Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

COMPLETE CHARACTERIZATION OF ODD FACTORS VIA THE SIZE, SPECTRAL RADIUS OR DISTANCE SPECTRAL RADIUS OF GRAPHS

  • Li, Shuchao (Faculty of Mathematics and Statistics Central China Normal University) ;
  • Miao, Shujing (Faculty of Mathematics and Statistics Central China Normal University)
  • Received : 2021.08.18
  • Accepted : 2021.12.06
  • Published : 2022.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

Given a graph G, a {1, 3, …, 2n-1}-factor of G is a spanning subgraph of G, in which each degree of vertices is one of {1, 3, …, 2n-1}, where n is a positive integer. In this paper, we first establish a lower bound on the size (resp. the spectral radius) of G to guarantee that G contains a {1, 3, …, 2n-1}-factor. Then we determine an upper bound on the distance spectral radius of G to ensure that G has a {1, 3, …, 2n-1}-factor. Furthermore, we construct some extremal graphs to show all the bounds obtained in this contribution are best possible.

Keywords

Acknowledgement

This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 12171190, 11671164).

References

  1. A. Amahashi, On factors with all degrees odd, Graphs Combin. 1 (1985), no. 2, 111-114. https://doi.org/10.1007/BF02582935
  2. R. B. Bapat, Graphs and Matrices, second edition, Universitext, Springer, London, 2014. https://doi.org/10.1007/978-1-4471-6569-9
  3. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier Publishing Co., Inc., New York, 1976.
  4. E.-K. Cho, J. Y. Hyun, S. O, and J. R. Park, Sharp conditions for the existence of an even [a, b]-factor in a graph, Bull. Korean Math. Soc. 58 (2021), no. 1, 31-46. https://doi.org/10.4134/BKMS.b191050
  5. C. Godsil and G. Royle, Algebraic Graph Theory, Graduate Texts in Mathematics, 207, Springer-Verlag, New York, 2001. https://doi.org/10.1007/978-1-4613-0163-9
  6. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985. https://doi.org/10.1017/CBO9780511810817
  7. S. Li and S. Miao, Characterizing P≥-factor and P≥-factor covered graphs with respect to the size or the spectral radius, Discrete Math. 344 (2021), no. 11, Paper No. 112588, 12 pp. https://doi.org/10.1016/j.disc.2021.112588
  8. Z. Liu, On spectral radius of the distance matrix, Appl. Anal. Discrete Math. 4 (2010), no. 2, 269-277. https://doi.org/10.2298/AADM100428020L
  9. H. Minc, Nonnegative Matrices, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1988.
  10. V. Nikiforov, Merging the A- and Q-spectral theories, Appl. Anal. Discrete Math. 11 (2017), no. 1, 81-107. https://doi.org/10.2298/AADM1701081N
  11. S. O, Spectral radius and matchings in graphs, Linear Algebra Appl. 614 (2021), 316-324. https://doi.org/10.1016/j.laa.2020.06.004
  12. W. T. Tutte, The factorization of linear graphs, J. London Math. Soc. 22 (1947), 107-111. https://doi.org/10.1112/jlms/s1-22.2.107
  13. M. L. Vergnas, An extension of Tutte's 1-factor theorem, Discrete Math. 23 (1978), no. 3, 241-255. https://doi.org/10.1016/0012-365X(78)90006-7
  14. D. B. West, Introduction to Graph Theory, Prentice Hall, Inc., Upper Saddle River, NJ, 1996.
  15. L. You, M. Yang, W. So, and W. Xi, On the spectrum of an equitable quotient matrix and its application, Linear Algebra Appl. 577 (2019), 21-40. https://doi.org/10.1016/j.laa.2019.04.013
  16. C. Yuting and M. Kano, Some results on odd factors of graphs, J. Graph Theory 12 (1988), no. 3, 327-333. https://doi.org/10.1002/jgt.3190120305