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Investigation and application of Laplace spectra of orgraphs with the ring structure

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Abstract

The Laplace matrix is a square matrix L = (ℓ ij ) ∈ ℝn×n in which all nondiagonal elements are nonpositive and all row sums are equal to zero. Each Laplace matrix corresponds to a weighted orgraph with positive arc weights. The problem of reality of Laplace matrix spectrum for orgraphs of a special type consisting of two “counter” Hamiltonian cycles in one of which one or two arcs are removed is studied. Characteristic polynomials of Laplace matrices of these orgraphs are expressed through polynomials Z n (x) that can be obtained from Chebyshev second-kind polynomials P 2n (y) by the substitution of y 2 = x. The obtained results relate to properties of the product of Chebyshev second-kind polynomials. A direct method for computing the spectrum of Laplace circuit matrix is given. The obtained results can be used for computing the number of spanning trees in orgraphs of the studied type. One of the possible practical applications of these results is the investigation of topology and development of new Internet protocols.

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References

  1. Cvetkovic, D.M., Doob, M., and Sachs, H., Spectra of Graphs, New York: Academic, 1980. Translated under the title Spectry grafov: teoriya i primenenie, Kiev: Naukova Dumka, 1984.

    Google Scholar 

  2. Agaev, R.P. and Chebotarev, P.Yu., Laplace Spectra of Orgraphs and Their Applications, Avtom. Telemekh., 2005, no. 5, pp. 47–62.

  3. Fax, J.A. and Murray, R.M., Graph Laplacians and Stabilization of Vehicle Formations. Engineering and Applied Science, Pasadena: California Inst. of Tech., 2001.

    Google Scholar 

  4. Veerman, J.J.P., Lafferriere, G., Caughman, J.S., and Williams, A., Flocks and Formations, J. Statist. Physics, 2005, vol. 121, nos. 5–6. pp. 901–936.

    Article  MATH  MathSciNet  Google Scholar 

  5. Stankevich, M.I., Stankevich, I.V., and Zefirov, N.S., Topological Indices in Organic Chemistry, Usp. Khim., 1988, vol. 57, no. 3, pp. 337–366.

    Google Scholar 

  6. Vukadinovic, D., Huang, P., and Erlebach, Th., On the Spectrum and Structure of Internet Topology Graphs, Innovative Int. Comput. Syst. Second Int. Workshop on Innovative Int. Comput. Syst. (IICS); Lecture Notes Comput. Sci., 2002, vol. 2346, pp. 83–95.

    Google Scholar 

  7. Kelmans, A.K., On the Number of Trees of a Graph. I, Avtom. Telemekh., 1965, no. 12, pp. 2118–2129.

  8. Kelmans, A.K., On the Number of Trees of a Graph. II, Avtom. Telemekh., 1966, no. 2, pp. 233–241.

  9. Kelmans, A.K. and Chelnokov, V.M., A Certain Polynomial of a Graph and Graphs with an Extremal Number of Trees, J. Combinat. Theory. Ser. B, 1974, vol. 16, pp. 197–214.

    Article  MATH  MathSciNet  Google Scholar 

  10. Grone, R., Merris, R., and Sunder, V.S., The Laplacian Spectrum of a Graph, SIAM J. Matrix Anal. Appl., 1990, vol. 11, pp. 218–238.

    Article  MATH  MathSciNet  Google Scholar 

  11. Merris, R., Laplacian Matrices of Graphs: A Survey, Lin. Alg. Appl., 1994, vol. 197–198, pp. 143–176.

    Article  MathSciNet  Google Scholar 

  12. Merris, R., A Survey of Graph Laplacians, Lin. Multilin. Alg., 1995, vol. 39, pp. 19–31.

    Article  MATH  MathSciNet  Google Scholar 

  13. Mohar, B., Laplace Eigenvalues of Graphs-A Survey, Discret. Math., 1992, vol. 109, pp. 171–183.

    Article  MATH  MathSciNet  Google Scholar 

  14. Chaiken, S., A Combinatorial Proof of the All Minors Matrix Tree Theorem, SIAM J. Algebraic Discret. Methods, 1982, vol. 3, pp. 319–329.

    Article  MATH  MathSciNet  Google Scholar 

  15. Fiedler, M. and Sedláček, J., O W-basích orientovaných grafů, Časopis Pěst. Mat., 1958, vol. 83, pp. 214–225.

    MATH  Google Scholar 

  16. Chebotarev, P. Yu. and Shamis, E.V., Matrix-Forest Theorem and Measurement of Relations in Small Social Groups, Avtom. Telemekh., 1997, no. 9, pp. 124–136.

  17. Chebotarev, P.Yu. and Shamis, E.V., Proximity Indices of Graph Vertexes, Avtom. Telemekh., 1998, no. 10, pp. 113–133.

  18. Chebotarev, P. and Agaev, R., Forest Matrices around the Laplacian Matrix, Lin. Alg. Appl., 2002, vol. 356, pp. 253–274.

    MATH  MathSciNet  Google Scholar 

  19. Wu, Ch.W., Algebraic Connectivity of Directed Graphs, Lin. Multilin. Alg., 2005, vol. 53, no. 3, pp. 203–223.

    Article  MATH  Google Scholar 

  20. Caughman, J.S. and Veerman, J.J.P., Kernels of Directed Graph Laplacians, The Electron. J. Combinat., 2006, vol. 13, no. 1, R39.

    MathSciNet  Google Scholar 

  21. Pashkovskii, S., Vychislitel’nye primeneniya mnogochlenov i ryadov Chebysheva (Computational Applications of Chebyshev Polynomials and Series), Moscow: Nauka, 1983.

    Google Scholar 

  22. Mason, J.C. and Handscomb, D.C., Chebyshev Polynomials, New York: CRC Press, 2002.

    Book  Google Scholar 

  23. Fiedler, M., Bounds for Eigenvalues of Doubly Stochastic Matrices, Lin. Alg. Appl., 1972, vol. 5, pp. 299–310.

    Article  MATH  MathSciNet  Google Scholar 

  24. Anderson, W.N. and Morley, T.D., Eigenvalues of the Laplacian of a Graph, Lin. Multilin. Alg., 1985, vol. 18, pp. 141–145.

    Article  MATH  MathSciNet  Google Scholar 

  25. Boesch, F.T. and Prodinger, H., Spanning Tree Formulas and Chebyshev Polynomials, Graphs Combinat., 1986, vol. 2, pp. 191–200.

  26. Zhang, Y., Yong, X., and Golin, M., Chebyshev Polynomials and Spanning Tree Formulas for Circulant and Related Graphs, Discret. Math., 2005, vol. 298, pp. 334–364.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    R. P. Agaev

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Original Russian Text © R.P. Agaev, 2008, published in Avtomatika i Telemekhanika, 2008, No. 2, pp. 17–25.

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Agaev, R.P. Investigation and application of Laplace spectra of orgraphs with the ring structure. Autom Remote Control 69, 177–188 (2008). https://doi.org/10.1134/S000511790802001X

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