Hypergraph Extensions of Spectral Turán Theorem

Abstract

The spectral Turán theorem states that the $k$-partite Turán 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 Turán theorem. In this paper, we consider hypergraph extensions of spectral Turán theorem. For $k\ge r\ge 2$, 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: $\{1,2,\dots ,r\}=:[r]$ and $E_{ij}\cup \{i,j\}$ over all pairs $\{i,j\}\in \left(\genfrac{}{}{0ex}{}{[k+1]}{2}\right)\setminus \left(\genfrac{}{}{0ex}{}{[r]}{2}\right)$, where $E_{ij}$ are pairwise disjoint $(r-2)$-sets disjoint from $[k+1]$. Generalizing the Turán 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 Turán 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 Turán theorem and hypergraph spectral Turán theorem in a unified form via the $p$-spectral radius.

34 pages