Coronae Graphs and Their \(\alpha \)-Eigenvalues


Let \(G_1\) and \(G_2\) be two simple connected graphs. The properties of \(\alpha -\)eigenvalues of four coronal graphs of \(G_1\) and \(G_2\): \(G_1 \circ G_2, G_1\lozenge G_2\), \(G_1 \odot G_2\) and \(G_1 \circleddash G_2\) have been studied. As an application, we construct infinitely many pairs of non-isomorphic graphs with the same \(\alpha \)-eigenvalues.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3


  1. 1.

    Barik, S., Pati, S., Sarma, B.K.: The spectrum of the corona of two graphs. SIAM J. Discrete Math. 21(1), 47–56 (2007)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Springer, New York (2012)

    Google Scholar 

  3. 3.

    Cvetković, D.M., Rowlinson, P., Simíc, S.: An Introduction to the Theory of Graph Spectra. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  4. 4.

    Cvetković, D., Sachs, H.: Spectra of Graphs, Theory and Application, 3rd edn. Johann Ambrosius Barth, Heidelberg (1995)

    Google Scholar 

  5. 5.

    Cui, S.Y., Tian, G.X.: The spectrum and the signless Laplacian spectrum of coronae. Linear Algebra Appl. 437, 1692–1703 (2012)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Frucht, R., Harary, F.: On the corona of two graphs. Aequationes Math. 4, 322–325 (1970)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)

    Google Scholar 

  8. 8.

    Hou, Y., Shiu, W.C.: The spectrum of the edge corona of two graphs. Electron. J. Linear Algebra. 20, 586–594 (2010)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Lan, J., Zhou, B.: Spectra of graph operations based on \(R\)-graph. Linear Multilinear Algebra 63, 1401–1422 (2015)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Liu, X., Liu, S.: On the \(A_\alpha \)-characteristic polynomial of a graph. Linear Algebra Appl. 546, 274–288 (2018)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Liu, X., Zhang, Z.: Spectra of subdivision-vertex and subdivision-edge joins of graphs. Linear Algebra Appl. 438, 3547–3559 (2013)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Liu, X., Zhou, S.: Spectra of the neighbourhood corona of two graphs. Linear Multilinear Algebra 62, 1205–1219 (2014)

    MathSciNet  Article  Google Scholar 

  13. 13.

    McLeman, C., McNicholas, E.: Spectra of coronae. Linear Algebra Appl. 435, 998–1007 (2011)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Nevov, N.: Application of the corona-product of two graphs in Ramsey theory. Annuaire Univ. Sofia Fac. Math. Inform. 79, 349–355 (1985)

    MathSciNet  Google Scholar 

  15. 15.

    Nikiforov, V.: Merging the A- and Q- spectral theories. Appl. Anal. Discrete Math. 11, 81–107 (2017)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Wang, S., Zhou, B.: The signless Laplacian spectra of the corona and edge corona of two graph. Linear Multilinear Algebra 61, 197–204 (2013)

    MathSciNet  Article  Google Scholar 

Download references


The authors would like to thank the anonymous referees for careful reading and suggestions to an earlier version of this paper, which results in an improvement.

Author information



Corresponding author

Correspondence to Xiao-Dong Zhang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by the National Natural Science Foundation of China (Nos.11971311 and 11531001), the Montenegrin-Chinese Science and Technology Cooperation Project (No.3-12) and the Joint NSFC-ISF Research Program (jointly funded by the National Natural Science Foundation of China and the Israel Science Foundation (No. 11561141001)).

Communicated by Sanming Zhou.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tahir, M.A., Zhang, X. Coronae Graphs and Their \(\alpha \)-Eigenvalues. Bull. Malays. Math. Sci. Soc. 43, 2911–2927 (2020).

Download citation


  • \(\alpha \)-Eigenvalue
  • Corona
  • Edge corona
  • R-vertex corona
  • R-edge corona
  • Isospectral graph

Mathematics Subject Classification

  • 05C50
  • 05C76