Bulletin of the Iranian Mathematical Society

, Volume 44, Issue 1, pp 43–51 | Cite as

On Two Conjectures of Spectral Graph Theory

Original Paper


Let \(G=(V,\,E)\) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the Laplacian matrix and the signless Laplacian matrix of G are \(L(G)=D(G)-A(G)\) and \(Q(G)=D(G)+A(G)\), respectively. Also denote by \(\lambda _1(G)\), a(G), \(q_1(G)\) and \(\delta (G)\) the largest eigenvalue of A(G), the second smallest eigenvalue of L(G), the largest eigenvalue of Q(G) and the minimum degree of G, respectively. In this paper, we give partial proofs to the following two conjectures:

Aouchiche (Comparaison Automatisée d’Invariants en Théorie des Graphes, 2006) if G is a connected graph, then \(a(G)/\delta (G)\) is minimum for graph composed of 2 triangles linked with a path.


Aouchiche et al. (Linear Algebra Appl 432:2293–2322, 2010) and Cvetković et al. (Publ Inst Math Beogr 81(95):11–27, 2007) if G is a connected graph with \(n\ge 4\) vertices, then \(q_1(G)-2\,\lambda _1(G)\le n-2\,\sqrt{n-1}\) with equality holding if and only if \(G\cong K_{1,n-1}\).


Graph Spectral radius Signless Laplacian spectral radius Algebraic connectivity 

Mathematics Subject Classification

05C50 15A18 15A36 



The authors are grateful to the anonymous referee for helpful suggestions and valuable comments, which led to an improvement of the original manuscript. The first author is supported by the Sungkyun research fund, Sungkyunkwan University, 2017, and National Research Foundation of the Korean government with Grant no. 2017R1D1A1B03028642. The second author is partially supported by NSFC (No. 11571123), the Training Program for Outstanding Young Teachers in University of Guangdong Province (No. YQ2015027) and Guangdong Engineering Research Center for Data Science (No. 2017A-KF02).


  1. 1.
    Aouchiche M.: Comparaison Automatisée d’Invariants en Théorie des Graphes, Ph.D. Thesis, École Polytechnique de Montréal (2006)Google Scholar
  2. 2.
    Aouchiche, M., Hansen, P.: A survey of automated conjectures in spectral graph theory. Linear Algebra Appl. 432, 2293–2322 (2010)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cvetković, D., Doob, M., Sachs, H.: Spectra of Graphs, 3rd edn. Barth, Heidelberg (1995)MATHGoogle Scholar
  4. 4.
    Cvetković, D., Rowlinson, P., Simić, S.K.: Eigenvalue bounds for the signless Laplacian. Publ. Inst. Math. ( Beogr.) 81(95), 11–27 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Das, K.C., Kumar, P.: Bounds on the greatest eigenvalue of graphs. Indian J. Pure Appl. Math. 34(6), 917–925 (2003)MathSciNetMATHGoogle Scholar
  6. 6.
    Das, K.C., Liu, M., Shan, H.: Upper bounds on the (signless) Laplacian eigenvalues of graphs. Linear Algebra Appl. 459, 334–341 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Feng, L.: The signless Laplacian spectral radius for bicyclic graphs with \(k\) pendant vertices. Kyungpook Math. J. 50, 109–116 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fiedler, M.: Algebraic connectivity of graphs. Czechoslovak Math. J. 23, 298–305 (1973)MathSciNetMATHGoogle Scholar
  9. 9.
    Merris, R.: Laplacian matrices of graphs: a survey. Linear Algebra Appl. 197–198, 143–176 (1994)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Li, J., Guo, J.-M., Shiu, W.C.: The smallest values of algebraic connectivity for trees. Acta Math. Sin. (Engl. Ser.) 28(10), 2021–2032 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Shao, J.-Y., Guo, J.-M., Shan, H.-Y.: The ordering of trees and connected graphs by algebraic connectivity. Linear Algebra Appl. 428, 1421–1438 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Stein, et al., W.A.: Sage Mathematics Software (Version 6.8), The Sage Development Team, (2015).
  13. 13.
    Yang, J., You, L.: On a conjecture for the signless Laplacian eigenvalues. Linear Algebra Appl. 446, 115–132 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Yuan, X.-Y., Shao, J.-Y., Zhang, L.: The six classes of trees with the largest algebraic connectivity. Discrete Appl. Math. 156, 757–769 (2008)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Zhai, M., Shu, J., Hong, Y.: On the Laplacian spread of graphs. Appl. Math. Lett. 24, 2097–2101 (2011)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  1. 1.Department of MathematicsSungkyunkwan UniversitySuwonRepublic of Korea
  2. 2.Department of Applied MathematicsSouth China Agricultural UniversityGuangzhouPeople’s Republic of China
  3. 3.College of Mathematics and StatisticsShenzhen UniversityShenzhenPeople’s Republic of China

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