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Seidel Integral Complete r-Partite Graphs

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Abstract

A graph is S-integral (or Seidel integral) if the spectrum of its Seidel matrix consists entirely of integers. In this paper, we give a sufficient and necessary condition for complete r-partite graphs to be S-integral, from which we construct infinitely many new classes of S-integral graphs. We also present an upper bound and a lower bound for the smallest S-eigenvalue (or Seidel eigenvalue) of a complete multipartite graph.

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References

  1. Arsić B., Cvetković D., Simić S.K., Škarić M.: Graph spectral techniques in computer sciences. Appl. Anal. Discret. Math. 6, 1–30 (2012)

    Article  MATH  Google Scholar 

  2. Balińska K.T., Cvetković D., Radosavljević Z., Simić S.K., Stevanović D.: A survey on integral graphs. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat 13, 42–65 (2002)

    MATH  MathSciNet  Google Scholar 

  3. Borwein, P., Erdślyi, T.: Polynomials and Polynomial Inequalities, pp. 106–107. Springer, New York (1995)

  4. Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Springer, New York (2012)

  5. Christandl, M., Datta, N., Ekert, A., Landahl, A.J.: Perfect state transfer in quantum spin networks. Phys. Rev. Lett.92, 187902 (2004)

    Google Scholar 

  6. Cvetković D.: The main part of the spectrum, divisors and switching of graphs. Publ. Inst. Math. (Beograd) 23(37), 31–38 (1978)

    MathSciNet  Google Scholar 

  7. Cvetković, D., Doob, M., Sachs, H.: Spectra of Graphs—Theory and Application. Academic Press, New York (1980)

  8. Cvetković D., Rowlinson P., Simi{\'{c S.K.: Signless Laplacians of finite graphs. Linear Algebra Appl. 423, 155–171 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cvetković, D., Rowlinson, P., Simić, S.K.: An Introduction to the Theory of Graph Spectra. Cambridge University Press, Cambridge (2010)

  10. de Freitas M.A.A., Abreu N.M.M., Del-Vecchio R.R., Jurkiewicz S.: Infinite families of Q-integral graphs. Linear Algebra Appl. 432, 2352–2360 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Delorme C.: Spectra and suts. Australas. J. Comb. 26, 183–191 (2002)

    MATH  MathSciNet  Google Scholar 

  12. Grone R., Merris R.: The Laplacian spectrum of a graph II. SIAM J. Discrete Math. 7, 221–229 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  13. Harary, F., Schwenk, A.J.: Which graphs have integral spectra? In: Bari, R., Harary, F. (eds.) Graphs and Combinatorics. Lecture Notes in Mathematics, vol.406, pp. 45–51. Springer, Berlin (1974)

  14. Kirkland S.: Constructably Laplacian integral graphs. Linear Algebra Appl. 423, 3–21 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kirkland, S., Severini, S.: Spin systems dynamics and fault detection in threshold networks. Phys. Rev. A 83(3), # 012310 (2011)

  16. Lv, S.M., Wei, L., Zhao, H.X.: On the Seidel integral complete multipartite graphs (2011, preprint)

  17. Lepović M.: On spectral complementary graphs. Novi Sad. J. Math. 30(3), 83–91 (2000)

    MathSciNet  Google Scholar 

  18. Seidel, J.J.: Graphs and two-graphs. In: Hoffman, F. et al. (eds.) Proc. of the 5th Southeast. Conf. Comb., Graph Th., Comp. 1974, pp. 125–143. Utilitas Mathematica Pub., Winnipeg (1974)

  19. Wang L.G., Li X.L., Hoede C.: Integral complete r-partite graphs. Discrete Math. 283, 231–241 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi’an, 710072, Shaanxi, People’s Republic of China

    Ligong Wang, Guopeng Zhao & Ke Li

Authors
  1. Ligong Wang
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  2. Guopeng Zhao
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  3. Ke Li
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Corresponding author

Correspondence to Ligong Wang.

Additional information

This work was supported by the National Natural Science Foundation of China (No.11171273), the Natural Science Basic Research Plan in Shaanxi Province of China (No.SJ08A01).

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Wang, L., Zhao, G. & Li, K. Seidel Integral Complete r-Partite Graphs. Graphs and Combinatorics 30, 479–493 (2014). https://doi.org/10.1007/s00373-012-1276-6

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  • DOI: https://doi.org/10.1007/s00373-012-1276-6

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