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Hypergraph Extensions of Spectral Turán Theorem · arXivDesk

2408.03122Aug 6, 202434 pages

Hypergraph Extensions of Spectral Turán Theorem

Lele Liu, Zhenyu Ni, Jing Wang, Liying Kang

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 $k\geq r\geq 2$

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, 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 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.

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