Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth

Abstract

In this paper we consider the following problem: Over the class of all simple connected unicyclic graphs on n vertices with girth g (n, g being fixed), which graph minimizes the Laplacian spectral radius? Let U n,g be the lollipop graph obtained by appending a pendent vertex of a path on ng (n > g) vertices to a vertex of a cycle on g ⩾ 3 vertices. We prove that the graph U n,g uniquely minimizes the Laplacian spectral radius for n ⩾ 2g − 1 when g is even and for n ⩾ 3g − 1 when g is odd.

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References

  1. [1]

    S. M. Fallat, S. Kirkland, S. Pati: Minimizing algebraic connectivity over connected graphs with fixed girth. Discrete Math. 254 (2002), 115–142.

    Article  MATH  MathSciNet  Google Scholar 

  2. [2]

    S. M. Fallat, S. Kirkland, S. Pati: Maximizing algebraic connectivity over unicyclic graphs. Linear Multilinear Algebra 51 (2003), 221–241.

    Article  MATH  MathSciNet  Google Scholar 

  3. [3]

    M. Fiedler: Algebraic connectivity of graphs. Czech. Math. J. 23 (1973), 298–305.

    MathSciNet  Google Scholar 

  4. [4]

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

    Article  MATH  MathSciNet  Google Scholar 

  5. [5]

    R. Grone, R. Merris, V. S. Sunder: The Laplacian spectrum of a graph. SIAM J. Matrix Anal. Appl. 11 (1990), 218–238.

    Article  MATH  MathSciNet  Google Scholar 

  6. [6]

    J.-M. Guo: The effect on the Laplacian spectral radius of a graph by adding or grafting edges. Linear Algebra Appl. 413 (2006), 59–71.

    Article  MATH  MathSciNet  Google Scholar 

  7. [7]

    J.-M. Guo: The Laplacian spectral radius of a graph under perturbation. Comput. Math. Appl. 54 (2007), 709–720.

    Article  MATH  MathSciNet  Google Scholar 

  8. [8]

    R. A. Horn, C.R. Johnson: Matrix Analysis. Reprinted with corrections, Cambridge University Press, Cambridge, 1990.

    Google Scholar 

  9. [9]

    A. K. Lal, K. L. Patra: Maximizing Laplacian spectral radius over trees with fixed diameter. Linear Multilinear Algebra 55 (2007), 457–461.

    Article  MATH  MathSciNet  Google Scholar 

  10. [10]

    R. Merris: Laplacian matrices of graphs: A survey. Linear Algebra Appl. 197–198 (1994), 143–176.

    Article  MathSciNet  Google Scholar 

  11. [11]

    R. Merris: A survey of graph Laplacians. Linear Multilinear Algebra 39 (1995), 19–31.

    Article  MATH  MathSciNet  Google Scholar 

  12. [12]

    B. Mohar: The Laplacian spectrum of graphs. Graph Theory, Combinatorics and Applications, Kalamazoo, MI, 1988, Vol. 2 (Alavi, Y., ed.). Wiley-Intersci. Publ., Wiley, New York, 1991, pp. 871–898.

    Google Scholar 

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Correspondence to Kamal Lochan Patra.

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Patra, K.L., Sahoo, B.K. Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth. Czech Math J 63, 909–922 (2013). https://doi.org/10.1007/s10587-013-0061-x

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Keywords

  • Laplacian matrix
  • Laplacian spectral radius
  • girth
  • unicyclic graph

MSC 2010

  • 05C50