不含长Berge圈的谱极值超图
Spectral extremal hypergraphs without long Berge cycles
- School of Mathematics and Statistics, Qinghai Minzu University（青海民族大学数学与统计学院）
- HNP-LAMA Central South University（中南大学）
- Changsha 410083, China（长沙）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文确定了不含长度至少为$k$的Berge圈的$r$-均匀超图的最大谱半径，并给出了奇偶两种情形下的唯一极值结构及渐近公式。
AI中文摘要：
设$r\ge 3$和$k\ge 2r+1$为固定整数。我们确定了对于所有足够大的$n$，不含长度至少为$k$的Berge圈的$n$顶点$r$-均匀超图的最大邻接张量谱半径。记$s=\left\lfloor\frac{k-1}{2}\right\rfloor$。若$k=2s+1$为奇数，唯一的极值超图由所有至多包含一个在固定$s$-集之外的顶点的$r$-集组成。若$k=2s+2$为偶数，则额外包含所有在$s$-集之外包含一个固定对，并与$s$-集内$r-2$个顶点结合的$r$-集。因此，最大谱半径由\\[ \operatorname{spex}_r(n,k) = \left[ \binom{s}{r-1} \left(\frac{r-1}{s}\right)^{(r-1)/r} +o(1) \right]n^{(r-1)/r} \\]给出。
英文摘要：
Let $r\ge 3$ and $k\ge 2r+1$ be fixed integers. We determine, for all sufficiently large $n$, the maximum adjacency-tensor spectral radius of an $n$-vertex $r$-uniform hypergraph containing no Berge cycle of length at least $k$. Write $s=\left\lfloor\frac{k-1}{2}\right\rfloor$. If $k=2s+1$ is odd, the unique extremal hypergraph consists of all $r$-sets containing at most one vertex outside a fixed $s$-set. If $k=2s+2$ is even, one additionally includes all $r$-sets containing a fixed pair outside the $s$-set together with $r-2$ vertices inside it. Consequently, the maximum spectral radius is given as \[ \operatorname{spex}_r(n,k) = \left[ \binom{s}{r-1} \left(\frac{r-1}{s}\right)^{(r-1)/r} +o(1) \right]n^{(r-1)/r}. \]
