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Spectra of partitioned matrices and the \({\mathcal {M}}\)-join of graphs

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Abstract

In this paper, first we consider some partitioned matrices having some special kind of blocks, and we obtain their eigenvalues and eigenvectors. From these, we deduce several results on the eigenvalues of partitioned matrices in the literature. Next we consider an extension of a generalized join of graphs introduced by Hedetniemi (On classes of graphs defined by special cutsets of lines. In: Many facets of graph theory. Proceedings of the conference held at Western MiGammagan University, Kalamazoo/Mi, 1968. Lecture notes in mathematics, vol 110, pp 171–189, 1969), we call it the \({\mathcal {M}}\)-join of \({\mathcal {H}}_k\). The spectra of partitioned matrices allow to detect the adjacency, the Laplacian and the signless Laplacian spectrum of old and new type of join of graphs. Further, we have constructed several pairs of simultaneously A-cospectral, L-cospectral and Q-cospectral graphs.

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References

  1. Adiga, C., Rakshith, B.R.: Spectra of graph operations based on corona and neighborhood corona of graph \(G\) and \(K_1\). J. Int. Math. Virtual Inst. 5, 55–69 (2015)

    MathSciNet  Google Scholar 

  2. Barik, S., Kalita, D., Pati, S., Sahoo, G.: Spectra of graphs resulting from various graph operations and products: a survey. Spec. Matrices 6, 323–342 (2018)

    Article  MathSciNet  Google Scholar 

  3. Bapat, R.B., Karimi, M.: Construction of cospectral regular graphs. Matematićki Vesnik 68(1), 66–76 (2016)

    MathSciNet  Google Scholar 

  4. Butler, S.: A note about cospectral graphs for the adjacency and normalized Laplacian matrices. Linear Multilinear Algebra 58(3), 38–390 (2010)

    Article  MathSciNet  Google Scholar 

  5. Cardoso, D.M., de Freitas, M.A., Martins, E.A., Robbiano, M.: Spectra of graphs obtained by a generalization of the join graph operation. Discrete Math. 313, 733–741 (2013)

    Article  MathSciNet  Google Scholar 

  6. Cardoso, D.M., Martins, E.A., Robbiano, M., Rojo, O.: Eigenvalues of a \(H\)-generalized join graph operation constrained by vertex subsets. Linear Algebra Appl. 438, 3278–3290 (2013)

    Article  MathSciNet  Google Scholar 

  7. Chang, T.-C., Tam, B.-S., Wu, S.-H.: Theorems on partitioned matrices revisited and their applications to graph spectra. Linear Algebra Appl. 434, 559–581 (2011)

    Article  MathSciNet  Google Scholar 

  8. Chen, Y., Chen, H.: The characteristic polynomial of generalized join of graphs. Appl. Math. Comput. 348, 456–464 (2019)

    MathSciNet  Google Scholar 

  9. Cvetković, D., Rowlinson, P., Simić, S.: An Introduction to Theory of Graph Spectra. Cambridge University Press, New York (2010)

    Google Scholar 

  10. Das, A., Panigrahi, P.: Spectra of \(R\)-vertex join and \(R\)-edge join of two graphs. Discuss. Math. Gen. Algebra Appl. 38, 19–31 (2018)

    Article  MathSciNet  Google Scholar 

  11. Das, A., Panigrahi, P.: New classes of simultaneous cospectral graphs for adjacency, the Laplacian and normalized Laplacian matrices. Discuss. Math. Gen. Algebra Appl. 43(2), 303–323 (2019)

    MathSciNet  Google Scholar 

  12. Gallian, J.A.: A dynamic survey of graph labeling. Electron. J. Combin. DS6, 1–535 (2019)

    Google Scholar 

  13. Gayathri, M., Rajkumar, R.: Adjacency and Laplacian spectra of variants of neighbourhood corona of graphs constrained by vertex subsets. Discrete Math. Algorithms Appl. 11(6), Article No. 1950073 (2019)

  14. Goddard, L.S., Schneider, H.: Matrices with nonzero commutator. Proc. Camb. Philos. Soc. 51(4), 551–553 (1955)

    Article  ADS  Google Scholar 

  15. Haynsworth, E.V.: Applications of a theorem on partitioned matrices. J. Res. Nat. Bureau Stand. 62(2), 73–78 (1959)

    MathSciNet  Google Scholar 

  16. Haynsworth, E.V.: A reduction formula for partitioned matrices. J. Res. Nat. Bureau Stand. 64(3), 73–78 (1960)

    MathSciNet  Google Scholar 

  17. Hedetniemi, S.: On classes of graphs defined by special cutsets of lines. In: Many Facets of Graph Theory. Proceedings of the Conference Held at Western MiGammagan University, Kalamazoo/Mi, 1968. Lecture Notes in Mathematics, vol. 110, pp. 171–189 (1969)

