Journal of Algebraic Combinatorics

, Volume 32, Issue 3, pp 459–464 | Cite as

A relation between the Laplacian and signless Laplacian eigenvalues of a graph

  • Saieed Akbari
  • Ebrahim Ghorbani
  • Jack H. Koolen
  • Mohammad Reza Oboudi
Article

Abstract

Let G be a graph of order n such that \(\sum_{i=0}^{n}(-1)^{i}a_{i}\lambda^{n-i}\) and \(\sum_{i=0}^{n}(-1)^{i}b_{i}\lambda^{n-i}\) are the characteristic polynomials of the signless Laplacian and the Laplacian matrices of G, respectively. We show that a i b i for i=0,1,…,n. As a consequence, we prove that for any α, 0<α≤1, if q 1,…,q n and μ 1,…,μ n are the signless Laplacian and the Laplacian eigenvalues of G, respectively, then \(q_{1}^{\alpha}+\cdots+q_{n}^{\alpha}\geq\mu_{1}^{\alpha}+\cdots+\mu _{n}^{\alpha}\).

Keywords

Laplacian Signless Laplacian Incidence energy Laplacian-like energy 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Saieed Akbari
    • 1
    • 2
  • Ebrahim Ghorbani
    • 1
    • 2
  • Jack H. Koolen
    • 3
    • 4
  • Mohammad Reza Oboudi
    • 1
    • 2
  1. 1.Department of Mathematical SciencesSharif University of TechnologyTehranIran
  2. 2.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran
  3. 3.Department of MathematicsPOSTECHPohangSouth Korea
  4. 4.Pohang Mathematics Institute (PMI)POSTECHPohangSouth Korea

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