The maximum spectral radius of uniform hypergraphs whose shadow excludes a complete or complete bipartite minor
Abstract
For a k-uniform hypergraph ℋ, the shadow of ℋ is the graph whose edges are the pairs covered by a hyperedge. In this paper, for all sufficiently large n, we determine the n-vertex k-uniform hypergraphs of maximum adjacency-tensor spectral radius whose shadow has no Kt minor, for every t≥ k+1, and those whose shadow has no Ks,t minor, for every 2≤ s≤ t with s+t≥ k+1 and every residue of n−s+1 modulo t; outside these ranges the problems are trivial. This extends to uniform hypergraphs the theorem of Tait on graphs with no Kr or Ks,t minor, whose remaining residues were settled by Zhai and Lin. In each case the extremal hypergraph is unique, and it is the k-clique hypergraph of the join of a clique with a graph that we call the light part. For Ks,t the answer depends on j=k−s+1. When j≤1, the maximum has order n(k−1)/k, and the light part is the one found by Zhai and Lin for the adjacency matrix, including its exceptional components. When j≥2, a regime that does not occur for graphs, the maximum has order n(s−1)/k and t enters its leading constant. The light part then consists of copies of Kt and one smaller clique, with a single exception: for (k,s,t)=(9,8,8) and n−s+1≡2pmod 8, the complement of the Petersen graph appears. When the smaller clique has between $1$ and j−1 vertices, the extremal graph is not unique. In particular, for t=8, 4≤ s≤7, k=s+1 and n−s+1≡2pmod 8, the clique hypergraph of the extremal graph of Zhai and Lin is not extremal. For j≥2 the light part is determined by a weighted clique inequality, which for j≥3 follows from a weighted form of the closed-neighborhood counting of Chao and Dong.
Related
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BibTeX
@misc{liu2026maximum,
title = {{The maximum spectral radius of uniform hypergraphs whose shadow excludes a complete or complete bipartite minor}},
author = {Pei Liu and Suil O},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22370},
eprint = {2609.22370},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22370},
}
RIS
TY - UNPB
TI - The maximum spectral radius of uniform hypergraphs whose shadow excludes a complete or complete bipartite minor
AU - Pei Liu
AU - Suil O
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.22370
UR - https://arxiv.org/abs/2609.22370
AB - For a k-uniform hypergraph ℋ, the shadow of ℋ is the graph whose edges are the pairs covered by a hyperedge. In this paper, for all sufficiently large n, we determine the n-vertex k-uniform hypergraphs of maximum adjacency-tensor spectral radius whose shadow has no K_t minor, for every t≥ k+1, and those whose shadow has no K_s,t minor, for every 2≤ s≤ t with s+t≥ k+1 and every residue of n−s+1 modulo t; outside these ranges the problems are trivial. This extends to uniform hypergraphs the theorem of Tait on graphs with no K_r or K_s,t minor, whose remaining residues were settled by Zhai and Lin. In each case the extremal hypergraph is unique, and it is the k-clique hypergraph of the join of a clique with a graph that we call the light part. For K_s,t the answer depends on j=k−s+1. When j≤1, the maximum has order n^(k−1)/k, and the light part is the one found by Zhai and Lin for the adjacency matrix, including its exceptional components. When j≥2, a regime that does not occur for graphs, the maximum has order n^(s−1)/k and t enters its leading constant. The light part then consists of copies of K_t and one smaller clique, with a single exception: for (k,s,t)=(9,8,8) and n−s+1≡2pmod 8, the complement of the Petersen graph appears. When the smaller clique has between $1$ and j−1 vertices, the extremal graph is not unique. In particular, for t=8, 4≤ s≤7, k=s+1 and n−s+1≡2pmod 8, the clique hypergraph of the extremal graph of Zhai and Lin is not extremal. For j≥2 the light part is determined by a weighted clique inequality, which for j≥3 follows from a weighted form of the closed-neighborhood counting of Chao and Dong.
ER -