Distribution of signless Laplacian eigenvalues and degree sequence
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914605498892288 |
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| author | Akbari, Saieed Darougheh, M. de Lima, L. S. Tracina, D. |
| author_facet | Akbari, Saieed Darougheh, M. de Lima, L. S. Tracina, D. |
| contents | Let $G$ be a graph of order $n$ with degree sequence $d_1 \geq \cdots \geq d_{n}$. Let $m_{G}I$ be the number of signless Laplacian eigenvalues in an interval $I$. In this paper, we characterize the distribution of the signless Laplacian eigenvalues in terms of the degree sequence of a graph within specific subintervals of $[0, \, 2n-2].$ We determine all graphs $G$ such that $m_{G}[d_n, 2n-2] \leq 2, \; m_{G}[d_{n-1}, 2n-2] = 1, \; m_{G}[0, d_1] \le 2.$ We also prove that there is no graph such that $m_{G}[0, d_3]=1$. In addition, we obtain all disconnected graphs such that $m_{G}[0, d_1] = 3$. Finally, we propose two open problems for future research. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27405 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Distribution of signless Laplacian eigenvalues and degree sequence Akbari, Saieed Darougheh, M. de Lima, L. S. Tracina, D. Combinatorics Spectral Theory 05C07, 05C50, 15A18 Let $G$ be a graph of order $n$ with degree sequence $d_1 \geq \cdots \geq d_{n}$. Let $m_{G}I$ be the number of signless Laplacian eigenvalues in an interval $I$. In this paper, we characterize the distribution of the signless Laplacian eigenvalues in terms of the degree sequence of a graph within specific subintervals of $[0, \, 2n-2].$ We determine all graphs $G$ such that $m_{G}[d_n, 2n-2] \leq 2, \; m_{G}[d_{n-1}, 2n-2] = 1, \; m_{G}[0, d_1] \le 2.$ We also prove that there is no graph such that $m_{G}[0, d_3]=1$. In addition, we obtain all disconnected graphs such that $m_{G}[0, d_1] = 3$. Finally, we propose two open problems for future research. |
| title | Distribution of signless Laplacian eigenvalues and degree sequence |
| topic | Combinatorics Spectral Theory 05C07, 05C50, 15A18 |
| url | https://arxiv.org/abs/2605.27405 |