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The maximum Laplacian separators of bicyclic and tricyclic graphs

  • 1.

    School of Mathematics and Computation Sciences, Anqing Normal University, Anqing 246133, China

  • 2.

    School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China

Funds:  Supported by National Natural Science Foundation of China (11371028), NSF of Anhui Province (11040606M14), NSF of Department of Education of Anhui Province (KJ2015ZD27, KJ2017A362).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2017.09.004
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  • Corresponding author: YU Guidong(corresponding author), female, born in 1973, PhD/Prof. Research field: Graph theory. E-mail: guidongy@163.com
  • Received Date: 14 March 2016
  • Accepted Date: 29 December 2016
  • Rev Recd Date: 29 December 2016
  • Publish Date: 30 September 2017
    Abstract

  • Abstract

    Let G be an undirected simple graph of order n, L(G) be the Laplacian matrix of G, and μ1(G)≥μ2(G) ≥…≥μn(G) be the eigenvalues of L(G). The Laplacian separator of G is defined as SL(G)=μ1(G)-μ2(G). Here the maximum Laplacian separators of bicyclic and tricyclic graphs of a given order were studied, and the corresponding extremal graphs were characterized.

    Abstract

    Let G be an undirected simple graph of order n, L(G) be the Laplacian matrix of G, and μ1(G)≥μ2(G) ≥…≥μn(G) be the eigenvalues of L(G). The Laplacian separator of G is defined as SL(G)=μ1(G)-μ2(G). Here the maximum Laplacian separators of bicyclic and tricyclic graphs of a given order were studied, and the corresponding extremal graphs were characterized.
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    • References(15)
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    HELLESETH T, KUMAR P V. Sequences with low correlation[C]// Handbook of Coding Theory. Amsterdam: Elsevier, 1998.
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    ZINOVIEV D V, SOL P. Quaternary codes and biphase sequences from Z8-codes[J]. Problems of Information Transmission, 2004, 40(2): 147-158 (translated from Problemy Peredachi Informatsii, 2004, 2: 50-62).
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    LAHTONEN J, LING S, SOL P, ZINOVIEV D V. Z8-Kerdock codes and pseudo-random binary sequences[J]. J Complexity, 2004, 20: 318-330.
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    WAN Z X. Finite Fields and Galois Rings[M]. Singapore: World Scientific, 2003.
    [12]
    LIDL R, NIEDERREITER H. Finite Fields[M]. Cambridge, UK: Cambridge University Press, 1997.
    [13]
    GOLOMB S W, GONG G. Signal Design for Good Correlation for Wireless Communication, Cryptography and Radar[M]. New York: Cambridge University Press, 2005.
    [14]
    LING S, BLACKFORD J T. Zpk+1-linear codes[J]. IEEE Trans Inform Theory, 2002, 48(9): 2592-2605.
    [15]
    HAMMONS A R Jr, KUMAR P V, CALDERBANK A R, et al. The Z4-linearity of Kerdock, Preparata, Goethals, and related codes[J]. IEEE Trans Inform Theory, 1994, 40(2): 301-309.
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    [1]
    HELLESETH T, KUMAR P V. Sequences with low correlation[C]// Handbook of Coding Theory. Amsterdam: Elsevier, 1998.
    [2]
    LI N, TANG X H, HELLESETH T. New M-ary sequences with low autocorrelation from interleaved technique[J]. Des Codes Crytogr, 2014, 73: 237-249.
    [3]
    DING C. Codes From Difference Sets[M]. Singapore: World Scientific, 2015.
    [4]
    ZHU S X, WANG Y, SHI M J. Some results on cyclic codes over F2+vF2[J]. IEEE Trans Inform Theory, 2010, 56(4): 1680-1684.
    [5]
    KUMAR P V, HELLESETH T, CALDERBANK A R. An upper bound for Weil exponential sums over Galois rings and applications[J]. IEEE Trans Inform Theory, 1995, 41(2): 456-468.
    [6]
    KUMAR P V, HELLESETH T, CALDERBANK A R, et al. Large families of quaternary sequences with low correlation[J]. IEEE Trans Inform Theory, 1996, 42(2): 579-592.
    [7]
    SHANBHAG A, KUMAR P V, HELLESETH T. Improved binary codes and sequences families from Z4-linear codes[J]. IEEE Trans Inform Theory, 1996, 42(5): 1582-1586.
    [8]
    ZINOVIEV D V, SOL P. Quaternary codes and biphase sequences from Z8-codes[J]. Problems of Information Transmission, 2004, 40(2): 147-158 (translated from Problemy Peredachi Informatsii, 2004, 2: 50-62).
    [9]
    HU H G, FENG D G, WU W L. Incomplete exponential sums over Galois rings with application to some binary sequences derived from Z2l[J]. IEEE Trans Inform Theory, 2006, 52(5): 2260-2265.
    [10]
    LAHTONEN J, LING S, SOL P, ZINOVIEV D V. Z8-Kerdock codes and pseudo-random binary sequences[J]. J Complexity, 2004, 20: 318-330.
    [11]
    WAN Z X. Finite Fields and Galois Rings[M]. Singapore: World Scientific, 2003.
    [12]
    LIDL R, NIEDERREITER H. Finite Fields[M]. Cambridge, UK: Cambridge University Press, 1997.
    [13]
    GOLOMB S W, GONG G. Signal Design for Good Correlation for Wireless Communication, Cryptography and Radar[M]. New York: Cambridge University Press, 2005.
    [14]
    LING S, BLACKFORD J T. Zpk+1-linear codes[J]. IEEE Trans Inform Theory, 2002, 48(9): 2592-2605.
    [15]
    HAMMONS A R Jr, KUMAR P V, CALDERBANK A R, et al. The Z4-linearity of Kerdock, Preparata, Goethals, and related codes[J]. IEEE Trans Inform Theory, 1994, 40(2): 301-309.
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