On Distribution of Laplacian Eigenvalues of Graphs
Bilal A. Rather
备注：
P.hD Thesis submitted in mathematics at University of Kashmir, Srinagar Kashmir 190006, India. The thesis is also available at http://hdl.handle.net/10603/329592
本文主要研究了图的Laplacian矩阵、归一化Laplacian矩阵、无符号Laplacian矩阵和距离无符号Laplacian矩阵的特征值。在第一章中，我们对谱图理论做了一个简单的介绍，并给出了一些定义。第四章$2$讨论了图$ k $的最大拉普拉斯特征值$ S_{k}(G) $和问题以及Brouwer猜想。我们得到了几类图的$ S_{k}(G) $的上界，并用它们来验证Brouwer猜想。此外，我们证明了更一般的类图的布劳威尔猜想。在$3$章中，我们研究了图的拉普拉斯特征值和树的拉普拉斯能量猜想。证明了直径为$ 4 $的树的Laplacian能量猜想。此外，我们证明了这一猜想的所有树有最多$ \frac{9n}{25}-2 $非悬挂顶点。同时，我们还得到了$ n $阶树猜想为真的充分条件。在第$4$章中，我们确定了正则图的并图的正规化Laplacian谱，并得到了一些著名图的谱。作为联合并的结果，我们得到了有限循环群的幂图的归一化拉普拉斯谱。在第二章PDMATHAMV中，我们给出了有限交换环上正则图和零因子图的距离无符号Laplacian谱。同时，我们还得到了广义距离矩阵的谱半径的界。进一步，我们得到了二部图和二部树的广义距离能量。证明了在所有连通二部图中，完全二部图具有最小广义距离能量。此外，对于$ α\in \big(0, \frac{2n}{3n-2}\big) $，我们证明了星图在所有树中具有最小的广义距离能量。
英文摘要：
The work in this thesis concerns the investigation of eigenvalues of the Laplacian matrix, normalized Laplacian matrix, signless Laplacian matrix and distance signless Laplacian matrix of graphs. In Chapter 1, we present a brief introduction of spectral graph theory with some definitions.
Chapter $2$ deals with the sum of $ k $ largest Laplacian eigenvalues $ S_{k}(G) $ of graph $ G $ and Brouwer's conjecture. We obtain the upper bounds for $ S_{k}(G) $ for some classes of graphs and use them to verify Brouwer's conjecture for these classes of graphs. Also, we prove Brouwer's conjecture for more general classes of graphs.
In Chapter $3$, we investigate the Laplacian eigenvalues of graphs and the Laplacian energy conjecture for trees. We prove the Laplacian energy conjecture completely for trees of diameter $ 4 $. Further, we prove this conjecture for all trees having at most $ \frac{9n}{25}-2 $ non-pendent vertices. Also, we obtain the sufficient conditions for the truth of conjecture for trees of order $ n $.
In Chapter $4$, we determine the normalized Laplacian spectrum of the joined union of regular graphs and obtain the spectrum of some well known graphs. As consequences of joined union, we obtain the normalized Laplacian spectrum of power graphs associated to finite cyclic groups.
In Chapter $5$, we find the distance signless Laplacian spectrum of regular graphs and zero-divisor graphs associated to finite commutative ring. Also, we find the bounds for spectral radius of generalized distance matrix. Further, we obtain the generalized distance energy for bipartite graphs and trees. We prove that the complete bipartite graph has minimum generalized distance energy among all connected bipartite graphs. Besides, for $ α\in \big(0, \frac{2n}{3n-2}\big) $, we show that the star graph has minimum generalized distance energy among all trees.
On graphs with distance Laplacian eigenvalues of multiplicity $n-4$
设$G$是具有$n$顶点的连通简单图.距离拉普拉斯矩阵$D^{L}(G)$被定义为$D^L(G)=Diag(Tr)-D(G)$，其中$Diag(Tr)$是顶点传输的对角矩阵，而$D(G)$是$G$的距离矩阵。$D^{L}(G)$的特征值是$G$的距离拉普拉斯特征值，并由$\partial_{1}^{L}(G)\geq \partial_{2}^{L}(G)\geq \dots \geq \partial_{n}^{L}(G)$表示。最大的特征值$\partial_{1}^{L}(G)$称为距离拉普拉斯谱半径。Lu et al.（2017），Fernandes et al.（2018）和Ma et al.（2018）完全刻画了具有多重数$n-3$的距离拉普拉斯特征值的图。本文刻画了具有重数为$n-4$的距离Laplacian谱半径和重数为3或2的距离Laplacian特征值之一为$n$的图。此外，我们完全确定了距离拉普拉斯特征值$n$重数为$n-4$的图。
On graphs with some normalized Laplacian eigenvalue of extremal multiplicity
设$G$是一个以$n$为顶点的连通单图.设$\mathcal{L}(G)$为$G$的归一化拉普拉斯矩阵，$ρ_{n-1}(G)$为$\mathcal{L}(G)$的次小特征值.用$ν(G)$表示$G$的独立数。最近，文献[Characterization of graphs with some normalized Laplacian eigenvalues of multiplicity $n-3$，arXiv：1912.13227]讨论了具有某些正规化Laplacian eigenvalues of multiplicity $n-3$的图.然而，还有一个剩余病例（带有$ρ_{n-1}(G)\neq 1$和$ν(G)= 2$的图表）未被考虑。本文主要研究直径为3的上图和图，研究了重数为$n-3$的正则拉普拉斯特征值的图。
