Spectral Generalized Turán Problems for Expansion Hypergraphs
Abstract
Let Q be an r-graph on q vertices and let α>1. The (α,Q)-spectral radius λα,Q(ℋ) of an r-graph ℋ is defined as the maximum value of the Q-Lagrangian polynomial over all vectors with unit α-norm. Given a family ℱ of r-graphs, the (α, Q)-spectral Turán number spexα(n, Q, ℱ) of ℱ is the maximum (α,Q) spectral radius λα, Q(ℋ) among all n-vertex ℱ-free r-graphs ℋ. This spectral parameter provides a spectral counterpart of the generalized Turán problem and contains many classical spectral extremal problems as special cases. In this paper, we first prove that for any α>1 and every $2$-covered r-graph Q with q≤ ℓ vertices, the complete balanced ℓ-partite r-graph Tℓ,nr uniquely maximizes λα,Q(ℋ) among all n-vertex ℓ-partite r-graphs. Based on this result and spectral stability theorems, we further determine the exact value of spexα(n,Q,HFr) for all sufficiently large n, where HFr stands for the expansion of an edge-critical graph F with chromatic number ℓ+1. As corollaries, we obtain several Turán-type results for graphs and r-graphs, including spectral bounds for the clique spectral radius of F-free graphs and the α-spectral radius of HFr-free r-graphs, where F is an edge-critical graph.
Related
- Spectral Turán problems for complete bipartite expansions in linear hypergraphs
- Turán problems with bounded matching number in k-uniform hypergraphs
- Generalized Turán problems for shorter even cycles
- NP-Hardness of Generalized-Turán Graph Modification Problems
- Spectral moments and characteristic polynomials of vertex expansion hypergraphs of graphs
BibTeX
@article{cheng2026spectral,
title = {{Spectral Generalized Turán Problems for Expansion Hypergraphs}},
author = {Zijuan Cheng and Yongchun Lu and Liying Kang},
journal = {The Electronic Journal of Combinatorics},
year = {2026},
month = oct,
doi = {10.37236/15808},
url = {https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p19},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Spectral Generalized Turán Problems for Expansion Hypergraphs
AU - Zijuan Cheng
AU - Yongchun Lu
AU - Liying Kang
PY - 2026
DA - 2026/10/23
JO - The Electronic Journal of Combinatorics
DO - 10.37236/15808
UR - https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p19
AB - Let Q be an r-graph on q vertices and let α>1. The (α,Q)-spectral radius λ_α,Q(ℋ) of an r-graph ℋ is defined as the maximum value of the Q-Lagrangian polynomial over all vectors with unit α-norm. Given a family ℱ of r-graphs, the (α, Q)-spectral Turán number spex_α(n, Q, ℱ) of ℱ is the maximum (α,Q) spectral radius λ_α, Q(ℋ) among all n-vertex ℱ-free r-graphs ℋ. This spectral parameter provides a spectral counterpart of the generalized Turán problem and contains many classical spectral extremal problems as special cases. In this paper, we first prove that for any α>1 and every $2$-covered r-graph Q with q≤ ℓ vertices, the complete balanced ℓ-partite r-graph T_ℓ,n^r uniquely maximizes λ_α,Q(ℋ) among all n-vertex ℓ-partite r-graphs. Based on this result and spectral stability theorems, we further determine the exact value of spex_α(n,Q,H_F^r) for all sufficiently large n, where H_F^r stands for the expansion of an edge-critical graph F with chromatic number ℓ+1. As corollaries, we obtain several Turán-type results for graphs and r-graphs, including spectral bounds for the clique spectral radius of F-free graphs and the α-spectral radius of H_F^r-free r-graphs, where F is an edge-critical graph.
SN - 1077-8926
ER -