Spectral Stability Theorems for Hypergraphs and Applications
Organizer
Speaker
Lele Liu
Time
Tuesday, December 30, 2025 4:00 PM - 5:30 PM
Venue
Online
Online
Zoom 787 662 9899
(BIMSA)
Abstract
Spectral stability results are powerful tools for solving spectral extremal problems, which say roughly that a near-extremal (with respect to spectral radius) $n$-vertex $F$-free graph must be structurally close to the extremal graphs. Such stability results are crucial in resolving spectral Tur\'an-type problems. In this paper, we study spectral stability results for hypergraphs and their applications. For $k\geq r\geq 2$, let $H_{k+1}^{(r)}$ denote 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 $[k+1]$. The main contributions of this paper are twofold. First, we establish a general criterion that can obtain spectral stability results easily. Utilizing this criterion, we then derive spectral stability results for $H_{k+1}^{(r)}$ and $F_{k+1}^{(r)}$, respectively. Second, we determine the unique extremal hypergraph having the maximum $p$-spectral radius among all $n$-vertex $F_{k+1}^{(r)}$-free (resp. $H_{k+1}^{(r)}$) $r$-uniform hypergraphs for sufficiently large $n$.
Our results offer $p$-spectral analogues of the results by Mubayi-Pikhurko [J. Combin. Theory Ser. B, 97 (2007) 669--678] and Pikhurko [J. Combin. Theory Ser. B, 103 (2013) 220--225], and connect both hypergraph Tur\'an theorem and hypergraph spectral Tur\'an theorem in a unified form via the $p$-spectral radius.
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 $[k+1]$. The main contributions of this paper are twofold. First, we establish a general criterion that can obtain spectral stability results easily. Utilizing this criterion, we then derive spectral stability results for $H_{k+1}^{(r)}$ and $F_{k+1}^{(r)}$, respectively. Second, we determine the unique extremal hypergraph having the maximum $p$-spectral radius among all $n$-vertex $F_{k+1}^{(r)}$-free (resp. $H_{k+1}^{(r)}$) $r$-uniform hypergraphs for sufficiently large $n$.
Our results offer $p$-spectral analogues of the results by Mubayi-Pikhurko [J. Combin. Theory Ser. B, 97 (2007) 669--678] and Pikhurko [J. Combin. Theory Ser. B, 103 (2013) 220--225], and connect both hypergraph Tur\'an theorem and hypergraph spectral Tur\'an theorem in a unified form via the $p$-spectral radius.