Spectral extremal problems on [a,b]-factors in binding graphs, and beyond

@misc{xu2026spectral,
  title = {{Spectral extremal problems on $[a,b]$-factors in binding graphs, and beyond}},
  author = {Ting Xu and Ruifang Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16712},
  eprint = {2610.16712},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16712},
}

TY  - UNPB
TI  - Spectral extremal problems on [a,b]-factors in binding graphs, and beyond
AU  - Ting Xu
AU  - Ruifang Liu
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16712
UR  - https://arxiv.org/abs/2610.16712
AB  - The binding number b(G) of a graph G is the minimum value of |N_G(X)|/|X| taken over all nonempty sets X⊆ V(G) such that N_G(X)≠ V(G). A graph G is 1-binding if b(G)≥ 1. Fan and Lin [Electron. J. Combin. 31 (2024) P1.30] characterized the unique spectral extremal graphs among 1-binding graphs with δ(G)≥ k that contain no k-factor for k=1, $2$. Subsequently, Tang and Zhang solved the problem for k≥ 3. Note that δ(G)≥ a is a necessary condition for the existence of an [a,b]-factor in a graph G. For 1≤ a<b, Fan et al. [Discrete Math. 349 (2026) 115103] determined the unique spectral extremal graph H_n^a,b among connected graphs with δ(G)≥ a that contain no [a,b]-factor. Note that b(H_n^a,b)<1. This motivates the corresponding spectral extremal problem for 1-binding graphs. Note that every 1-binding graph with δ(G)≥ 1 contains a [1,b]-factor for b>1. For 2≤ a<b, we in this paper determine the unique extremal graph ℬ_n^a,b with the maximum spectral radius among 1-binding graphs of order n with δ(G)≥ a that contain no [a,b]-factor. The toughness of a non-complete graph G is t(G)=min{|X|/(c(G−X))| X⊆ V(G) and c(G−X)≥ 2}. A graph G is 1-tough if c(G−X)≤ |X| for every X⊆ V(G) with c(G−X)≥ 2. Every 1-tough graph of order n≥2 is 1-binding. Note that t(ℬ_n^a,b)<1. This leads us to investigate the corresponding spectral extremal problem for 1-tough graphs. Note that every 1-tough graph with δ(G)≥ 1 contains a [1,b]-factor for b>1. For 2≤ a<b, we in this paper characterize the unique extremal graph with the maximum spectral radius among 1-tough graphs of order n with δ(G)≥ a containing no [a,b]-factor.
ER  -