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$\mathbb{Z}_{n}$同极大图的Laplacian谱

Laplacian Spectra of Comaximal Graph of $\mathbb{Z}_{n}$

Subarsha Banerjee

发表信息：
Special Matrices, 2021
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摘要：

本文主要研究了环$n> 2$的环$\mathbb Z_n$的共极大图$Γ(\mathbb Z_n)$的Laplacian矩阵的特征值。我们确定了不同$n$的$Γ(\mathbb Z_n)$的特征值，并提供了一个程序来找到任何$n> 2$的$Γ(\mathbb Z_n)$的特征值。证明了$Γ(\mathbb Z_n)$是$n=p^αq^β$的Laplacian积分，其中$p,q$是素数，$α, β$是非负整数. $Γ(\mathbb Z_n)$的代数连通度和顶点连通度对于所有的$n> 2$都是相等的。得到了$Γ(\mathbb Z_n)$第二大特征值的一个上界，并确定了其相等的一个充要条件.最后讨论了$Γ(\mathbb Z_n)$的谱半径的重数和代数连通度的重数。文章最后对今后研究中的一些问题进行了讨论。

英文摘要：

This article focuses on finding the eigenvalues of the Laplacian matrix of the comaximal graph $Γ(\mathbb Z_n)$ of the ring $\mathbb Z_n$ for $n> 2$. We determine the eigenvalues of $Γ(\mathbb Z_n)$ for various $n$ and also provide a procedure to find the eigenvalues of $Γ(\mathbb Z_n)$ for any $n> 2$. We show that $Γ(\mathbb Z_n)$ is Laplacian Integral for $n=p^αq^β$ where $p,q$ are primes and $α, β$ are non-negative integers. The algebraic and vertex connectivity of $Γ(\mathbb Z_n)$ have been shown to be equal for all $n> 2$. An upper bound on the second largest eigenvalue of $Γ(\mathbb Z_n)$ has been obtained and a necessary and sufficient condition for its equality has also been determined. Finally we discuss the multiplicity of the spectral radius and the multiplicity of the algebraic connectivity of $Γ(\mathbb Z_n)$. Some problems have been discussed at the end of this article for further research.

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