On the spectra of prefix-reversal graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916787512147968 |
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| author | Blanco, Saúl A. Buehrle, Charles |
| author_facet | Blanco, Saúl A. Buehrle, Charles |
| contents | In this paper, we study spectral properties of prefix-reversal graphs. These graphs are obtained by connecting two elements of $C_m\wr S_n$ via prefix reversals. If $m=1,2$, the corresponding prefix-reversal graphs are the classic pancake and burnt pancake graphs. If $m>2$, then one can consider the directed and undirected versions of these graphs. We prove that the spectrum of the undirected prefix-reversal graph $\mathbb{P}_m(n)$ contains all even integers in the interval $[0,2n]\setminus\{2\lfloor n/2\rfloor\}$ and if $m\equiv0\pmod4$, we then show that the spectrum contains all even integers in $[0,2n]$. In the directed case, we show that the spectrum of the directed prefix-reversal graph $P(m,n)$ contains all integers in the interval $[0,n]\setminus\{\lfloor n/2\rfloor\}$. As a consequence, we show that in either case, the prefix-reversal graphs have a small spectral gap. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_08345 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the spectra of prefix-reversal graphs Blanco, Saúl A. Buehrle, Charles Combinatorics Discrete Mathematics 05C50, 68R10 G.2.1; G.2.2 In this paper, we study spectral properties of prefix-reversal graphs. These graphs are obtained by connecting two elements of $C_m\wr S_n$ via prefix reversals. If $m=1,2$, the corresponding prefix-reversal graphs are the classic pancake and burnt pancake graphs. If $m>2$, then one can consider the directed and undirected versions of these graphs. We prove that the spectrum of the undirected prefix-reversal graph $\mathbb{P}_m(n)$ contains all even integers in the interval $[0,2n]\setminus\{2\lfloor n/2\rfloor\}$ and if $m\equiv0\pmod4$, we then show that the spectrum contains all even integers in $[0,2n]$. In the directed case, we show that the spectrum of the directed prefix-reversal graph $P(m,n)$ contains all integers in the interval $[0,n]\setminus\{\lfloor n/2\rfloor\}$. As a consequence, we show that in either case, the prefix-reversal graphs have a small spectral gap. |
| title | On the spectra of prefix-reversal graphs |
| topic | Combinatorics Discrete Mathematics 05C50, 68R10 G.2.1; G.2.2 |
| url | https://arxiv.org/abs/2506.08345 |