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Asymptotic Properties of the Spectra of a Strongly Regular Graph

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Innovation, Engineering and Entrepreneurship (HELIX 2018)

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 505))

Abstract

Let G be a strongly regular graph with three distinct eigenvalues and A his matrix of adjacency. In this work we associate a three dimensional real Euclidean Jordan algebra V with rank three to A and next we consider a Jordan frame B of idempotents of V. Next we analyse the spectra of a particular convergent Hadamard series of \(A^{2}\) and establish asymptotic inequalities over the spectra and the parameters of G.

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References

  1. Cardoso, D.M., Vieira, L.A.: On the optimal parameter of a self-concordant barrier over a symmetric cone. Eur. J. Oper. Res. 169, 1148–1157 (2006)

    Article  MathSciNet  Google Scholar 

  2. Faybusovich, L.: Linear systems in Jordan algebras and primal-dual interior-point algorithms. J. Comput. Appl. Math. 86, 149–175 (1997)

    Article  MathSciNet  Google Scholar 

  3. Faybusovich, L.: Euclidean Jordan Algebras and Interior-point Algorithms. Positivity 1, 331–357 (1997)

    Article  MathSciNet  Google Scholar 

  4. Jordan, P., Neuman, J.V., Wigner, E.: On an algebraic generalization of the quantum mechanical formalism. Ann. Math. 35, 29–64 (1934)

    Article  MathSciNet  Google Scholar 

  5. Mano, V.M., Vieira, L.A.: Admissibility conditions and asymptotic behaviour of strongly regular graph. Int. J. Math. Models Methods Appl. Sci. 5(6), 1027–1033 (2011)

    Google Scholar 

  6. Mano, V.M., Martins, E.A., Vieira, L.A.: On generalized binomial series and strongly regular graphs. Proyecciones J. Math. 4, 393–408 (2013)

    Article  MathSciNet  Google Scholar 

  7. Mano, V.M., Vieira, L.A.: Alternating schur series and necessary conditions for the existence of strongly graphs. Int. J. Math. Models Methods Appl. Sci. Methods. 8, 256–261 (2014)

    Google Scholar 

  8. Massam, H., Neher, E.: Estimation and testing for lattice condicional independence models on Euclidean Jordan algebras. Ann. Stat. 26, 1051–1082 (1998)

    Article  Google Scholar 

  9. Vieira, L.A.: Euclidean Jordan algebras and inequalities on the parameters of a strongly regular graph. In: AIP Conference Proceedings, vol. 1168, pp. 995–998 (2009)

    Google Scholar 

  10. Vieira, L.A., Mano, V.M.: Generalized Krein parameters of a strongly regular graph. Appl. Math. 6, 37–45 (2015)

    Article  Google Scholar 

  11. Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Oxford Mathematical Monographs. Clarendon Press, Oxford (1994)

    MATH  Google Scholar 

  12. Koecher, M.: The Minnesota Notes on Jordan Algebras and Their Applications. Springer, Berlin (1999)

    Book  Google Scholar 

  13. Bose, R.C.: Strongly regular graphs, partial geometries and partially balanced designs. Pac. J. Math. 13, 389–419 (1963)

    Article  MathSciNet  Google Scholar 

  14. Godsil, C., Royle, G.: Algebraic Graph Theory. Chapman & Hall, New York (1993)

    MATH  Google Scholar 

  15. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Engineering of Porto, Section of Mathematics, Porto, Portugal

    Luís Vieira

  2. University of Porto, Porto, Portugal

    Luís Vieira

  3. Center of Mathematics of University of Porto, Street Dr Roberto Frias, 0351, 4200-465, Porto, Portugal

    Luís Vieira

Authors
  1. Luís Vieira
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Correspondence to Luís Vieira .

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Editors and Affiliations

  1. School of Engineering, University of Minho, Guimarães, Portugal

    José Machado

  2. School of Engineering, University of Minho, Guimarães, Portugal

    Filomena Soares

  3. Faculty of Engineering, University of Porto, Porto, Portugal

    Germano Veiga

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Vieira, L. (2019). Asymptotic Properties of the Spectra of a Strongly Regular Graph. In: Machado, J., Soares, F., Veiga, G. (eds) Innovation, Engineering and Entrepreneurship. HELIX 2018. Lecture Notes in Electrical Engineering, vol 505. Springer, Cham. https://doi.org/10.1007/978-3-319-91334-6_109

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