Spectral characterizations of anti-regular graphs
Spectral characterizations of anti-regular graphs
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DOI：
10.1016/j.laa.2018.07.028
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发表时间：
2018-07
影响因子：
1.1
通讯作者：
Cesar O. Aguilar;Joon-yeob Lee;Eric Piato;Barbara J. Schweitzer
中科院分区：
文献类型：
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作者：
Cesar O. Aguilar;Joon-yeob Lee;Eric Piato;Barbara J. Schweitzer
我们研究唯一连通反正则图 A n 的特征值。利用第二类切比雪夫多项式，我们得到一个以特征值为根的三角方程，并进行初等分析以获得特征值的几乎完整的表征。特别是，我们证明区间 Ω=[− 1− 2 2,− 1+ 2 2] 仅包含平凡特征值 λ=− 1 或 λ= 0，并且任何严格大于 Ω 的闭区间将包含对于所有 n 足够大的 A n 特征值。我们还获得了最大和最小特征值的界限，并且对于所有其他特征值，我们获得了随着 n 增加而改善的区间界限。此外，我们的方法揭示了 A n 特征值的二分特征的更完整的图景，即，随着 n 的增加，特征值关于数字− 1 2 （近似）对称。我们还获得了特征值的渐近分布，因为 n→∞ 。最后讨论了A n 的特征值与一般阈值图的特征值之间的关系。
We study the eigenvalues of the unique connected anti-regular graph A n. Using Chebyshev polynomials of the second kind, we obtain a trigonometric equation whose roots are the eigenvalues and perform elementary analysis to obtain an almost complete characterization of the eigenvalues. In particular, we show that the interval Ω=[− 1− 2 2,− 1+ 2 2] contains only the trivial eigenvalues λ=− 1 or λ= 0, and any closed interval strictly larger than Ω will contain eigenvalues of A n for all n sufficiently large. We also obtain bounds for the maximum and minimum eigenvalues, and for all other eigenvalues we obtain interval bounds that improve as n increases. Moreover, our approach reveals a more complete picture of the bipartite character of the eigenvalues of A n, namely, as n increases the eigenvalues are (approximately) symmetric about the number− 1 2. We also obtain an asymptotic distribution of the eigenvalues as n→∞. Finally, the relationship between the eigenvalues of A n and the eigenvalues of a general threshold graph is discussed.