Abstract
Let \(P_k\) be the path of order k. The square \(P_k^2\) of \(P_k\) is obtained by joining all pairs of vertices with distance no more than two in \(P_k\). A graph is called H-free if it does not contain H as a subgraph. In this paper, we consider a Brualdi-Solheid-Turán type problem for \(P_5^2\)-free graphs, and determine the maximum spectral radius among \(P_5^2\)-free graphs of order n.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availibility of data and material
Not applicable.
Change history
01 September 2022
The original version is updated due to changes in spacing and font
References
Babai, L., Guiduli, B.: Spectral extrema for graphs: the Zarankiewicz problem. Electron. J. Combin. 16(1), #R123 (2009)
Brouwer, A., Haemers, W.: Spectra of Graphs. Springer, Berlin (2011)
Chen, M., Liu, A., Zhang, X.: Spectral extremal results with forbidding linear forests. Graphs Combin. 35, 335–351 (2019)
Cioabă, S., Desai, D., Tait, M.: The spectral radius of graphs with no odd wheels. Eur. J. Combin. 99, 103420 (2022)
Cioabă, S., Feng, L., Tait, M., Zhang, X.: The maximum spectral radius of graphs without friendship subgraphs. Electron. J. Combin. 27(4), 4–22 (2020)
Conlon, D., Fox, J.: Graph removal lemmas, surveys in combinatorics. Lond. Math. Soc. Lect. Note Ser. 409, 1–49 (2013)
Dirac, G.: Extensions of Turán’s theorem on graphs. Acta. Math. Acad. Sci. Hunga. 14, 417–422 (1963)
Godsil, C., Royle, G.: Algebraic Graph Theory, Graduate Texts in Mathematics, 207th edn. Springer-Verlag, New York (2001)
Haemers, W.: Interlacing eigenvalues and graphs. Linear Algebra Appl. 226(228), 593–616 (1995)
Li, Y., Liu, W., Feng, L.: A survey on spectral conditions for some extremal graph problems. https://arxiv.org/abs/2111.03309 (2021)
Mantel, W.: Problem 28, soln. by H. Gouventak, W. Mantel, J. Teixeira de Mattes, F. Schuh and W.A. Wythoff. Wiskundige Opgaven 10, 60–61 (1907)
Nikiforov, V.: Bounds on graph eigenvalues II. Linear Algebra Appl. 427, 183–189 (2007)
Nikiforov, V.: The spectral radius of graphs without paths and cycles of specified length. Linear Algebra Appl. 432, 2243–2256 (2010)
Nikiforov, V.: A contribution to the Zarankiewicz problem. Linear Algebra Appl. 432, 1405–1411 (2010)
Nikiforov, V.: Some new results in extremal graph theory, in: surveys in combinatorics. London Math. Soc. Lect. Note Ser. 392, 141–181 (2011)
Sidoreno, A.: What we know and what we do not know about Turán numbers. Graphs Combin. 11(2), 179–199 (1995)
Stein, W.: SageMath, The sage mathematics software system. http://www.sagemath.org (2020)
Wilf, H.: Spectral bounds for the clique and independence numbers of graphs. J. Combin. Theory Ser. B. 40, 113–117 (1986)
Xiao, C., Katona, G., Xiao, J., Oscar, Z.: The Turán number of the square of a path. Discrete Appl. Math. 307, 1–14 (2022)
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)
Zhai, M., Wang, B.: Proof of a conjecture on the spectral radius of \(C_4\)-free graphs. Linear Algebra Appl. 437, 1641–1647 (2012)
Zhai, M., Lin, H.: Spectral extrema of graphs: Forbidden hexagon. Discrete Math. 343(10), 112028 (2020)
Zhai, M., Lin, H.: A strengthening of the spectral chromatic critical edge theorem: books and theta graphs. https://arxiv.org/abs/2102.04041v2 (2021)
Zhao, Y., Huang, X., Lin, H.: The maximum spectral radius of wheel-free graphs. Discrete Math. 344(5), 112341 (2021)
Acknowledgements
Yanhua Zhao gratefully acknowledge the support of this work by the China Scholarship Council (CSC) (No. 202106740049). The authors would like to thank the reviewers for their careful reading and valuable comments.
Funding
This work was supported in part by the National Natural Science Foundation of China (Grant numbers 11771141 and 12011530064), and in part by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (Grant number NRF-2020R1A2C1A01101838).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have 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 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
Zhao, Y., Park, J. A Spectral Condition for the Existence of the Square of a Path. Graphs and Combinatorics 38, 126 (2022). https://doi.org/10.1007/s00373-022-02529-4
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00373-022-02529-4