Integral Laplacian graphs with a unique double Laplacian eigenvalue, I
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866913915600896000 |
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| author | Hameed, Abdul Tyaglov, Mikhail |
| author_facet | Hameed, Abdul Tyaglov, Mikhail |
| contents | 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 $S_{\{i,n\}_{n}^{m}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_00980 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Integral Laplacian graphs with a unique double Laplacian eigenvalue, I Hameed, Abdul Tyaglov, Mikhail Combinatorics 05C50, 05C76 (Primary) 15A1 (Secondary) 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 $S_{\{i,n\}_{n}^{m}}$. |
| title | Integral Laplacian graphs with a unique double Laplacian eigenvalue, I |
| topic | Combinatorics 05C50, 05C76 (Primary) 15A1 (Secondary) |
| url | https://arxiv.org/abs/2206.00980 |