  18. Heinze, A.: Applications of Schur rings in algebraic combinatorics: graphs, partial difference sets and cyclotomic schemes (Ph.D. dissertation). Universitat Oldenburg (2001)

  19. Indulal, G.: Spectrum of two new joins of graphs and infinite families of integral graphs. Kragujev. J. Math. 36, 133–139 (2012)

    MathSciNet  Google Scholar 

  20. Lee, S.M., Liu, A.: A construction of k-graceful graphs from complete bipartitite graphs. Southeast Asian Bull. Math. 12, 23–30 (1988)

    MathSciNet  Google Scholar 

  21. Liu, X., Zhou, J., Bu, C.: Resistance distance and Kirchhoff index of R-vertex join and R-edge join of two graphs. Discrete Appl. Math. 187, 130–139 (2015)

    Article  MathSciNet  Google Scholar 

  22. Lu, P., Gao, K., Yang, Y.: Generalized characteristic polynomials of join graphs and its applications. Discrete Dyn. Nat. Soc. 17, 931–942 (2017)

    Google Scholar 

  23. Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia (2000)

    Book  Google Scholar 

  24. Paul, S.: The spectra of a new join of graphs. Discrete Math. Algorithms Appl. 10(6), Article No. 1850074 (2018)

  25. Rajkumar, R., Gayathri, M.: Spectra of \((H_1, H_2)\)-merged subdivision graph of a graph. Indag. Math. 30, 1061–1076 (2019)

    Article  MathSciNet  Google Scholar 

  26. Rajkumar, R., Gayathri, M.: \(M\)-generalized corona of graphs constrained by vertex subsets: generalized characteristic polynomial, adjacency spectrum and Laplacian spectrum (Submitted)

  27. Rajkumar, R., Pavithra, R.: Spectra of M-rooted product of graphs. Linear Multilinear Algebra (2020). https://doi.org/10.1080/03081087.2019.1709407

    Article  Google Scholar 

  28. Sampath Kumar, E.: On duplicate graphs. J. Indian Math. Soc. 37, 285–293 (1973)

    MathSciNet  Google Scholar 

  29. Sampath Kumar, E., Walikar, H.B.: On the splitting graph of a graph. J. Karnatak Univ. Sci. 25, 13–16 (1980)

  30. Sayama, H.: Estimation of Laplacian spectra of direct and strong product graphs. Discrete Appl. Math. 205, 160–170 (2016)

    Article  MathSciNet  Google Scholar 

  31. Schwenk, A.J.: Computing the characteristic polynomial of a graph. In: Bary, R., Harary, F. (eds.) Graphs Combinatorics, Lecture Notes in Mathematics, vol. 406, pp. 153–172. Springer, Berlin (1974)

    Chapter  Google Scholar 

  32. Tian, G.X., He, J.X., Cui, S.Y.: On the Laplacian spectra of some double join operations of graphs. Bull. Malays. Math. Sci. Soc. 42(4), 1555–1566 (2019)

    Article  MathSciNet  Google Scholar 

  33. Varghese, R.P., Rejikumar, K.: Spectra of a new join in duplication graph. Math. Combin. Book. Ser. 4, 53–64 (2018)

    Google Scholar 

  34. Yager, R.R.: On the theory of bags. Int. J. Gen. Syst. 13, 23–37 (1986)

    Article  MathSciNet  Google Scholar 

  35. You, L., Yang, M., So, W., Xi, W.: On the spectrum of an equitable quotient matrix and its application. Linear Algebra Appl. 577, 21–40 (2019)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the referee for his/her useful comments which helped to improve the paper.

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  1. Department of Mathemtics, The Gandhigram Rural Institute (Deemed to be University), Gandhigram, 624 302, India

    Gayathri Murugesan & Rajkumar Rajendran

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  1. Gayathri Murugesan
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  2. Rajkumar Rajendran
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Correspondence to Gayathri Murugesan.

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The first author is supported by INSPIRE Fellowship, Department of Science and Technology, Government of India under the Grant No. DST/INSPIRE Fellowship/[IF150651]/2015.

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Murugesan, G., Rajendran, R. Spectra of partitioned matrices and the \({\mathcal {M}}\)-join of graphs. Ricerche mat 73, 213–260 (2024). https://doi.org/10.1007/s11587-021-00589-x

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  • DOI: https://doi.org/10.1007/s11587-021-00589-x

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