Integral Laplacian graphs with a unique repeated Laplacian eigenvalue, II
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초록
The multiset~$S_{\{i,j\}_{n}^{m}}=\{0,1,2,\ldots,m-1,m,m,m+1,\ldots,\linebreak n-1,n\}\setminus\{i,j\},\quad 0<i<j\leqslant n$, is called Laplacian realizable if there exists a simple connected graph $G$ whose Laplacian spectrum is~$S_{\{i,j\}_{n}^{m}}$. In this case, the graph $G$ is said to realize $S_{\{i,j\}_{n}^{m}}$. In this paper, we completely describe graphs realizing the multisets~$S_{\{i,j\}_{n}^{m}}$ with $m=1,2$ and determine the structure of these graphs.
Abstract
The multiset~$S_{\{i,j\}_{n}^{m}}=\{0,1,2,\ldots,m-1,m,m,m+1,\ldots,\linebreak n-1,n\}\setminus\{i,j\},\quad 0<i<j\leqslant n$, is called Laplacian realizable if there exists a simple connected graph $G$ whose Laplacian spectrum is~$S_{\{i,j\}_{n}^{m}}$. In this case, the graph $G$ is said to realize $S_{\{i,j\}_{n}^{m}}$. In this paper, we completely describe graphs realizing the multisets~$S_{\{i,j\}_{n}^{m}}$ with $m=1,2$ and determine the structure of these graphs.