Graphs with small third largest Laplacian eigenvalue
Ruanqin Gui
College of Sciences, China Jiliang University
DOI: https://doi.org/10.59429/esta.v13i3.15551
Keywords: Laplacian matrix; characteristic polynomial; Laplacian eigenvalue
Abstract
Let L(G) be the Laplacian matrix of G. In this paper, all connected graphs with third largest Laplacian eigenvalue no more than ℓ are characterized, where ℓ = 1.3820 is the smallest root of the equation x2 -5x+5=0 is characterized. Moreover, these results are used to characterize all connected graphs with the third largest Laplacian eigenvalue is 1.
References
[1] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, The Macmillan Press Ltd., 1976.
[2] D.M. Cevtkovic, M. Doob, H. Sachs, Spectra of Graphs Theory and Applications, Jo-hann Ambrosius Barth Verlag, 2013.
[3] R. Grone, R. Merris, V. Sunder, The Laplacian spectrum of a graph, SIAM J. Matrix Anal. Appl. 11(1990), 218-238.
[4] I. Gutman, V. Gineityte, M. Lepovic, M. Petrovic, The high-energy band in the photo- electron spectrum of alkanes and its dependence on molecular structure, J. Serb. Chem. Soc. 64(1999), 673-680.
[5] R. Merris. The number of eigenvalues greater than two in the Laplacian spectrum of a graph, Port. Math. 48(1991), 345-349.
[6] R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra Appl. 197/198(1994), 143-176.
[7] B. Mohar, Some applications of Laplace eigenvalues of graphs, in: G. Hahn, G. S- abidussi (Eds.), Graph Symmetry, Kluwer Academic Press, Dordrecht, 1997, 225-275.
[8] G.R. Omidi, On a Laplacian spectral characterization of graphs of index less than 2, Linear Algebra Appl. 429(2008), 2724-2731.
[9] M. Petrovic, I. Gutman, M. Lepovtic, On bipartite graphs with small number of Lapla- cian eigenvalues greater than two and three, Linear Multilinear Algebra 47(2000), 205- 215.
[10] L. Voza,s, J. Pelinka,n, On the eigenvalues of trees, Period. Math. Hungar. 3(1973), 175-182.
[11] Y. Wu, G. Yu, J. Shu, Graphs with small second largest Laplacian eigenvalue, Euro- pean J. Combin. 36(2014)190-197.