发表机构
Sungkyunkwan University; The State University of New York, Korea（成均馆大学; 纽约州立大学韩国分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了阴影不含完全或完全二部子式的均匀超图的最大谱半径，推广了Tait及Zhai-Lin的图论结果，并给出了极值超图的唯一结构及其阶的渐近估计。
AI 中文摘要
对于$k$-均匀超图$\mathcal H$，其阴影是指以超边所覆盖的二元对为边的图。本文中，对所有充分大的$n$，我们确定了在所有$n$个顶点的$k$-均匀超图中，其阴影不含$K_t$子式（对所有$t\ge k+1$）以及其阴影不含$K_{s,t}$子式（对所有$2\le s\le t$且$s+t\ge k+1$，以及$n-s+1$模$t$的每个余数）时，具有最大邻接张量谱半径的超图；在这些范围之外，问题是平凡的。这将对Tait关于不含$K_r$或$K_{s,t}$子式的图的定理推广到均匀超图，其中Tait定理中剩余的余数情形由Zhai和Lin解决。在每种情况下，极值超图是唯一的，它是某个团与一个我们称为“轻部”的图的联图的$k$-团超图。对于$K_{s,t}$，答案取决于$j=k-s+1$。当$j\le1$时，最大值具有阶$n^{(k-1)/k}$，且轻部是Zhai和Lin针对邻接矩阵所找到的那个，包括其例外分量。当$j\ge2$时（这是图中不会出现的范围），最大值具有阶$n^{(s-1)/k}$，且$t$进入其首项常数。此时轻部由若干$K_t$的拷贝和一个较小的团组成，只有一个例外：对于$(k,s,t)=(9,8,8)$且$n-s+1\equiv2\pmod 8$，出现Petersen图的补图。当较小的团具有$1$到$j-1$个顶点时，极值图不唯一。特别地，对于$t=8$，$4\le s\le7$，$k=s+1$且$n-s+1\equiv2\pmod 8$，Zhai和Lin的极值图的团超图不是极值的。对于$j\ge2$，轻部由一个加权团不等式确定，对于$j\ge3$，该不等式由Chao和Dong的闭邻域计数的加权形式得出。
英文摘要
For a $k$-uniform hypergraph $\mathcal H$, the shadow of $\mathcal H$ is the graph whose edges are the pairs covered by a hyperedge. In this paper, for all sufficiently large $n$, we determine the $n$-vertex $k$-uniform hypergraphs of maximum adjacency-tensor spectral radius whose shadow has no $K_t$ minor, for every $t\ge k+1$, and those whose shadow has no $K_{s,t}$ minor, for every $2\le s\le t$ with $s+t\ge k+1$ and every residue of $n-s+1$ modulo $t$; outside these ranges the problems are trivial. This extends to uniform hypergraphs the theorem of Tait on graphs with no $K_r$ or $K_{s,t}$ minor, whose remaining residues were settled by Zhai and Lin. In each case the extremal hypergraph is unique, and it is the $k$-clique hypergraph of the join of a clique with a graph that we call the light part. For $K_{s,t}$ the answer depends on $j=k-s+1$. When $j\le1$, the maximum has order $n^{(k-1)/k}$, and the light part is the one found by Zhai and Lin for the adjacency matrix, including its exceptional components. When $j\ge2$, a regime that does not occur for graphs, the maximum has order $n^{(s-1)/k}$ and $t$ enters its leading constant. The light part then consists of copies of $K_t$ and one smaller clique, with a single exception: for $(k,s,t)=(9,8,8)$ and $n-s+1\equiv2\pmod 8$, the complement of the Petersen graph appears. When the smaller clique has between $1$ and $j-1$ vertices, the extremal graph is not unique. In particular, for $t=8$, $4\le s\le7$, $k=s+1$ and $n-s+1\equiv2\pmod 8$, the clique hypergraph of the extremal graph of Zhai and Lin is not extremal. For $j\ge2$ the light part is determined by a weighted clique inequality, which for $j\ge3$ follows from a weighted form of the closed-neighborhood counting of Chao and Dong.
Comments50 pages
