Spectral Erdős--Gallai Theorems for the 𝒜α-Tensor of the s-Clique Hypergraph
Abstract
The Erdős--Gallai theorem determines the maximum number of edges in a graph with bounded matching number; its clique-counting extension replaces edges by s-cliques, and a spectral analogue in terms of the s-clique tensor has recently been established. We study the corresponding 𝒜α-tensor of the s-uniform clique hypergraph. For 0≤α≤1 and 3≤ s≤2t−1, we determine the maximum α-s-clique spectral radius among n-vertex graphs containing no matching of t edges: when 3≤ s≤ t and n is sufficiently large, the maximum is attained by the join of a clique of order t−1 and an independent set, and when t<s≤ 2t−1 and n≥2t−1, it is attained by a clique of order 2t−1 together with isolated vertices. At α=0, these statements recover the known result for the s-clique spectral radius; at α=1, they yield the corresponding statement for the maximum s-clique degree. For s=t≥3, we obtain the maximum for every n≥2t−1; at α=0, this removes the requirement that n be sufficiently large from the known result.
Related
- Vertex-colored Turán theorems with applications in extremal hypergraph problems
- Stronger Spectral Bound for Clique Number
- Smooth Tensor Multi-View Clustering on Well Logs via Hypergraph Topology and Concurrent Group Sparsity
- Clique and chromatic numbers of graph powers: spectral bounds and anticodes
- The 𝒜α-Spectral Turán Problem for s-Clique Tensors of Kr+1-Free Graphs
BibTeX
@misc{liu2026spectral,
title = {{Spectral Erdős--Gallai Theorems for the \(\mathcal A_α\)-Tensor of the \(s\)-Clique Hypergraph}},
author = {Xiaoqi Liu and Haiying Shan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08605},
eprint = {2610.08605},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08605},
}
RIS
TY - UNPB
TI - Spectral Erdős--Gallai Theorems for the 𝒜_α-Tensor of the s-Clique Hypergraph
AU - Xiaoqi Liu
AU - Haiying Shan
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08605
UR - https://arxiv.org/abs/2610.08605
AB - The Erdős--Gallai theorem determines the maximum number of edges in a graph with bounded matching number; its clique-counting extension replaces edges by s-cliques, and a spectral analogue in terms of the s-clique tensor has recently been established. We study the corresponding 𝒜_α-tensor of the s-uniform clique hypergraph. For 0≤α≤1 and 3≤ s≤2t−1, we determine the maximum α-s-clique spectral radius among n-vertex graphs containing no matching of t edges: when 3≤ s≤ t and n is sufficiently large, the maximum is attained by the join of a clique of order t−1 and an independent set, and when t<s≤ 2t−1 and n≥2t−1, it is attained by a clique of order 2t−1 together with isolated vertices. At α=0, these statements recover the known result for the s-clique spectral radius; at α=1, they yield the corresponding statement for the maximum s-clique degree. For s=t≥3, we obtain the maximum for every n≥2t−1; at α=0, this removes the requirement that n be sufficiently large from the known result.
ER -