Let G be a unicyclic graph of order n, λ1(G) and λ2(G) be the largest eigenvalue and second largest eigenvalue of the adjacent matrix of G, μ1(G) and μ2(G) be the largest eigenvalue and second largest eigenvalue of the Laplacian matrix of G, respectively. The separator of G is defined as SA(G)=λ1(G) -λ2(G). The Laplacian separator of G is defined as SL(G)=μ1(G) -μ2(G). In this paper, we study the (Laplacian) separator of unicyclic graphs, and give the extremal graphs which attain the second maximum separator and second maximum Laplacian separator respectively.
YU Guidong, RUAN Zheng, SHU Axiu, YU Tao
. The second maximum (Laplacian) separator of unicyclic graphs[J]. Operations Research Transactions, 2020
, 24(4)
: 128
-134
.
DOI: 10.15960/j.cnki.issn.1007-6093.2020.04.011
[1] Li J X, Shiu W C, Chan W H. Some results on the Laplacian spectrum of unicyclic graphs[J]. Linear Algebra and Application, 2009, 430:2080-2093.
[2] You Z F, Liu B L. The signless Laplacian separator of graphs[J]. Electronic Journal of Linear Algebra, 2011, 22:151-160.
[3] Li J X, Guo J M, Shiu W C. On the second largest Laplacian eigenvalues of graphs[J]. Linear Algebra and Application, 2013, 438:2438-2446.
[4] Yu G D, Huang D M, Zhang W X, et al. The maximum Laplacian separators of bicyclic and tricyclic graphs[J]. 中国科学技术大学学报, 2017, 9:733-737.
[5] 陈爱莲. 单圈图的最大特征值序[J]. 数学研究期刊, 2003, 36(1):87-94.
[6] 吴小军, 张健. 单圈图的第二特征值的下确界[J]. 同济大学学报, 1995, 23(2):206-209.
[7] Cvetković D, Doob D, Sachs M H. Spectra of Graphs-Theory and Application[M]. HeidelbergLeipzig:Johann Ambrosius Barth Verlag, 1995.
[8] Cvetkovic D M, Doob M, Sacks H. Spectra of Graphs-Theory and Applications[M]. New York:Academic Press, 1980, 6:191-195.
[9] Merris R. A note on Laplacian graph eigenvalues[J]. Linear Algebra and Application, 1998, 285:33-35.
[10] Yuan X Y. A note on the Laplacian spectral radii of bicyclic graphs[J]. Advances in Mathematics, 2010, 6(2):703-708.
