Spectral extremal graphs for W5-free graphs with odd size
Abstract
For a fixed integer k≥ 2, let W2k+1=K1∨ C2k be an odd wheel graph. The fixed-size spectral extremal problem aims to determine spex(m,W2k+1):=max{ρ(G): e(G)=m, G is W2k+1−free}, where ρ(G) denotes the adjacency spectral radius. Based on this problem, Yu, Li, and Peng [12] proposed the following conjecture: When m−binomk2 is divisible by k and m is large, every W2k+1-free graph of size m satisfies ρ(G)2−(k−1)ρ(G)≤ m−binomk2 with equality precisely for Kk∨ qK1. For nonzero residue class, Yu, Zhang, and Zhang [13] proposed the following conjecture: Let r be a nonzero remainder when m−binomk2 is divided by k and m is large. Then Sk,m is the unique graph among W2k+1-free graphs of size m having maximum spectral radius, where Sk,m is obtained from Kk∨ qK1 by adding a vertex z and joining it to exactly r vertices of the Kk. Very recently, Fang, Zhai and Zhang [4] confirmed the Yu--Li--Peng conjecture for k≥ 2. Chen, Gao and Li [2] confirmed Yu-Zhang-Zhang conjecture for k≥ 3. When k=2, then W5=K1∨ C4. For large odd m, determining spex(m,W5) is still open. In this paper we address the odd-size problem. Our result disproved Yu-Zhang-Zhang conjecture for k= 2. In our proof, a universal defect bound shows that only O(1) edges can lie outside the dense core. Perron localization then reduces this to at most one edge. A discrete defect inequality forces the complete bipartite crossing and quantizes the two matching deficiencies. Exact quotient-polynomial comparisons eliminate the remaining cross-edge and odd--odd candidates.
Related
- Extremal spectral radius of nonregular graphs with a fixed odd maximum degree
- Extremal Graphs with Prescribed Parameters for the Spectral Radius of Weighted Adjacency Matrices with Property P∗
- Spectral Extremal Graphs without a Kk-Factor
- Strict spectral supersaturation for cliques: extremal graphs and sharp thresholds
- Extremal spectral result of outerplanar graphs without P3· l
BibTeX
@misc{gao2026spectral,
title = {{Spectral extremal graphs for $W_5$-free graphs with odd size}},
author = {Jing Gao and Xianya Geng and Shuchao Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.28556},
eprint = {2609.28556},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28556},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Spectral extremal graphs for W_5-free graphs with odd size
AU - Jing Gao
AU - Xianya Geng
AU - Shuchao Li
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2609.28556
UR - https://arxiv.org/abs/2609.28556
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 [12] proposed the following conjecture: When m−binomk2 is divisible by k and m is large, 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. For nonzero residue class, Yu, Zhang, and Zhang [13] proposed the following conjecture: Let r be a nonzero remainder when m−binomk2 is divided by k and m is large. 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. Very recently, Fang, Zhai and Zhang [4] confirmed the Yu--Li--Peng conjecture for k≥ 2. Chen, Gao and Li [2] confirmed Yu-Zhang-Zhang conjecture for k≥ 3. When k=2, then W_5=K_1∨ C_4. For large odd m, determining spex(m,W_5) is still open. In this paper we address the odd-size problem. Our result disproved Yu-Zhang-Zhang conjecture for k= 2. In our proof, a universal defect bound shows that only O(1) edges can lie outside the dense core. Perron localization then reduces this to at most one edge. A discrete defect inequality forces the complete bipartite crossing and quantizes the two matching deficiencies. Exact quotient-polynomial comparisons eliminate the remaining cross-edge and odd--odd candidates.
ER -