Using digraphs to compute determinant, permanent, and Drazin inverse of circulant matrices with two parameters

Authors

  • Andrés M. Encinas Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain
  • Daniel A. Jaume Departamento de Matemáticas. Facultad de Ciencias Físico-Matemáticas y Naturales, Universidad Nacional de San Luis, Instituto de Matemáticas Aplicadas de San Luis, IMASL-CONICET, San Luis, Argentina
  • Cristian Panelo Departamento de Matemáticas, Facultad de Ciencias Físico-Matemáticas y Naturales, Universidad Nacional de San Luis, San Luis, Argentina
  • Denis E. Videla FaMAF - CIEM (CONICET), Universidad Nacional de Córdoba, Argentina

DOI:

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

Abstract

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

Downloads

Download data is not yet available.

References

SageMath, the Sage Mathematics Software System (Version 8.6), The Sage Developers, 2019. Available at https://sagemath.org.

J. Bang-Jensen and G. Gutin, Digraphs: theory, algorithms and applications, second ed., Springer Monographs in Mathematics, Springer-Verlag, London, 2009.  DOI  MR  Zbl

R. B. Bapat, Graphs and matrices, second ed., Universitext, Springer, London; Hindustan Book Agency, New Delhi, 2014.  DOI  MR  Zbl

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

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

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

S. L. Campbell and C. D. Meyer, Generalized inverses of linear transformations, Classics in Applied Mathematics 56, SIAM, Philadelphia, PA, 2009.  DOI  MR  Zbl

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

P. J. Davis, Circulant matrices, second ed., AMS Chelsea Publishing, 1994.  Zbl

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

L. Fuyong, The inverse of circulant matrix, Appl. Math. Comput. 217 no. 21 (2011), 8495–8503.  DOI  MR  Zbl

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

A. W. Ingleton, The rank of circulant matrices, J. London Math. Soc. 31 (1956), 445–460.  DOI  MR  Zbl

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

H. Minc, Permanents, Encyclopedia of mathematics and its applications 6, Cambridge University Press, Cambridge-New York, 1984.  DOI  MR  Zbl

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

B. Radičić, On $k$-circulant matrices (with geometric sequence), Quaest. Math. 39 no. 1 (2016), 135–144.  DOI  MR  Zbl

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

B. E. Sagan, The symmetric group, second ed., Graduate Texts in Mathematics 203, Springer-Verlag, New York, 2001.  DOI  MR  Zbl

J. W. Sander, On the kernel of integral circulant graphs, Linear Algebra Appl. 549 (2018), 79–85.  DOI  MR  Zbl

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

W. So, Integral circulant graphs, Discrete Math. 306 no. 1 (2006), 153–158.  DOI  MR  Zbl

W.-H. Steeb and Y. Hardy, Matrix calculus and Kronecker product, second ed., World Scientific, Hackensack, NJ, 2011.  DOI  MR  Zbl

G. Wang, Weighted Moore-Penrose, Drazin, and group inverses of the Kronecker product $A ⊗ B$, and some applications, Linear Algebra Appl. 250 (1997), 39–50.  DOI  MR  Zbl

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

Downloads

Published

2024-03-12

Issue

Vol. 67 No. 1 (2024)

Section

Article

License

Copyright (c) 2024 Andrés M. Encinas, Daniel A. Jaume, Cristian Panelo, Denis E. Videla

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Authors who publish with this journal agree to the following terms:

Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal. The Journal may retract the paper after publication if clear evidence is found that the findings are unreliable as a result of misconduct or honest error.