Article
| Title: | On the multiplicity of Laplacian eigenvalues for unicyclic graphs (English) |
| Author: | Wen, Fei |
| Author: | Huang, Qiongxiang |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 72 |
| Issue: | 2 |
| Year: | 2022 |
| Pages: | 371-390 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $G$ be a connected graph of order $n$ and $U$ a unicyclic graph with the same order. We firstly give a sharp bound for $m_{G}(\mu )$, the multiplicity of a Laplacian eigenvalue $\mu $ of $G$. As a straightforward result, $m_{U}(1)\le n-2$. We then provide two graph operations (i.e., grafting and shifting) on graph $G$ for which the value of $m_{G}(1)$ is nondecreasing. As applications, we get the distribution of $m_{U}(1)$ for unicyclic graphs on $n$ vertices. Moreover, for the two largest possible values of $m_{U}(1)\in \{n-5,n-3\}$, the corresponding graphs $U$ are completely determined. (English) |
| Keyword: | unicyclic graph |
| Keyword: | Laplacian eigenvalue |
| Keyword: | multiplicity |
| Keyword: | bound |
| MSC: | 05C50 |
| idZBL: | Zbl 07547210 |
| idMR: | MR4412765 |
| DOI: | 10.21136/CMJ.2022.0499-20 |
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| Date available: | 2022-04-21T19:00:04Z |
| Last updated: | 2022-09-08 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/150407 |
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| Reference: | [1] Akbari, S., Kiani, D., Mirzakhah, M.: The multiplicity of Laplacian eigenvalue two in unicyclic graphs.Linear Algebra Appl. 445 (2014), 18-28. Zbl 1292.05164, MR 3151261, 10.1016/j.laa.2013.11.022 |
| Reference: | [2] Akbari, S., Dam, E. R. van, Fakharan, M. H.: Trees with a large Laplacian eigenvalue multiplicity.Linear Algebra Appl. 586 (2020), 262-273. Zbl 1429.05118, MR 4027756, 10.1016/j.laa.2019.10.011 |
| Reference: | [3] Andrade, E., Cardoso, D. M., Pastén, G., Rojo, O.: On the Faria's inequality for the Laplacian and signless Laplacian spectra: A unified approach.Linear Algebra Appl. 472 (2015), 81-86. Zbl 1307.05136, MR 3314367, 10.1016/j.laa.2015.01.026 |
| Reference: | [4] Barik, S., Lal, A. K., Pati, S.: On trees with Laplacian eigenvalue one.Linear Multilinear Algebra 56 (2008), 597-610. Zbl 1149.05029, MR 2457687, 10.1080/03081080600679029 |
| Reference: | [5] Brouwer, A. E., Haemers, W. H.: Spectra of Graphs.Universitext. Springer, New York (2012). Zbl 1231.05001, MR 2882891, 10.1007/978-1-4614-1939-6 |
| Reference: | [6] Cvetković, D. M., Rowlinson, P., Simić, S.: An Introduction to the Theory of Graph Spectra.London Mathematical Society Student Texts 75. Cambridge University Press, Cambridge (2010). Zbl 1211.05002, MR 2571608, 10.1017/CBO9780511801518 |
| Reference: | [7] Das, K. C.: Sharp lower bounds on the Laplacian eigenvalues of trees.Linear Algebra Appl. 384 (2004), 155-169. Zbl 1047.05027, MR 2055349, 10.1016/j.laa.2004.01.012 |
| Reference: | [8] Doob, M.: Graphs with a small number of distinct eigenvalues.Ann. N. Y. Acad. Sci. 175 (1970), 104-110. Zbl 0241.05112, MR 0263674, 10.1111/j.1749-6632.1970.tb56460.x |
| Reference: | [9] Faria, I.: Permanental roots and the star degree of a graph.Linear Algebra Appl. 64 (1985), 255-265. Zbl 0559.05041, MR 0776531, 10.1016/0024-3795(85)90281-2 |
| Reference: | [10] Grone, R., Merris, R.: Algebraic connectivity of trees.Czech. Math. J. 37 (1987), 660-670. Zbl 0681.05022, MR 0913997, 10.21136/CMJ.1987.102192 |
| Reference: | [11] Grone, R., Merris, R., Sunder, V. S.: The Laplacian spectrum of a graph.SIAM J. Matrix Anal. Appl. 11 (1990), 218-238. Zbl 0733.05060, MR 1041245, 10.1137/0611016 |
| Reference: | [12] Guo, J.-M., Feng, L., Zhang, J.-M.: On the multiplicity of Laplacian eigenvalues of graphs.Czech. Math. J. 60 (2010), 689-698. Zbl 1224.05297, MR 2672410, 10.1007/s10587-010-0063-x |
| Reference: | [13] Huang, X., Huang, Q.: On regular graphs with four distinct eigenvalues.Linear Algebra Appl. 512 (2017), 219-233. Zbl 1348.05125, MR 3567523, 10.1016/j.laa.2016.09.043 |
| Reference: | [14] Kirkland, S.: A bound on algebra connectivity of a graph in terms of the number cutpoints.Linear Multilinear Algebra 47 (2000), 93-103. Zbl 0947.05052, MR 1752168, 10.1080/03081080008818634 |
| Reference: | [15] Lu, L., Huang, Q., Huang, X.: On graphs with distance Laplacian spectral radius of multiplicity $n-3$.Linear Algebra Appl. 530 (2017), 485-499. Zbl 1367.05134, MR 3672973, 10.1016/j.laa.2017.05.044 |
| Reference: | [16] Rowlinson, P.: On graphs with just three distinct eigenvalues.Linear Algebra Appl. 507 (2016), 462-473. Zbl 1343.05096, MR 3536969, 10.1016/j.laa.2016.06.031 |
| Reference: | [17] Dam, E. R. van: Nonregular graphs with three eigenvalues.J. Comb. Theory, Ser. B 73 (1998), 101-118. Zbl 0917.05044, MR 1631983, 10.1006/jctb.1998.1815 |
| Reference: | [18] Dam, E. R. van, Koolen, J. H., Xia, Z.-J.: Graphs with many valencies and few eigenvalues.Electron. J. Linear Algebra 28 (2015), 12-24. Zbl 1320.05082, MR 3386384, 10.13001/1081-3810.2987 |
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