush-type Hadamard matrices in terms of digraphs. We present some normal digraphs whose vertex set is decomposed into disjoint cocliques attaining the bound. The digraphs provided here are relation graphs of some nonsymmetric association schemes.

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References

  1. Brouwer, A.E.: Distance regular graphs of diameter \(3\) and strongly regular graphs. Discrete Math. 49, 101–103 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brouwer, A.E., Haemers, W.H.: Spectra of graphs. Universitext. Springer, New York (2012). xiv+250 pp

    Google Scholar 

  3. Chang, Y.: Imprimitive Symmetric Association Schemes of Rank 4. University of Michigan, Thesis (1994)

    Google Scholar 

  4. Cvetković, D.M., Doob, M., Sachs, H.: Spectra of graphs. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1980)

    Google Scholar 

  5. Goldbach, R.W., Claasen, H.L.: \(3\)-class association schemes and Hadamard matrices of a certain block form. Europ. J. Combin. 19, 943–951 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Haemers, W.H., Tonchev, V.D.: Spreads in strongly regular graphs. Des. Codes Crypt. 8, 145–157 (1996)

    MathSciNet  MATH  Google Scholar 

  7. Holzmann, W.H., Kharaghani, H., Suda, S.: Mutually unbiased biangular vectors and association schemes. In Colbourn, C. J. (ed.) Algebraic Design Theory and Hadamard Matrices, vol. 133, pp. 149–157. Springer International Publishing (2015)

    Google Scholar 

  8. Ionin, Y.J., Kharaghani, H.: Doubly regular digraphs and symmetric designs. J. Combin. Theory Ser. A 101(1), 35–48 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jorgensen, L.K., Jones, G.A., Klin, M.H., Song, S.Y.: Normally regular digraphs, association schemes and related combinatorial structures. Sém. Lothar. Combin. 71, Art. B71c, 39pp (2013/14)

    Google Scholar 

  10. Kharaghani, H.: New class of weighing matrices. Ars. Combin. 19, 69–72 (1985)

    MathSciNet  MATH  Google Scholar 

  11. Kharaghani, H., Sasani,S., Suda, S.: Mutually unbiased Bush-type Hadamard matrices and association schemes. Elec. J. Combin. 22 P3. 10 (2015)

    Google Scholar 

  12. Muzychuk, M., Xiang, Q.: Symmetric Bush-type Hadamard matrices of order \(4m^4\) exist for all odd \(m\). Proc. Amer. Math. Soc. 134(8), 2197–2204 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Wallis, W.D.: On a problem of K. A. Bush concerning Hadamard matrices. Bull. Aust. Math. Soc. 6, 321–326 (1971)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Hadi Kharaghani is supported by an NSERC Discovery Grant. Sho Suda is supported by JSPS KAKENHI Grant Number 15K21075.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB, T1K 3M4, Canada

    Hadi Kharaghani

  2. Department of Mathematics Education, Aichi University of Education, 1 Hirosawa, Igaya-cho, Kariya, Aichi, 448-8542, Japan

    Sho Suda

Authors
  1. Hadi Kharaghani
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  2. Sho Suda
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Corresponding author

Correspondence to Hadi Kharaghani .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, American University of Sharjah, Sharjah, United Arab Emirates

    Taher Abualrub

  2. Department of Mathematics and Statistics, American University of Sharjah , Sharjah, United Arab Emirates

    Abdul Salam Jarrah

  3. Department of Mathematics, Université de Lille 1, Lille, France

    Sadok Kallel

  4. Department of Mathematics and Statistics, American University of Sharjah, Sharjah, United Arab Emirates

    Hana Sulieman

Appendices

Appendix 1: Parameters of the association Scheme in Theorem 5.3(i)

