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New upper and lower bound for the signless Laplacian spectral radius

  • 1.

    School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

  • 2.

    Department of Mathematics, Taizhou University, Linhai 317000, China

Funds:  Supported by the National Science Foundation of Zhejiang (Y6110054).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2015.12.002
More Information
  • Corresponding author: ZHAO Hongting (corresponding author), male born in 1979, master/lecturer. Research field: combinatorial analysis.
  • Received Date: 03 November 2014
  • Accepted Date: 01 June 2015
  • Rev Recd Date: 01 June 2015
  • Publish Date: 30 December 2015
    Abstract

  • Abstract

    Let D be the degree diagonal matrix of G, A be the adjacency matrix of G, Q=D+A be the signless Laplacian matrix of G. Let ξ(G) be the signless Laplacian spectral radius of G. Here the degree of graph was extended to k-degree, and average degree to k-average degree of a graph. A new upper and a new lower bound for the signless spectral radius of a graph G was obtained. Comparisons were made of the result with several classical results on the ξ(G).

    Abstract

    Let D be the degree diagonal matrix of G, A be the adjacency matrix of G, Q=D+A be the signless Laplacian matrix of G. Let ξ(G) be the signless Laplacian spectral radius of G. Here the degree of graph was extended to k-degree, and average degree to k-average degree of a graph. A new upper and a new lower bound for the signless spectral radius of a graph G was obtained. Comparisons were made of the result with several classical results on the ξ(G).
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    • References(16)
  • [1]
    Bondy J A, Murty U S R. Graph Theory and its Application[M]. New York: North Holland, 1976.
    [2]
    Minc H. Nonnegative Matrices[M]. New York: John Wiley and Sons Inc, 1988.
    [3]
    Cvetkovi D, Rowlinson P, Simi S. An Introduction to the Theory of Graph Spectra[J]. Cambridge: Cambridge Univ Press, 2010.
    [4]
    Brauldi R A, Hoffman A J. On the spectral radius of a (0,1) matrix[J].Linear Algebra Appl, 1985, 65: 133-146.
    [5]
    Stanley R P. A bound on the spectral radius of graphs with e edges[J].Linear Algebra Appl, 1987, 67: 267-269.
    [6]
    Hong Y. A bound on the spectral radius of graphs[J]. Linear Algebra Appl, 1988, 108: 135-140.
    [7]
    Hong Y, Shu J L, Fang K. A sharp upper bound of the spectral radius of graphs[J]. J Combin Theory Ser B, 2001, 81: 177-183.
    [8]
    Das K C, Kumar P. Some new bounds radius of graphs[J]. Discrete Math, 2004, 281: 149-161.
    [9]
    Shu J L, Wu Y R. Sharp upper bounds on the spectral radius of graphs[J]. Linear Algebra Appl, 2004, 377: 241-248.
    [10]
    Merris R. Laplacian matries of graphs: A survey[J]. Linear Algebra Appl, 1994, 197-198: 143-176.
    [11]
    Mohar B. Laplace eigenvalue of graphs: A survey[J]. Discrete Math, 1992,109: 171-183.
    [12]
    Grone R, Merris R. The Laplacian spectrum of a graph[J]. SIAM J Disctete Math, 1994, 7(2): 221-229.
    [13]
    Li J S, Pan Y L. De Caens inequality and bounds on the largest Laplacian eigenvalue of a graph[J]. Linear Algebra Appl, 2001,328: 153-160.
    [14]
    Pan Yongliang. Sharp upper bounds for the Laplacian graph eigenvalues[J]. Linear Algebra Appl, 2002, 355: 287-295.
    [15]
    Zhang X D. Two sharp upper bounds for the Laplacian eigenvalues[J]. Linear Algebra Appl, 2004, 376: 207-213.
    [16]
    Das K C. A characterization on graphs which achieve the upper bound for the largest Laplacian eigenvalue of graphs[J]. Linear Algebra Appl, 2004, 376: 173-186.
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    [1]
    Bondy J A, Murty U S R. Graph Theory and its Application[M]. New York: North Holland, 1976.
    [2]
    Minc H. Nonnegative Matrices[M]. New York: John Wiley and Sons Inc, 1988.
    [3]
    Cvetkovi D, Rowlinson P, Simi S. An Introduction to the Theory of Graph Spectra[J]. Cambridge: Cambridge Univ Press, 2010.
    [4]
    Brauldi R A, Hoffman A J. On the spectral radius of a (0,1) matrix[J].Linear Algebra Appl, 1985, 65: 133-146.
    [5]
    Stanley R P. A bound on the spectral radius of graphs with e edges[J].Linear Algebra Appl, 1987, 67: 267-269.
    [6]
    Hong Y. A bound on the spectral radius of graphs[J]. Linear Algebra Appl, 1988, 108: 135-140.
    [7]
    Hong Y, Shu J L, Fang K. A sharp upper bound of the spectral radius of graphs[J]. J Combin Theory Ser B, 2001, 81: 177-183.
    [8]
    Das K C, Kumar P. Some new bounds radius of graphs[J]. Discrete Math, 2004, 281: 149-161.
    [9]
    Shu J L, Wu Y R. Sharp upper bounds on the spectral radius of graphs[J]. Linear Algebra Appl, 2004, 377: 241-248.
    [10]
    Merris R. Laplacian matries of graphs: A survey[J]. Linear Algebra Appl, 1994, 197-198: 143-176.
    [11]
    Mohar B. Laplace eigenvalue of graphs: A survey[J]. Discrete Math, 1992,109: 171-183.
    [12]
    Grone R, Merris R. The Laplacian spectrum of a graph[J]. SIAM J Disctete Math, 1994, 7(2): 221-229.
    [13]
    Li J S, Pan Y L. De Caens inequality and bounds on the largest Laplacian eigenvalue of a graph[J]. Linear Algebra Appl, 2001,328: 153-160.
    [14]
    Pan Yongliang. Sharp upper bounds for the Laplacian graph eigenvalues[J]. Linear Algebra Appl, 2002, 355: 287-295.
    [15]
    Zhang X D. Two sharp upper bounds for the Laplacian eigenvalues[J]. Linear Algebra Appl, 2004, 376: 207-213.
    [16]
    Das K C. A characterization on graphs which achieve the upper bound for the largest Laplacian eigenvalue of graphs[J]. Linear Algebra Appl, 2004, 376: 173-186.
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