The second largest eigenvalue of some nonnormal Cayley graphs on symmetric groups
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866918173719134208 |
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| author | Li, Yuxuan Xia, Binzhou Zhou, Sanming |
| author_facet | Li, Yuxuan Xia, Binzhou Zhou, Sanming |
| contents | A Cayley graph on the symmetric group $S_n$ is said to have the Aldous property if its strictly second largest eigenvalue (that is, the largest eigenvalue strictly smaller than the degree) is attained by the standard representation of $S_n$. For $1\leq r < k < n$, let $C(n,k;r)$ be the set of $k$-cycles of $S_n$ moving every point in $\{1, \ldots, r\}$. Recently, Siemons and Zalesski [J. Algebraic Combin. 55 (2022) 989--1005] posed a conjecture which is equivalent to saying that for any $n \ge 5$ and $1\leq r<k<n$ the nonnormal Cayley graph $\mathrm{Cay}(S_n, C(n,k;r))$ on $S_n$ with connection set $C(n,k;r)$ has the Aldous property. Solving this conjecture, we prove that all these graphs have the Aldous property except when (i) $(n, k, r) = (6, 5, 1)$ or (ii) $n$ is odd, $k = n-1$, and $1 \le r < \frac{n}{2}$. Along the way we determine all irreducible representations of $S_n$ that can achieve the strictly second largest eigenvalue of $\mathrm{Cay}(S_n, C(n,n-1;r))$ as well as the smallest eigenvalue of this graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02427 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The second largest eigenvalue of some nonnormal Cayley graphs on symmetric groups Li, Yuxuan Xia, Binzhou Zhou, Sanming Combinatorics 05C50, 05C25 A Cayley graph on the symmetric group $S_n$ is said to have the Aldous property if its strictly second largest eigenvalue (that is, the largest eigenvalue strictly smaller than the degree) is attained by the standard representation of $S_n$. For $1\leq r < k < n$, let $C(n,k;r)$ be the set of $k$-cycles of $S_n$ moving every point in $\{1, \ldots, r\}$. Recently, Siemons and Zalesski [J. Algebraic Combin. 55 (2022) 989--1005] posed a conjecture which is equivalent to saying that for any $n \ge 5$ and $1\leq r<k<n$ the nonnormal Cayley graph $\mathrm{Cay}(S_n, C(n,k;r))$ on $S_n$ with connection set $C(n,k;r)$ has the Aldous property. Solving this conjecture, we prove that all these graphs have the Aldous property except when (i) $(n, k, r) = (6, 5, 1)$ or (ii) $n$ is odd, $k = n-1$, and $1 \le r < \frac{n}{2}$. Along the way we determine all irreducible representations of $S_n$ that can achieve the strictly second largest eigenvalue of $\mathrm{Cay}(S_n, C(n,n-1;r))$ as well as the smallest eigenvalue of this graph. |
| title | The second largest eigenvalue of some nonnormal Cayley graphs on symmetric groups |
| topic | Combinatorics 05C50, 05C25 |
| url | https://arxiv.org/abs/2402.02427 |