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The distance signless Laplacian spectral radius of trees with n-3 pendent vertices

  • School of Mathematics and Computation Sciences, Anqing Normal College, Anqing 246011, China

Funds:  Supported by National Natural Science Foundation of China (11071002), NFS of Anhui Province (11040606M14), NSF of Department of Education of Anhui Province (KJ2011A195, KJ2010B136).
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https://doi.org/10.3969/j.issn.0253-2778.2014.03.002
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  • Corresponding author: YU Guidong (corresponding author), female, born in 1973, PhD. Research field: Graph theory.
  • Received Date: 26 November 2013
  • Accepted Date: 18 February 2014
  • Rev Recd Date: 18 February 2014
  • Publish Date: 30 March 2014
    Abstract

  • Abstract

    The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G, defined as Q(G)=Tr(G)+D(G), where Tr(G) is the diagonal matrix of vertex transmissions of G, and D(G) is the distance matrix of G. It was investigated that the minimum of the distance signless Laplacian spectral radius among all trees with n-3 pendent vertices, and characterized that the unique tree whose distance signless Laplacian spectral radius is the maximum (minimum) among some trees with n-3 pendent vertices.

    Abstract

    The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G, defined as Q(G)=Tr(G)+D(G), where Tr(G) is the diagonal matrix of vertex transmissions of G, and D(G) is the distance matrix of G. It was investigated that the minimum of the distance signless Laplacian spectral radius among all trees with n-3 pendent vertices, and characterized that the unique tree whose distance signless Laplacian spectral radius is the maximum (minimum) among some trees with n-3 pendent vertices.
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    • References(10)
  • [1]
    Bose S S, Nath M, Paul S. Distance spectral radius of graphs with r pendent vertices[J]. Linear Algebra Appl,2011, 435: 2 828-2 836.
    [2]
    Graham R L, Lovsz L. Distance matrix polynomials of trees[J]. Adv Math, 1978, 29: 60-88.
    [3]
    Ilic′ A. Distance spectral radius of trees with given matching number[J]. Discrete Appl Math, 2010, 158: 1 799-1 806.
    [4]
    Merris R. The distance spectrum of a tree[J]. J Graph Theory, 1990, 14: 365-369.
    [5]
    Ruzieh S N, Powers D L. The distance spectrum of the path Pn and the first distance eigenvector of connected graphs[J]. Linear Multilinear Algebra, 1990, 28: 75-81.
    [6]
    Zhou B, Trinajstic′ N. On the largest eigenvalue of the distance matrix of a connected graph[J]. Chem Phys Lett, 2007, 447: 384-387.
    [7]
    Aouchiche M, Hansen P. A signless Laplacian for the distance matrix of a graph[Z]. Preprint.
    [8]
    Xing R, Zhou B, Li J. On the distance signless Laplacian spectral radius of graphs[DB/OL]. Linear and Multilinear Algebra, http://dx.doi.org/10.1080/03081087.2013.828720.
    [9]
    Xing R, Zhou B. On the distance and distance signless Laplacian spectral radii of bicyclic graphs[J]. Linear Algebra and Its Applications, 2013, 439: 3 955-3 963.
    [10]
    Cvetkovic′ D, Rowlinson P, Simic′ S. Eigenspaces of Graphs[M]. Cambridge: Cambridge University Press, 1997.
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    [1]
    Bose S S, Nath M, Paul S. Distance spectral radius of graphs with r pendent vertices[J]. Linear Algebra Appl,2011, 435: 2 828-2 836.
    [2]
    Graham R L, Lovsz L. Distance matrix polynomials of trees[J]. Adv Math, 1978, 29: 60-88.
    [3]
    Ilic′ A. Distance spectral radius of trees with given matching number[J]. Discrete Appl Math, 2010, 158: 1 799-1 806.
    [4]
    Merris R. The distance spectrum of a tree[J]. J Graph Theory, 1990, 14: 365-369.
    [5]
    Ruzieh S N, Powers D L. The distance spectrum of the path Pn and the first distance eigenvector of connected graphs[J]. Linear Multilinear Algebra, 1990, 28: 75-81.
    [6]
    Zhou B, Trinajstic′ N. On the largest eigenvalue of the distance matrix of a connected graph[J]. Chem Phys Lett, 2007, 447: 384-387.
    [7]
    Aouchiche M, Hansen P. A signless Laplacian for the distance matrix of a graph[Z]. Preprint.
    [8]
    Xing R, Zhou B, Li J. On the distance signless Laplacian spectral radius of graphs[DB/OL]. Linear and Multilinear Algebra, http://dx.doi.org/10.1080/03081087.2013.828720.
    [9]
    Xing R, Zhou B. On the distance and distance signless Laplacian spectral radii of bicyclic graphs[J]. Linear Algebra and Its Applications, 2013, 439: 3 955-3 963.
    [10]
    Cvetkovic′ D, Rowlinson P, Simic′ S. Eigenspaces of Graphs[M]. Cambridge: Cambridge University Press, 1997.
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