Spectrum of the Transposition graph
| Autor: | Elena V. Konstantinova, Artem Kravchuk |
|---|---|
| Rok vydání: | 2022 |
| Předmět: |
Numerical Analysis
Algebra and Number Theory Mathematics::Commutative Algebra FOS: Mathematics Discrete Mathematics and Combinatorics Mathematics - Combinatorics Mathematics::General Topology Condensed Matter::Strongly Correlated Electrons Geometry and Topology Combinatorics (math.CO) Mathematics::Spectral Theory |
| DOI: | 10.48550/arxiv.2204.03153 |
| Popis: | Transposition graph $T_n$ is defined as a Cayley graph over the symmetric group generated by all transpositions. It is known that all eigenvalues of $T_n$ are integers. However, an explicit description of the spectrum is unknown. In this paper we prove that for any integer $k\geqslant 0$ there exists $n_0$ such that for any $n\geqslant n_0$ and any $m \in \{0, \dots, k\}$, $m$ is an eigenvalue of $T_n$. In particular, it is proved that zero is an eigenvalue of $T_n$ for any $n\neq2$, and one is an eigenvalue of $T_n$ for any odd $n\geqslant 7$ and for any even $n \geqslant 14$. We also present exact values of the third and the fourth largest eigenvalues of $T_n$ with their multiplicities. |
| Databáze: | OpenAIRE |
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