一类树的主特征值数目刻画
Characterization of the Number of Main Eigenvalues of a Class of Trees
DOI: 10.12677/ORF.2021.112016, PDF,  被引量   
作者: 杜泽楠, 于 倩：长安大学，理学院，陕西 西安
关键词: 邻接矩阵；主特征值；树图；Adjacency Matrix； Main Eigenvalues； Tree
摘要: 一个图的邻接矩阵的特征值称为图的特征值，图的所有特征值的多重集称为图的谱。如果图的一个特征值所对应的特征空间与全1向量不正交，则称其为主特征值，主特征值对于刻画图及研究图的性质都有重要的意义。刻画恰有k(2≤k≤n)个主特征值的图是一个存在已久的问题，本文给出了一类树的主特征值数目的下界。
Abstract: The eigenvalues of the adjacency matrix of a graph are called the eigenvalues of a graph, the multi-set of all eigenvalues of a graph is called the spectrum of a graph. An eigenvalue of a graph is a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector, the main eigenvalues are significant to the characterization of graphs and the properties of graphs. Characterizing graphs with k(2≤k≤n) number of main eigenvalues are a long-standing problem. In this paper, the lower bound of main eigenvalues of a class of trees is determined.
文章引用：杜泽楠, 于倩. 一类树的主特征值数目刻画[J]. 运筹与模糊学, 2021, 11(2): 131-136. https://doi.org/10.12677/ORF.2021.112016

参考文献

[1] Cvetković, D. (1978) The Main Part of the Spectrum, Divisors and Switching of Graphs. Publications de l’Institut Mathématique (Beograd), 23, 31-38.
[2] Butler, G. and McKay, J. (1983) The Transitive Groups of Degree up to Eleven. Communications in Algebra, 11, 836-911. [Google Scholar] [CrossRef] 
[3] Hagos, E.M. (2002) Some Results on Graph Spectra. Linear Algebra and Its Applications, 356, 103-111. [Google Scholar] [CrossRef] 
[4] Hayat, S., Koolen, J.H., Liu, F. and Qiao, Z. (2016) A Note on Graphs with Exactly Two Main Eigenvalues. Linear Algebra and Its Applications, 511, 318-327. [Google Scholar] [CrossRef] 
[5] Hou, Y.P., Tang, Z.K. and Shiu, W.C. (2012) Some Results on Graphs with Exactly Two Main Eigenvalues. Applied Mathematics Letters, 25, 1274-1278. [Google Scholar] [CrossRef] 
[6] Hou, Y.P. and Tian, F. (2006) Unicyclic Graphs with Exactly Two Main Eigenvalues. Applied Mathematics Letters, 19, 1143-1147. [Google Scholar] [CrossRef] 
[7] Hou, Y.P. and Zhou, H.Q. (2005) Trees with Exactly Two Main Eigenvalues. Journal of Natural Science of Hunan Normal University, 26, 1-3. (In Chinese)
[8] Lepović, M. (2001) Some Results on Graphs with Exactly Two Main Eigenvalues. Publikacija Elektrotehnikog Fakulteta Serija Matematika, 12, 68-84.
[9] Shi, L. (2009) On Graphs with Given Main Eigenvalues. Applied Mathematics Letters, 22, 1870-1874. [Google Scholar] [CrossRef] 
[10] Godsil, C. (2012) Controllable Subsets in Graphs. Annals of Com-binatorics, 16, 733-744. [Google Scholar] [CrossRef] 
[11] O’Rourke, S. and Touri, B. (2016) On a Conjecture of Godsil Concerning Controllable Random Graphs. SIAM Journal on Control and Optimization, 54, 3347-3378. [Google Scholar] [CrossRef] 
[12] 高随祥. 图论与网络流理论[M]. 北京: 高等教育出版社, 2009.
[13] Biggs, N. Algebraic Graph Theory [M]. 北京: 世界图书出版公司, 2004.
[14] Oboudi, R.M. (2018) On the Eigenvalues and Spectral Radius of Starlike Trees. Aequationes Mathematicae, 92, 683-694. [Google Scholar] [CrossRef] 

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