On the Laplacian and Signless Laplacian Spectra of Complete Multipartite Graphs
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Published
Nov 1, 2017
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Abstract
Let G be a finite simple graph with vertex set V(G) = {v1, v2, v3, …, vn} and edge set E(G). The adjacency matrix of G is an (n´n)-matrix A(G) = [aij] where aij = 1 if vivj Î E(G) and aij = 0 elsewhere, and the degree matrix of G is a diagonal (n´n)-matrix D(G) = [dij] where dii = degG(vi) and dij = 0 for i ≠ j. The Laplacian matrix of G is L(G) = D(G) – A(G) and the signless Laplacian matrix of G is Q(G) = D(G) + A(G). The study of spectrum of Laplacian and signless Laplacian matrix of graph are interesting topic till today. In this paper, we determine the Laplacian and signless Laplacian spectra of complete multipartite graphs.
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How to Cite
ABDUSSAKIR, Abdussakir; IKAWATI, Deasy Sandhya Elya; SARI, F. Kurnia Nirmala.
On the Laplacian and Signless Laplacian Spectra of Complete Multipartite Graphs.
Proceedings of the International Conference on Green Technology, [S.l.], v. 8, n. 1, p. 335-338, nov. 2017.
ISSN 2580-7099.
Available at: <http://conferences.uin-malang.ac.id/index.php/ICGT/article/view/635>. Date accessed: 26 apr. 2024.
doi: https://doi.org/10.18860/icgt.v8i1.635.
Section
Pure and Applied Mathematics
References
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[2] Abdussakir, N.N. Azizah, F.F. Novandika, Teori graf: Topik dasar untuk tugas akhir/skripsi, UIN Malang Press, Malang, 2009.
[3] Abdussakir, R.R. Elvierayani, M. Nafisah, On the spectra of commuting and non commuting graph on dihedral group, Cauchy-Jurnal Mat. Murni dan Apl. 4 (2017) 176–182.
[4] R. Grone, R. Merris, V.S. Sunder, The Laplacian spectrum of a graph, SIAM J. Matrix Anal. Appl. 11 (1990) 218–238.
[5] R. Grone, R. Merris, The Laplacian spectrum of a graph II, SIAM J. Discret. Math. 7 (1994) 221–229.
[6] A. Jamakovic, P. Van Mieghem, The Laplacian spectrum of complex networks, in: Proc. Eur. Conf. Complex Syst. Oxford, Sept. 25-29, 2006, 2006.
[7] S. Yin, Investigation on spectrum of the adjacency matrix and Laplacian matrix of graph Gl, WSEAS Trans. Syst. 7 (2008) 362–372.
[8] R.B. Bapat, A.K. Lal, S. Pati, Laplacian spectrum of weakly quasi-threshold graphs, Graphs Comb. 24 (2008) 273–290. doi:10.1007/s00373-008-0785-9.
[9] S.Y. Cui, G.X. Tian, The spectrum and the signless Laplacian spectrum of coronae, Linear Algebra Appl. 437 (2012) 1692–1703. doi:10.1016/j.laa.2012.05.019.
[10] R.R. Elvierayani, Abdussakir, Spectrum of the Laplacian matrix of non-commuting graph of dihedral group D2n, in: Proceeding Int. Conf. 4th Green Technol., Faculty of Science and Technology, Malang, 2013: pp. 321–323.
[11] D.M. Cardoso, E.A. Martins, M. Robbiano, V. Trevisan, Computing the Laplacian spectra of some graphs, Discret. Appl. Math. 160 (2012) 2645–2654. doi:10.1016/j.dam.2011.04.002.
[12] Y. Chen, R. Pan, X. Zhang, The Laplacian spectra of graphs and complex networks, J. Univ. Sci. Technol. China. 40 (2010) 1236–1244. doi:10.3696/j.issn.0253-2778.2010.12.004.
[13] G. Guo, G. Wang, On the (signless) Laplacian spectral characterization of the line graphs of lollipop graphs, Linear Algebra Appl. 438 (2013) 4595–4605. doi:10.1016/j.laa.2012.12.015.
[14] S.C. De Lange, M.A. De Reus, M.P. van den Heuvel, The Laplacian spectrum of neural networks, Front. Comput. Neurosci. 7 (2014) 1–12. doi:10.3389/fncom.2013.00189.
[15] S. Barik, R.B. Bapat, S. Pati, On the Laplacian spectra of product graphs, Appl. Anal. Discret. Math. 91 (2015) 39–58.
[16] W.N. Anderson, T.D. Morley, Eigenvalues of the Laplacian of a graph, Linear Multilinear Algebr. 18 (1985) 141–145. doi:10.1080/03081088508817681.
[17] B. Mohar, Laplace eigenvalues of graphs-a survey, Discrete Math. 109 (1992) 171–183. doi:10.1016/0012-365X(92)90288-Q.
[18] R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra Appl. 197–198 (1994) 143–176. doi:10.1016/0024-3795(94)90486-3.
[19] R. Merris, Laplacian graph eigenvectors, Linear Algebra Appl. 278 (1998) 221–236. doi:10.1016/S0024-3795(97)10080-5.
[20] X.D. Zhang, Bipartite graphs with small third Laplacian eigenvalue, Discrete Math. 278 (2004) 241–253. doi:10.1016/S0012-365X(03)00255-3.
[21] B. Jiao, Y. Chun, T.Y. Qiang, Signless laplacians of finite graphs, Int. Conf. Apperceiving Comput. Intell. Anal. ICACIA 2010 - Proceeding. 423 (2010) 440–443. doi:10.1109/ICACIA.2010.5709937.
[22] H. Deng, H. Huang, On the main signless Laplacian eigenvalues of a graph, Electron. J. Linear Algebr. 26 (2013) 381–393. doi:10.1016/j.laa.2013.01.023.
[23] J.M. Guo, J.Y. Ren, J.S. Shi, Maximizing the least signless Laplacian eigenvalue of unicyclic graphs, Linear Algebra Appl. 519 (2017) 136–145. doi:10.1016/j.laa.2016.12.041.