论文标题
具有唯一双拉拉斯特征值的整体拉普拉斯图，i
Integral Laplacian graphs with a unique double Laplacian eigenvalue, I
论文作者
论文摘要
The set $S_{i,n}=\{0,1,2,\ldots,n-1,n\}\setminus\{i\}$, $1\leqslant i\leqslant n$ is called Laplacian realizable if there exists an undirected simple graph whose Laplacian spectrum is $S_{i,n}$. S. Fallat等人建立了此类图的存在。在2005年。在本文中，我们调查了其拉普拉斯光谱具有$$的图形s _ {\ {i，j \} _ {n}^{m}} = \ {0,1,2，\ ldots，m-1，m，m，m，m+1，\ ldots，n-1，n \} $ m = n-1 $和$ m = n $。我们还显示图表之间的密切关系，实现$ s_ {i，n} $和$ s _ {\ {\ {i，j \} _ {n}^{m}} $，并讨论所谓的$ s_ {n，n} $ - $ s _ {\ {i，n \} _ {n}^{m}} $。
The set $S_{i,n}=\{0,1,2,\ldots,n-1,n\}\setminus\{i\}$, $1\leqslant i\leqslant n$ is called Laplacian realizable if there exists an undirected simple graph whose Laplacian spectrum is $S_{i,n}$. The existence of such graphs was established by S. Fallat et al. in 2005. In this paper, we investigate graphs whose Laplacian spectra have the form $$ S_{\{i,j\}_{n}^{m}}=\{0,1,2,\ldots,m-1,m,m,m+1,\ldots,n-1,n\}\setminus\{i,j\},\qquad 0<i<j\leqslant n, $$ and completely describe those ones with $m=n-1$ and $m=n$. We also show close relations between graphs realizing $S_{i,n}$ and $S_{\{i,j\}_{n}^{m}}$, and discuss the so-called $S_{n,n}$-conjecture and the correspondent conjecture for $S_{\{i,n\}_{n}^{m}}$.
