Revista de la
Unión Matemática Argentina
Using digraphs to compute determinant, permanent, and Drazin inverse of circulant matrices with two parameters
Andrés M. Encinas, Daniel A. Jaume, Cristian Panelo, and Denis E. Videla

Volume 67, no. 1 (2024), pp. 81–106    

Published online: March 12, 2024

https://doi.org/10.33044/revuma.2815

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Abstract

This work presents closed formulas for the determinant, permanent, inverse, and Drazin inverse of circulant matrices with two non-zero coefficients.

References

  1. SageMath, the Sage Mathematics Software System (Version 8.6), The Sage Developers, 2019. Available at https://sagemath.org.
  2. J. Bang-Jensen and G. Gutin, Digraphs: theory, algorithms and applications, second ed., Springer Monographs in Mathematics, Springer-Verlag, London, 2009.  DOI  MR  Zbl
  3. R. B. Bapat, Graphs and matrices, second ed., Universitext, Springer, London; Hindustan Book Agency, New Delhi, 2014.  DOI  MR  Zbl
  4. M. Bašić and A. Ilić, On the clique number of integral circulant graphs, Appl. Math. Lett. 22 no. 9 (2009), 1406–1411.  DOI  MR  Zbl
  5. A. Ben-Israel and T. N. E. Greville, Generalized inverses: theory and applications, second ed., CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 15, Springer-Verlag, New York, 2003.  MR  Zbl
  6. R. A. Brualdi and D. Cvetković, A combinatorial approach to matrix theory and its applications, Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL, 2009.  MR  Zbl
  7. S. L. Campbell and C. D. Meyer, Generalized inverses of linear transformations, Classics in Applied Mathematics 56, SIAM, Philadelphia, PA, 2009.  DOI  MR  Zbl
  8. A. Carmona, A. M. Encinas, S. Gago, M. J. Jiménez, and M. Mitjana, The inverses of some circulant matrices, Appl. Math. Comput. 270 (2015), 785–793.  DOI  MR  Zbl
  9. P. J. Davis, Circulant matrices, second ed., AMS Chelsea Publishing, 1994.  Zbl
  10. A. M. Encinas, D. A. Jaume, C. Panelo, and A. Pastine, Drazin inverse of singular adjacency matrices of directed weighted cycles, Rev. Un. Mat. Argentina 61 no. 2 (2020), 209–227.  DOI  MR  Zbl
  11. L. Fuyong, The inverse of circulant matrix, Appl. Math. Comput. 217 no. 21 (2011), 8495–8503.  DOI  MR  Zbl
  12. A. Ilić and M. Bašić, New results on the energy of integral circulant graphs, Appl. Math. Comput. 218 no. 7 (2011), 3470–3482.  DOI  MR  Zbl
  13. A. W. Ingleton, The rank of circulant matrices, J. London Math. Soc. 31 (1956), 445–460.  DOI  MR  Zbl
  14. Z. Jiang, Y. Gong, and Y. Gao, Invertibility and explicit inverses of circulant-type matrices with $k$-Fibonacci and $k$-Lucas numbers, Abstr. Appl. Anal. (2014), Art. ID 238953, 9 pp.  DOI  MR  Zbl
  15. H. Minc, Permanents, Encyclopedia of mathematics and its applications 6, Cambridge University Press, Cambridge-New York, 1984.  DOI  MR  Zbl
  16. M. D. Petković and M. Bašić, Further results on the perfect state transfer in integral circulant graphs, Comput. Math. Appl. 61 no. 2 (2011), 300–312.  DOI  MR  Zbl
  17. B. Radičić, On $k$-circulant matrices (with geometric sequence), Quaest. Math. 39 no. 1 (2016), 135–144.  DOI  MR  Zbl
  18. J. J. Rotman, An introduction to the theory of groups, fourth ed., Graduate Texts in Mathematics 148, Springer-Verlag, New York, 1995.  DOI  MR  Zbl
  19. B. E. Sagan, The symmetric group, second ed., Graduate Texts in Mathematics 203, Springer-Verlag, New York, 2001.  DOI  MR  Zbl
  20. J. W. Sander, On the kernel of integral circulant graphs, Linear Algebra Appl. 549 (2018), 79–85.  DOI  MR  Zbl
  21. S.-Q. Shen, J.-M. Cen, and Y. Hao, On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers, Appl. Math. Comput. 217 no. 23 (2011), 9790–9797.  DOI  MR  Zbl
  22. W. So, Integral circulant graphs, Discrete Math. 306 no. 1 (2006), 153–158.  DOI  MR  Zbl
  23. W.-H. Steeb and Y. Hardy, Matrix calculus and Kronecker product, second ed., World Scientific, Hackensack, NJ, 2011.  DOI  MR  Zbl
  24. G. Wang, Weighted Moore-Penrose, Drazin, and group inverses of the Kronecker product $A\otimes B$, and some applications, Linear Algebra Appl. 250 (1997), 39–50.  DOI  MR  Zbl
  25. Y. Yazlik and N. Taskara, On the inverse of circulant matrix via generalized $k$-Horadam numbers, Appl. Math. Comput. 223 (2013), 191–196.  DOI  MR  Zbl