arXiv 1807.075912018-07math.CO

反正则图的谱特征

Spectral characterizations of anti-regular graphs

Cesar O. Aguilar, Joon-yeob Lee, Eric Piato, Barbara J. Schweitzer

备注：: 20 pages, 6 figures, 1 table
发表信息：: Linear Algebra and Its Applications, Vol. 557, pp. 84-104, 2018
DOI：: 10.1016/j.laa.2018.07.028

摘要：

研究了唯一连通反正则图$A_n$的特征值.利用第二类切比雪夫多项式，得到了一个以特征值为根的三角方程，并进行了初等分析，得到了特征值的一个几乎完全的特征。特别是，我们表明，区间$Ω=[\tfrac{-1-\sqrt{2}}{2}, \tfrac{-1+\sqrt{2}}{2}]$只包含平凡的本征值$λ= -1$或$λ=0$，和任何封闭的区间严格大于$Ω$将包含$A_n$的本征值的所有$n$足够大。我们还获得了最大和最小特征值的界限，并为所有其他特征值，我们得到的区间界限，提高$n$增加。此外，我们的方法揭示了一个更完整的图片$A_n$的特征值的二分字符，即，作为$n$增加的特征值是（近似）对称的数量$-\tfrac{1}{2}$。我们还得到了一个渐近分布的特征值为$n\rightarrow\infty$。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。

英文摘要：

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 $Ω=[\tfrac{-1-\sqrt{2}}{2}, \tfrac{-1+\sqrt{2}}{2}]$ 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 $-\tfrac{1}{2}$. We also obtain an asymptotic distribution of the eigenvalues as $n\rightarrow\infty$. Finally, the relationship between the eigenvalues of $A_n$ and the eigenvalues of a general threshold graph is discussed.

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摘要：

英文摘要：

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