The edge spectral extremal problem for odd wheels in nonzero residue classes

Abstract

For a fixed integer k≥ 2, let W_2k+1=K₁∨ C_2k be an odd wheel graph. The fixed-size spectral extremal problem aims to determine spex(m,W_2k+1):=max{ρ(G): e(G)=m, G is W_2k+1−free}, where ρ(G) denotes the adjacency spectral radius. Based on this problem, Yu, Li, and Peng [18] proposed the following conjecture: For large m, every W_2k+1-free graph of size m satisfies ρ(G)²−(k−1)ρ(G)≤ m−binomk2 with equality precisely for K_k∨ qK₁ and m−binomk2=kq. Very recently, Fang, Zhai and Zhang [7] confirmed the Yu--Li--Peng conjecture. When m is large, k≥ 2 and m−binomk2 is not divisible by k, the exact solution for the above problem is still open. Regarding this problem, Yu, Zhang, and Zhang [19] proposed the following conjecture: Let r be a nonzero remainder when m−binomk2 is divided by k. Then S_k,m is the unique graph among W_2k+1-free graphs of size m having maximum spectral radius, where S_k,m is obtained from K_k∨ qK₁ by adding a vertex z and joining it to exactly r vertices of the K_k. In this paper we address this problem in each nonzero residue class. Our result completely settles the Yu-Zhang-Zhang conjecture for k≥ 3.

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BibTeX RIS

BibTeX

@misc{chen2026edge,
  title = {{The edge spectral extremal problem for odd wheels in nonzero residue classes}},
  author = {Honghao Chen and Jing Gao and Shuchao Li},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13666},
  eprint = {2609.13666},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13666},
  note = {CC BY-NC-SA 4.0},
}

RIS

TY  - UNPB
TI  - The edge spectral extremal problem for odd wheels in nonzero residue classes
AU  - Honghao Chen
AU  - Jing Gao
AU  - Shuchao Li
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.13666
UR  - https://arxiv.org/abs/2609.13666
AB  - For a fixed integer k≥ 2, let W_2k+1=K_1∨ C_2k be an odd wheel graph. The fixed-size spectral extremal problem aims to determine spex(m,W_2k+1):=max{ρ(G): e(G)=m, G is W_2k+1−free}, where ρ(G) denotes the adjacency spectral radius. Based on this problem, Yu, Li, and Peng [18] proposed the following conjecture: For large m, every W_2k+1-free graph of size m satisfies ρ(G)^2−(k−1)ρ(G)≤ m−binomk2 with equality precisely for K_k∨ qK_1 and m−binomk2=kq. Very recently, Fang, Zhai and Zhang [7] confirmed the Yu--Li--Peng conjecture. When m is large, k≥ 2 and m−binomk2 is not divisible by k, the exact solution for the above problem is still open. Regarding this problem, Yu, Zhang, and Zhang [19] proposed the following conjecture: Let r be a nonzero remainder when m−binomk2 is divided by k. Then S_k,m is the unique graph among W_2k+1-free graphs of size m having maximum spectral radius, where S_k,m is obtained from K_k∨ qK_1 by adding a vertex z and joining it to exactly r vertices of the K_k. In this paper we address this problem in each nonzero residue class. Our result completely settles the Yu-Zhang-Zhang conjecture for k≥ 3.
ER  -

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