PreprintCC0arXiv
Spectral extremal hypergraphs without long Berge cycles
Abstract
Let r≥ 3 and k≥ 2r+1 be fixed integers. We determine, for all sufficiently large n, the maximum adjacency-tensor spectral radius of an n-vertex r-uniform hypergraph containing no Berge cycle of length at least k. Write s=⌊(k−1)/2⌋. If k=2s+1 is odd, the unique extremal hypergraph consists of all r-sets containing at most one vertex outside a fixed s-set. If k=2s+2 is even, one additionally includes all r-sets containing a fixed pair outside the s-set together with r−2 vertices inside it. Consequently, the maximum spectral radius is given as spexr(n,k) = [ binomsr−1 ((r−1)/s)(r−1)/r +o(1) ]n(r−1)/r.
Related
BibTeX
@misc{feng2026spectral,
title = {{Spectral extremal hypergraphs without long Berge cycles}},
author = {Lihua Feng and Lu Lu and Tingzeng Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25030},
eprint = {2609.25030},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25030},
note = {CC0},
}
RIS
TY - UNPB
TI - Spectral extremal hypergraphs without long Berge cycles
AU - Lihua Feng
AU - Lu Lu
AU - Tingzeng Wu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.25030
UR - https://arxiv.org/abs/2609.25030
AB - Let r≥ 3 and k≥ 2r+1 be fixed integers. We determine, for all sufficiently large n, the maximum adjacency-tensor spectral radius of an n-vertex r-uniform hypergraph containing no Berge cycle of length at least k. Write s=⌊(k−1)/2⌋. If k=2s+1 is odd, the unique extremal hypergraph consists of all r-sets containing at most one vertex outside a fixed s-set. If k=2s+2 is even, one additionally includes all r-sets containing a fixed pair outside the s-set together with r−2 vertices inside it. Consequently, the maximum spectral radius is given as spex_r(n,k) = [ binomsr−1 ((r−1)/s)^(r−1)/r +o(1) ]n^(r−1)/r.
ER -