Spectral Turán problems for complete bipartite expansions in linear hypergraphs
Abstract
An r-uniform hypergraph is linear if any two edges share at most one vertex. For a family ℱ of r-graphs, let exlinr(n,ℱ) and spexlinr(n,ℱ) denote, respectively, the maximum number of edges and the maximum spectral radius of an n-vertex ℱ-free linear r-graph. In this paper, we show that for t≥ s≥3, spexlinr(n,Ks,t+)≤ ((t−s+1)1/s)/(r−1)n1−1/s+o(n1−1/s), where Ks,t+ is the r-graph obtained from the complete bipartite graph Ks,t by enlarging each edge with r−2 new vertices. This answers a problem of She, Fan and Kang in a stronger form. Moreover, our result implies that exlinr(n,Ks,t+) ≤ ((t−s+1)1/s)/(r(r−1))n2−1/s+o(n2−1/s), which improves the leading coefficient in the corresponding bound of Gao and Chang. We also give a construction which implies that spexlinr(n,Ks,t+)=Θ(n1−1/s) and exlinr(n,Ks,t+)=Θ(n2−1/s) for s≥ 4r−6 and t>(s−1)!.
Related
- The maximum spectral radius of uniform hypergraphs whose shadow excludes a complete or complete bipartite minor
- Spectral Generalized Turán Problems for Expansion Hypergraphs
- Negative edge correlation in complete bipartite graphs
- The domination number of the 2-token graph of the complete bipartite graph
- Maximal Linear Arrangement is NP complete on Bipartite Graphs
BibTeX
@misc{fang2026spectral,
title = {{Spectral Turán problems for complete bipartite expansions in linear hypergraphs}},
author = {Lusheng Fang and Guorong Gao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34065},
eprint = {2610.34065},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34065},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spectral Turán problems for complete bipartite expansions in linear hypergraphs
AU - Lusheng Fang
AU - Guorong Gao
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34065
UR - https://arxiv.org/abs/2610.34065
AB - An r-uniform hypergraph is linear if any two edges share at most one vertex. For a family ℱ of r-graphs, let ex^lin_r(n,ℱ) and spex^lin_r(n,ℱ) denote, respectively, the maximum number of edges and the maximum spectral radius of an n-vertex ℱ-free linear r-graph. In this paper, we show that for t≥ s≥3, spex^lin_r(n,K_s,t^+)≤ ((t−s+1)^1/s)/(r−1)n^1−1/s+o(n^1−1/s), where K_s,t^+ is the r-graph obtained from the complete bipartite graph K_s,t by enlarging each edge with r−2 new vertices. This answers a problem of She, Fan and Kang in a stronger form. Moreover, our result implies that ex^lin_r(n,K_s,t^+) ≤ ((t−s+1)^1/s)/(r(r−1))n^2−1/s+o(n^2−1/s), which improves the leading coefficient in the corresponding bound of Gao and Chang. We also give a construction which implies that spex^lin_r(n,K_s,t^+)=Θ(n^1−1/s) and ex^lin_r(n,K_s,t^+)=Θ(n^2−1/s) for s≥ 4r−6 and t>(s−1)!.
ER -