PreprintCC BY 4.0arXiv
Unbalanced spectral Turán problem for color-critical graphs with prescribed large maximum degree
Abstract
Let F be a connected color-critical graph with χ(F)=r+1≥4, let Sn,Δ(r)=(n−Δ)K1∨ T(Δ,r−1). We determine the graph of maximum adjacency spectral radius among all n-vertex F-free graphs with prescribed maximum degree Δ. There is a constant sF∈[0,1) such that, for all sufficiently large n, ⌈((r−1)n)/r⌉≤ Δ≤ n−Θ(nsF) implies that every n-vertex F-free graph G with Δ(G)=Δ satisfies ρ(G)≤ ρ(Sn,Δ(r)), with equality if and only if G≅ Sn,Δ(r). This is the spectral counterpart of the edge theorem of [European J. Combin. 106 (2022), 103576.] and extends the clique result in [arXiv:2608.26634, 2026.]. This result also provides a benchmark for unbalanced spectral Turán problems arising from other extremal parameters.
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BibTeX
@misc{liu2026unbalanced,
title = {{Unbalanced spectral Turán problem for color-critical graphs with prescribed large maximum degree}},
author = {Chang Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01114},
eprint = {2609.01114},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01114},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Unbalanced spectral Turán problem for color-critical graphs with prescribed large maximum degree
AU - Chang Liu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01114
UR - https://arxiv.org/abs/2609.01114
AB - Let F be a connected color-critical graph with χ(F)=r+1≥4, let S_n,Δ^(r)=(n−Δ)K_1∨ T(Δ,r−1). We determine the graph of maximum adjacency spectral radius among all n-vertex F-free graphs with prescribed maximum degree Δ. There is a constant s_F∈[0,1) such that, for all sufficiently large n, ⌈((r−1)n)/r⌉≤ Δ≤ n−Θ(n^s_F) implies that every n-vertex F-free graph G with Δ(G)=Δ satisfies ρ(G)≤ ρ(S_n,Δ^(r)), with equality if and only if G≅ S_n,Δ^(r). This is the spectral counterpart of the edge theorem of [European J. Combin. 106 (2022), 103576.] and extends the clique result in [arXiv:2608.26634, 2026.]. This result also provides a benchmark for unbalanced spectral Turán problems arising from other extremal parameters.
ER -