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The 𝒜α-Spectral Turán Problem for s-Clique Tensors of Kr+1-Free Graphs

  • Xiaoqi Liu
  • Haiying Shan
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Abstract

For fixed r≥3 and 0<α<1, we study the maximum α-s-clique spectral radius among n-vertex Kr+1-free graphs. Here 𝒜α,s(G)=α𝒟s(G)+(1−α)𝒜s(G), where 𝒜s(G) and 𝒟s(G) are the adjacency and degree tensors of the s-clique hypergraph of G. For s=r, put cr=((r−1)/r)r−1. We prove that, for all sufficiently large n, the unique extremal graph is the Turán graph Tr(n) if α≤ cr, and K1∨ Tr−1(n−1) if α>cr. For general 3≤ s≤ r and n≥ r, every extremal graph is complete r-partite. The asymptotic maximum α-r-clique spectral radius among n-vertex Kr+1-free graphs is max{r1−r,α(r−1)1−r}nr−1+o(nr−1). In every asymptotically extremal sequence of complete r-partite graphs, all parts have size n/r+o(n) when α<cr; when α>cr, one part has size o(n) and the remaining parts have size n/(r−1)+o(n). At α=cr, both part-size patterns are asymptotically extremal.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026𝒜αspectral,
  title = {{The \(\mathcal A_α\)-Spectral Turán Problem for \(s\)-Clique Tensors of \(K_r+1\)-Free Graphs}},
  author = {Xiaoqi Liu and Haiying Shan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34196},
  eprint = {2610.34196},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34196},
}
RIS
TY  - UNPB
TI  - The 𝒜_α-Spectral Turán Problem for s-Clique Tensors of K_r+1-Free Graphs
AU  - Xiaoqi Liu
AU  - Haiying Shan
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.34196
UR  - https://arxiv.org/abs/2610.34196
AB  - For fixed r≥3 and 0<α<1, we study the maximum α-s-clique spectral radius among n-vertex K_r+1-free graphs. Here 𝒜_α,s(G)=α𝒟_s(G)+(1−α)𝒜_s(G), where 𝒜_s(G) and 𝒟_s(G) are the adjacency and degree tensors of the s-clique hypergraph of G. For s=r, put c_r=((r−1)/r)^r−1. We prove that, for all sufficiently large n, the unique extremal graph is the Turán graph T_r(n) if α≤ c_r, and K_1∨ T_r−1(n−1) if α>c_r. For general 3≤ s≤ r and n≥ r, every extremal graph is complete r-partite. The asymptotic maximum α-r-clique spectral radius among n-vertex K_r+1-free graphs is max{r^1−r,α(r−1)^1−r}n^r−1+o(n^r−1). In every asymptotically extremal sequence of complete r-partite graphs, all parts have size n/r+o(n) when α<c_r; when α>c_r, one part has size o(n) and the remaining parts have size n/(r−1)+o(n). At α=c_r, both part-size patterns are asymptotically extremal.
ER  -