论文标题
具有某些归一化laplacian特征值$ n-3 $的图形的完整表征
Full characterization of graphs having certain normalized Laplacian eigenvalue of multiplicity $n-3$
论文作者
论文摘要
令$ g $为连接的简单订单$ n $的图表。令$ρ_1（g）\ geqρ_2（g）\ geq \ cdots \ geqρ_{n-1}（g）>ρ_n（g）= 0 $是归一化的laplacian matrix $ \ nathcal {l}（g）$ g $ g $的归一化laplacian矩阵的特征值。用$ m（ρ_i）$表示归一化laplacian eigenvalue $ρ_i$的多重性。令$ν（g）$为$ g $的独立数。在本文中，我们给出了一些标准化的laplacian特征值$ n-3 $的图表的完整表征，它回答了[S. S. Sun，K.C。 DAS，在归一化的Laplacian特征值的多重性上，线性代数应用。 609（2021）365-385]，$即，$没有$ m（ρ_1）= n-3 $（$ n \ geq 6 $）和$ν（g）= 2 $。此外，我们确认所有具有$ M（ρ_1）= n-3 $的图形由它们的归一化laplacian光谱确定。
Let $G$ be a connected simple graph of order $n$. Let $ρ_1(G)\geq ρ_2(G)\geq \cdots \geq ρ_{n-1}(G)> ρ_n(G)=0$ be the eigenvalues of the normalized Laplacian matrix $\mathcal{L}(G)$ of $G$. Denote by $m(ρ_i)$ the multiplicity of the normalized Laplacian eigenvalue $ρ_i$. Let $ν(G)$ be the independence number of $G$. In this paper, we give a full characterization of graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$, which answers a remaining problem in [S. Sun, K.C. Das, On the multiplicities of normalized Laplacian eigenvalues of graphs, Linear Algebra Appl. 609 (2021) 365-385], $i.e.,$ there is no graph with $m(ρ_1)=n-3$ ($n\geq 6$) and $ν(G)=2$. Moreover, we confirm that all the graphs with $m(ρ_1)=n-3$ are determined by their normalized Laplacian spectra.
