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On a poset of trees

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Abstract

We will prove that the path minimizes the number of closed walks of length ℓ among the connected graphs for all ℓ. Indeed, we will prove that the number of closed walks of length ℓ and many other properties such as the spectral radius, Estada index increase or decrease along a certain poset of trees. This poset is a leveled poset with path as the smallest element and star as the greatest element.

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References

  1. J. Brown, C. Colbourn and J. Devitt: Network transformations and bounding network reliability, Networks23 (1993), 1–17.

    Article  MATH  MathSciNet  Google Scholar 

  2. R. A. Brualdi and A. J. Hoffman: On the spectral radius of (0,1) matrices, Linear Algebra and its Applications65 (1985), 133–146.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. A. de la Peña, I. Gutnam and J. Rada: Estimating the Estrada index, Linear Algebra and its Applications427(1) (2007), 70–76.

    Article  MATH  MathSciNet  Google Scholar 

  4. H. Deng: A proof of a conjecture on the Estrada index, MATCH Commun. Math. Comput. Chem.62(3) (2009), 599–606.

    MATH  MathSciNet  Google Scholar 

  5. A. K. Kelmans: On graphs with randomly deleted edges, Acta. Math. Acad. Sci. Hung.37 (1981), 77–88.

    Article  MATH  MathSciNet  Google Scholar 

  6. L. Lovász: Combinatorial problems and exercises, North Holland, 2nd ed., (1993) (Problem 6.23).

  7. L. Lovász and J. Pelikán: On the eigenvalues of trees, Per. Mathematica Hungarica3 (1973), 175–182.

    Article  MATH  Google Scholar 

  8. V. Nikiforov: Some inequalities for the largest eigenvalue of a graph, Combinatorics, Probability and Computing11 (2001), 179–189.

    MathSciNet  Google Scholar 

  9. V. Nikiforov: Bounds on graph eigenvalues II, Linear Algebra and its Applications427(2–3) (2007), 183–189. Available at http://arxiv.org/abs/math.CO/0612461.

    Article  MATH  MathSciNet  Google Scholar 

  10. P. Rowlinson: On the maximal index of graphs with a prescribed number of edges, Linear Algebra and its Applications110 (1988), 43–53.

    Article  MATH  MathSciNet  Google Scholar 

  11. A. Satyanarayana, L. Schoppman and C. L. Suffel: A reliability-improving graph transformation with applications to network reliability, Networks22 (1992), 209–216.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Algebra and Number Theory, Eötvös Loránd University, H-1117, Budapest, Pázmány Péter sétány 1/C, Hungary

    Péter Csikvári

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  1. Péter Csikvári
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Correspondence to Péter Csikvári.

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While this paper was carried out the author was the guest of the University of Memphis.

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Csikvári, P. On a poset of trees. Combinatorica 30, 125–137 (2010). https://doi.org/10.1007/s00493-010-2516-0

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  • DOI: https://doi.org/10.1007/s00493-010-2516-0

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