Czechoslovak Mathematical Journal, first online, pp. 1-11


The fan graph is determined by its signless Laplacian spectrum

Muhuo Liu, Yuan Yuan, Kinkar Chandra Das

Received April 1, 2018.   Published online November 18, 2019.

Abstract:  Given a graph $G$, if there is no nonisomorphic graph $H$ such that $G$ and $H$ have the same signless Laplacian spectra, then we say that $G$ is $Q$-DS. In this paper we show that every fan graph $F_n$ is $Q$-DS, where $F_n=K_1\vee P_{n-1}$ and $n\geq3$.
Keywords:  signless Laplacian spectrum; join graph; graph determined by its spectrum
Classification MSC:  05C50, 15A18
DOI:  10.21136/CMJ.2019.0159-18

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] D. M. Cvetković, M. Doob, I. Gutman, A. Torgašev: Recent Results in the Theory of Graph Spectra. Annals of Discrete Mathematics 36, North-Holland, Amsterdam (1988). DOI 10.1016/S0167-5060(08)70277-2 | MR 0926481 | Zbl 0634.05054
[2] D. Cvetković, P. Rowlinson, S. K. Simić: Signless Laplacians of finite graphs. Linear Algebra Appl. 423 (2007), 155-171. DOI 10.1016/j.laa.2007.01.009 | MR 2312332 | Zbl 1113.05061
[3] K. Ch. Das: The Laplacian spectrum of a graph. Comput. Math. Appl. 48 (2004), 715-724. DOI 10.1016/j.camwa.2004.05.005 | MR 2105246 | Zbl 1058.05048
[4] K. Ch. Das: On conjectures involving second largest signless Laplacian eigenvalue of graphs. Linear Algebra Appl. 432 (2010), 3018-3029. DOI 10.1016/j.laa.2010.01.005 | MR 2639266 | Zbl 1195.05040
[5] K. Ch. Das, M. Liu: Complete split graph determined by its (signless) Laplacian spectrum. Discrete Appl. Math. 205 (2016), 45-51. DOI 10.1016/j.dam.2016.01.003 | MR 3478617 | Zbl 1333.05180
[6] M. A. A. de Freitas, N. M. M. de Abreu, R. R. Del-Vecchio, S. Jurkiewicz: Infinite families of $Q$-integral graphs. Linear Algebra Appl. 432 (2010), 2352-2360. DOI 10.1016/j.laa.2009.06.029 | MR 2599865 | Zbl 1219.05158
[7] W. H. Haemers: Interlacing eigenvalues and graphs. Linear Algebra Appl. 226-228 (1995), 593-616. DOI 10.1016/0024-3795(95)00199-2 | MR 1344588 | Zbl 0831.05044
[8] M. Liu: Some graphs determined by their (signless) Laplacian spectra. Czech. Math. J. 62 (2012), 1117-1134. DOI 10.1007/s10587-012-0067-9 | MR 3010260 | Zbl 1274.05299
[9] M. Liu, B. Liu: Extremal Theory of Graph Spectrum. Mathematical Chemistry Monographs 22, University of Kragujevac and Faculty of Science Kragujevac, Kragujevac (2018).
[10] X. Liu, Y. Zhang, X. Gui: The multi-fan graphs are determined by their Laplacian spectra. Discrete Math. 308 (2008), 4267-4271. DOI 10.1016/j.disc.2007.08.002 | MR 2427757 | Zbl 1225.05172
[11] E. R. van Dam, W. H. Haemers: Which graphs are determined by their spectrum? Linear Algebra Appl. 373 (2003), 241-272. DOI 10.1016/S0024-3795(03)00483-X | MR 2022290 | Zbl 1026.05079
[12] J. Wang, H. Zhao, Q. Huang: Spectral characterization of multicone graphs. Czech. Math. J. 62 (2012), 117-126. DOI 10.1007/s10587-012-0021-x | MR 2899739 | Zbl 1249.05256

Affiliations:   Muhuo Liu, Department of Mathematics, South China Agricultural University, 483 Wushan Road, Tianhe District, Guangzhou, 510642, P. R. China, and College of Mathematics and Statistics, Shenzhen University, 3688 Nanhai Boulevard, Nanshan District, Shenzhen, 518060, P. R. China, e-mail: liumuhuo@163.com; Yuan Yuan, School of Science, Hainan University, No. 58, Renmin Avenue, Haikou, 570228, P. R. China, e-mail: kuailenanshi@126.com; Kinkar Chandra Das (corresponding author), Department of Mathematics, Sungkyunkwan University, Seobu-Ro 2066, Jangan-Gu, Suwon 16419, Republic of Korea, e-mail: kinkardas2003@googlemail.com


 
PDF available at: