标题： 超图的谱Turán定理扩展 显示英文标题

标题： Hypergraph Extensions of Spectral Turán Theorem

摘要： 谱Turán定理指出，在所有不包含完全图$K_{k+1}$作为子图的阶为$n$的图中，$k$-分图Turán图是具有最大邻接谱半径的唯一图。 该结果已知比经典的Turán定理更强。 在本文中，我们考虑谱Turán定理的超图扩展。 对于$k\geq r\geq 2$，让$H_{k+1}^{(r)}$成为从$K_{k+1}$通过用新的 $(r-2)$个顶点扩展每条边得到的$r$-一致超图。 设$F_{k+1}^{(r)}$为一个$r$-一致超图，其边为$\{1,2,\ldots,r\} =: [r]$和$E_{ij} \cup\{i,j\}$，在所有对$\{i,j\}\in \binom{[k+1]}{2}\setminus\binom{[r]}{2}$上，其中$E_{ij}$是两两不相交的$(r-2)$-集，且与$[k+1]$不相交。 将Turán定理推广到超图，Pikhurko [J. Combin. Theory Ser. B, 103 (2013) 220--225] 和 Mubayi 和 Pikhurko [J. Combin. Theory Ser. B, 97 (2007) 669--678] 分别确定了$H_{k+1}^{(r)}$和$F_{k+1}^{(r)}$的精确Turán数，并刻画了相应的极值超图。 我们的主要结果表明，$T_r(n,k)$是一个在$n$个顶点上的完全$k$-部分$r$-一致超图，其中任意两个部分的大小相差不超过一，它是所有$n$个顶点的$H_{k+1}^{(r)}$-自由（分别）超图中具有最大$p$-谱半径的唯一超图。 $F_{k+1}^{(r)}$-free) $r$-uniform hypergraphs for sufficiently large $n$. 这些发现是通过建立$p$-spectral 版本的稳定性定理来实现的。 我们的结果提供了$p$-spectral 对 Mubayi 和 Pikhurko 的结果的类比，并通过$p$-spectral radius 将超图 Turán 定理和超图谱 Turán 定理以统一的形式联系起来。 显示英文摘要

摘要： The spectral Tur\'an theorem states that the $k$-partite Tur\'an graph is the unique graph attaining the maximum adjacency spectral radius among all graphs of order $n$ containing no the complete graph $K_{k+1}$ as a subgraph. This result is known to be stronger than the classical Tur\'an theorem. In this paper, we consider hypergraph extensions of spectral Tur\'an theorem. For $k\geq r\geq 2$, let $H_{k+1}^{(r)}$ be the $r$-uniform hypergraph obtained from $K_{k+1}$ by enlarging each edge with a new set of $(r-2)$ vertices. Let $F_{k+1}^{(r)}$ be the $r$-uniform hypergraph with edges: $\{1,2,\ldots,r\} =: [r]$ and $E_{ij} \cup\{i,j\}$ over all pairs $\{i,j\}\in \binom{[k+1]}{2}\setminus\binom{[r]}{2}$, where $E_{ij}$ are pairwise disjoint $(r-2)$-sets disjoint from $[k+1]$. Generalizing the Tur\'an theorem to hypergraphs, Pikhurko [J. Combin. Theory Ser. B, 103 (2013) 220--225] and Mubayi and Pikhurko [J. Combin. Theory Ser. B, 97 (2007) 669--678] respectively determined the exact Tur\'an number of $H_{k+1}^{(r)}$ and $F_{k+1}^{(r)}$, and characterized the corresponding extremal hypergraphs. Our main results show that $T_r(n,k)$, the complete $k$-partite $r$-uniform hypergraph on $n$ vertices where no two parts differ by more than one in size, is the unique hypergraph having the maximum $p$-spectral radius among all $n$-vertex $H_{k+1}^{(r)}$-free (resp. $F_{k+1}^{(r)}$-free) $r$-uniform hypergraphs for sufficiently large $n$. These findings are obtained by establishing $p$-spectral version of the stability theorems. Our results offer $p$-spectral analogues of the results by Mubayi and Pikhurko, and connect both hypergraph Tur\'an theorem and hypergraph spectral Tur\'an theorem in a unified form via the $p$-spectral radius.