Abstract
In this paper, we determine the graphs which have the minimal spectral radius among all the connected graphs of order n and the independence number \(\lceil \frac{n}{2}\rceil -1.\)
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Data sharing not applicable to this article as no data sets were generated or analysed during the current study.
References
Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Springer, New York (2011)
Cvetković, D., Rowlinson, P., Simić, S.: An Introduction to the Theory of Graph Spectra. Cambridge University Press, Cambridge (2010)
Das, J., Mohanty, S.: On the spectral radius of bi-block graphs with given independence number \(\alpha \). Appl. Math. Comput. 402, 125912 (2021)
Du, X., Shi, L.: Graphs with small independence number minimizing the spectral radius. Discrete Math. Algorithms Appl. 5, 1350017 (2013)
Godsil, C., Royle, G.: Algebraic Graph Theory, Graduate Texts in Mathematics, 207. Springer-Verlag, New York (2001)
Hoffman, A.J., Smith, J.H.: On the spectral radii of topologically equivalent graphs, in: Recent advances in Graph Theory, (Proceedings of the Symposium held in Prague, June 1974.)(ed. M. Fiedler) Academic, Prague, pp.273-281 (1975)
Hu, Y., Huang, Q., Lou, Z.: Graphs with the minimum spectral radius for given independence number, arXiv:2206.09152
Jin, Y.L., Zhang, X.D.: The sharp lower bound for the spectral radius of connected graphs with the independence number. Taiwan. J. Math. 19(2), 419–431 (2015)
Lou, Z., Guo, J.: The spectral radius of graphs with given independence number. Discrete Mathematics 345, 112778 (2022)
Simić, S.K.: On the largest eigenvalue of bicyclic graphs. Publ. Inst. Math. (Beograd) 46(60), 1–6 (1989)
Simić, S.K., Lj, V.: Kocić, On the largest eigenvalue of some homeomorphic graphs. Publ. Inst. Math. (Beograd) 40(54), 3–9 (1986)
Smith, J.H.: Some properties of the spectrum of a graph, in: R. Guy et al.(Eds.), Combinatorial Structures and their applications, Proc. Conf. Calgary, 1969, Gordon and Breach, New York, pp. 403-406 (1970)
Stevanović, D.: Spectral Radius of Graphs. Academic Press, Amsterdam (2015)
Wu, M.M., Hong, Y., Shu, J.L., Zhai, M.Q.: The minimum spectral radius of graphs with a given independence number. Linear Algebra and its Appl. 431, 937–945 (2009)
West, D.B.: Introduction to Graph Theory. Prentice Hall Inc, Upper Saddle River, NJ (2001)
Wu, B., Xiao, E., Hong, Y.: The spectral radius of trees on \(k\) pendant vertices. Linear Algebra and its Appl. 395, 343–349 (2005)
Funding
The first author was partially supported by NRF grant 2018R1C1B6005600.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author has no relevant financial or non-financial interests to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Choi, J., Park, J. The Minimal Spectral Radius with Given Independence Number. Results Math 79, 81 (2024). https://doi.org/10.1007/s00025-023-02117-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-023-02117-9