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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.00410 |
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| _version_ | 1866913292733120512 |
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| author | Kravchuk, Artem |
| author_facet | Kravchuk, Artem |
| contents | A Transposition graph $T_n$ is defined as a Cayley graph over the symmetric group $Sym_n$ generated by all transpositions. It is known that the spectrum of $T_n$ consists of integers, but it is not known exactly how these numbers are distributed. In this paper we prove that integers from the segment $[-n, n]$ lie in the spectrum of $T_n$ for any $n\geqslant 31$. Using this fact we also prove the main result of this paper that a segment of quadratic length with respect to $n$ lies in the spectrum of $T_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_00410 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constructing segments of quadratic length in $Spec(T_n)$ through segments of linear length Kravchuk, Artem Combinatorics A Transposition graph $T_n$ is defined as a Cayley graph over the symmetric group $Sym_n$ generated by all transpositions. It is known that the spectrum of $T_n$ consists of integers, but it is not known exactly how these numbers are distributed. In this paper we prove that integers from the segment $[-n, n]$ lie in the spectrum of $T_n$ for any $n\geqslant 31$. Using this fact we also prove the main result of this paper that a segment of quadratic length with respect to $n$ lies in the spectrum of $T_n$. |
| title | Constructing segments of quadratic length in $Spec(T_n)$ through segments of linear length |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2404.00410 |