Majorization and the spectral radius of starlike trees



A starlike tree is a tree with exactly one vertex of degree greater than two. The spectral radius of a graph G, that is denoted by \(\lambda (G)\), is the largest eigenvalue of G. Let k and \(n_1,\ldots ,n_k\) be some positive integers. Let \(T(n_1,\ldots ,n_k)\) be the tree T (T is a path or a starlike tree) such that T has a vertex v so that \(T{\setminus } v\) is the disjoint union of the paths \(P_{n_1-1},\ldots ,P_{n_k-1}\) where every neighbor of v in T has degree one or two. Let \(P=(p_1,\ldots ,p_k)\) and \(Q=(q_1,\ldots ,q_k)\), where \(p_1\ge \cdots \ge p_k\ge 1\) and \(q_1\ge \cdots \ge q_k\ge 1\) are integer. We say P majorizes Q and let \(P\succeq _M Q\), if for every j, \(1\le j\le k\), \(\sum _{i=1}^{j}p_i\ge \sum _{i=1}^{j}q_i\), with equality if \(j=k\). In this paper we show that if P majorizes Q, that is \((p_1,\ldots ,p_k)\succeq _M(q_1,\ldots ,q_k)\), then \(\lambda (T(q_1,\ldots ,q_k))\ge \lambda (T(p_1,\ldots ,p_k))\).


Starlike tree Spectral radius Majorization 

Mathematics Subject Classification

05C31 05C50 15A18 



This research was in part supported by a grant (No. 96050011) from School of Mathematics, Institute for Research in Fundamental Sciences (IPM).


  1. Arnold BC (1987) Majorization and the Lorenz order: a brief introduction. Lecture notes in statistics, vol 43. Springer, New YorkCrossRefGoogle Scholar
  2. Cvetković DM, Doob M, Sachs H (1980) Spectra of graphs, theory and application. Academic Press, New YorkMATHGoogle Scholar
  3. Cvetković D, Rowlinson P, Simić S (2010) An introduction to the theory of graph spectra. London mathematical society student texts, vol 75. Cambridge University Press, CambridgeMATHGoogle Scholar
  4. Das KCh, Kumar P (2004) Some new bounds on the spectral radius of graphs. Discrete Math 281:149–161MathSciNetCrossRefMATHGoogle Scholar
  5. Jacobs DP, Trevisan V (2011) Locating the eigenvalues of trees. Linear Algebra Appl 434:81–88MathSciNetCrossRefMATHGoogle Scholar
  6. Lovász L, Pelikán J (1973) On the eigenvalues of trees. Period Math Hung 3:175–182MathSciNetCrossRefMATHGoogle Scholar
  7. Ming GJ, Wang TSh (2001) On the spectral radius of trees. Linear Algebra Appl 329:1–8MathSciNetCrossRefMATHGoogle Scholar
  8. Oboudi MR (2013) On the largest real root of independence polynomials of graphs, an ordering on graphs, and starlike trees. arXiv:1303.3222
  9. Oboudi MR (2016a) Cospectrality of complete bipartite graphs. Linear Multilinear Algebra 64:2491–2497MathSciNetCrossRefMATHGoogle Scholar
  10. Oboudi MR (2016b) Energy and Seidel energy of graphs. MATCH Commun Math Comput Chem 75:291–303MathSciNetMATHGoogle Scholar
  11. Oboudi MR (2016c) On the third largest eigenvalue of graphs. Linear Algebra Appl 503:164–179MathSciNetCrossRefMATHGoogle Scholar
  12. Oboudi MR (2016d) Bipartite graphs with at most six non-zero eigenvalues. ARS Math Contemp 11:315–325MathSciNetMATHGoogle Scholar
  13. Oboudi MR (2017a) On the difference between the spectral radius and maximum degree of graphs. Algebra Discrete Math 24:302–307Google Scholar
  14. Oboudi MR (2017b) Characterization of graphs with exactly two non-negative eigenvalues. ARS Math Contemp 12:271–286MathSciNetMATHGoogle Scholar
  15. Oboudi MR (2018a) On the eigenvalues and spectral radius of starlike trees. Aequ Math. Google Scholar
  16. Oboudi MR (2018b) On the largest real root of independence polynomials of trees. Ars Comb. 137:149–164Google Scholar
  17. Shi L (2007) Bounds on the (Laplacian) spectral radius of graphs. Linear Algebra Appl 422:755–770MathSciNetCrossRefMATHGoogle Scholar
  18. Stevanović D (2003) Bounding the largest eigenvalue of trees in terms of the largest vertex degree. Linear Algebra Appl 360:35–42MathSciNetCrossRefMATHGoogle Scholar
  19. Stevanović D, Gutman I, Rehman MU (2015) On spectral radius and energy of complete multipartite graphs. ARS Math Contemp 9:109–113MathSciNetMATHGoogle Scholar
  20. Wu B, Xiao E, Hong Y (2005) The spectral radius of trees on \(k\) pendant vertices. Linear Algebra Appl 395:343–349MathSciNetCrossRefMATHGoogle Scholar
  21. Yu A, Lu M, Tian F (2004) On the spectral radius of graphs. Linear Algebra Appl 387:41–49MathSciNetCrossRefMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesShiraz UniversityShirazIran
  2. 2.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran

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