$$\begin{aligned} B_1&=\begin{pmatrix} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ \frac{n^2-2n}{2} &{} \frac{n^2-3n}{4} &{} \frac{n^2-3n}{4} &{} \frac{n^2-4n}{4} &{} \frac{n^2-2n}{4} \\ 0 &{} \frac{n^2-3n}{4} &{} \frac{n^2-3n}{4} &{} \frac{n^2}{4} &{} \frac{n^2-2n}{4} \\ 0 &{} \frac{n}{4}-1 &{} \frac{n}{4} &{} 0 &{} 0 \\ 0 &{} \frac{n}{4} &{} \frac{n}{4} &{} 0 &{} 0 \end{pmatrix}\\ B_2&=\begin{pmatrix} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} \frac{n^2-3n}{4} &{} \frac{n^2-3n}{4} &{} \frac{n^2}{4} &{} \frac{n^2-2n}{4} \\ \frac{n^2-2n}{2} &{} \frac{n^2-3n}{4} &{} \frac{n^2-3n}{4} &{} \frac{n^2-4n}{4} &{} \frac{n^2-2n}{4} \\ 0 &{} \frac{n}{4} &{} \frac{n-4}{4} &{} 0 &{} 0 \\ 0 &{} \frac{n}{4} &{} \frac{n}{4} &{} 0 &{} 0 \end{pmatrix}\\ B_3&=\begin{pmatrix} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} \frac{n}{4}-1 &{} \frac{n}{4} &{} 0 &{} 0 \\ 0 &{} \frac{n}{4} &{} \frac{n}{4}-1 &{} 0 &{} 0 \\ \frac{n}{2}-1 &{} 0 &{} 0 &{} \frac{n}{2}-2 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \frac{n}{2}-1 \end{pmatrix}\\ B_4&=\begin{pmatrix} 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} \frac{n}{4} &{} \frac{n}{4} &{} 0 &{} 0 \\ 0 &{} \frac{n}{4} &{} \frac{n}{4} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \frac{n}{2}-1 \\ \frac{n}{2} &{} 0 &{} 0 &{} \frac{n}{2} &{} 0 \end{pmatrix}\\ Q&=\begin{pmatrix} 1 &{} n-1 &{} n-2 &{} \frac{(n-1)(n-2)}{2} &{} \frac{(n-1)(n-2)}{2} \\ 1 &{} 0 &{} -1 &{} -\frac{n-1}{2} &{} \frac{n-1}{2} \\ 1 &{} 0 &{} -1 &{} \frac{n-1}{2} &{} -\frac{n-1}{2} \\ 1 &{} n-1 &{} n-2 &{} -n+1 &{} -n+1 \\ 1 &{} -n+1 &{} n-2 &{} 0 &{} 0 \end{pmatrix} \end{aligned}$$

Appendix 2: Parameters of the Association Scheme in Theorem 5.3(ii)

$$\begin{aligned} B_1&=\begin{pmatrix} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} \frac{n^2-3n}{4} &{} \frac{n^2-3n}{4} &{} \frac{n^2}{4} &{} \frac{n^2-2n}{4} \\ \frac{n^2-2n}{2} &{} \frac{n^2-3n}{4} &{} \frac{n^2-3n}{4} &{} \frac{n^2-4n}{4} &{} \frac{n^2-2n}{4} \\ 0 &{} \frac{n}{4}-1 &{} \frac{n}{4} &{} 0 &{} 0 \\ 0 &{} \frac{n}{4} &{} \frac{n}{4} &{} 0 &{} 0 \end{pmatrix}\\ B_2&=\begin{pmatrix} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ \frac{n^2-2n}{2} &{} \frac{n^2-3n}{4} &{} \frac{n^2-3n}{4} &{} \frac{n^2-4n}{4} &{} \frac{n^2-2n}{4} \\ 0 &{} \frac{n^2-3n}{4} &{} \frac{n^2-3n}{4} &{} \frac{n^2}{4} &{} \frac{n^2-2n}{4} \\ 0 &{} \frac{n}{4} &{} \frac{n-4}{4} &{} 0 &{} 0 \\ 0 &{} \frac{n}{4} &{} \frac{n}{4} &{} 0 &{} 0 \end{pmatrix}\\ B_3&=\begin{pmatrix} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} \frac{n}{4}-1 &{} \frac{n}{4} &{} 0 &{} 0 \\ 0 &{} \frac{n}{4} &{} \frac{n}{4}-1 &{} 0 &{} 0 \\ \frac{n}{2}-1 &{} 0 &{} 0 &{} \frac{n}{2}-2 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \frac{n}{2}-1 \end{pmatrix}\\ B_4&=\begin{pmatrix} 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} \frac{n}{4} &{} \frac{n}{4} &{} 0 &{} 0 \\ 0 &{} \frac{n}{4} &{} \frac{n}{4} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \frac{n}{2}-1 \\ \frac{n}{2} &{} 0 &{} 0 &{} \frac{n}{2} &{} 0 \end{pmatrix}\\ Q&=\begin{pmatrix} 1 &{} n-1 &{} n-2 &{} \frac{(n-1)(n-2)}{2} &{} \frac{(n-1)(n-2)}{2} \\ 1 &{} 0 &{} -1 &{} -\frac{\sqrt{-1}(n-1)}{2} &{} \frac{\sqrt{-1}(n-1)}{2} \\ 1 &{} 0 &{} -1 &{} \frac{\sqrt{-1}(n-1)}{2} &{} -\frac{\sqrt{-1}(n-1)}{2} \\ 1 &{} n-1 &{} n-2 &{} -n+1 &{} -n+1 \\ 1 &{} -n+1 &{} n-2 &{} 0 &{} 0 \end{pmatrix} \end{aligned}$$

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Kharaghani, H., Suda, S. (2017). Hoffman’s Coclique Bound for Normal Regular Digraphs, and Nonsymmetric Association Schemes. In: Abualrub, T., Jarrah, A., Kallel, S., Sulieman, H. (eds) Mathematics Across Contemporary Sciences. AUS-ICMS 2015. Springer Proceedings in Mathematics & Statistics, vol 190. Springer, Cham. https://doi.org/10.1007/978-3-319-46310-0_8

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