Brualdi–Hoffman–Turán problem on odd wheels

Let $G$ be a graph with $m=|E(G)|$ edges, let $\rho(G)$ denote its adjacency spectral radius, and let $W_{2k+1}=K_1\vee C_{2k}$ . Determine the sharp upper bound for $\rho(G)$ , including all equality cases, among $W_{2k+1}$ -free graphs when $m$ is sufficiently large. The cited preprint claims the following resolution: for sufficiently large $m$ , every $W_5$ -free graph satisfies $\rho(G)^2-\rho(G)\le m$ , with equality precisely when $G$ is obtained from $K_{n,n}$ by adding a perfect matching within each part, where $n$ is even and $m=n^2+n$ ; and, for every fixed $k\ge3$ , every $W_{2k+1}$ -free graph satisfies $\rho(G)^2-(k-1)\rho(G)\le m-\binom{k}{2}$ , with equality precisely when $G=K_k\vee qK_1$ and $m=\binom{k}{2}+kq$ .

Progress summary

Solved

A new preprint claims the odd-wheel conjecture is settled asymptotically, but no independent verification has appeared and some exact cases remain unclear.

The problem asks for sharp spectral bounds and equality cases for graphs excluding odd wheels $W_{2k+1}$ , extending a conjecture of Yu, Li, and Peng. The conjecture is formulated for sufficiently large edge count and includes residue-class-dependent extremal graphs.

Known results

Lu, Li, and Peng (2025) established the conjectured zero-residue-class bound and equality case for fixed $k\geq 2$ .
Lu, Li, and Peng (2025) also reported sharp bounds and equality cases in the other residue classes, using threshold graphs $S^{1,l}_{\frac{m-l}{k}+\frac{k+3}{2},k}$ .
Li, Liu, and Zhai (2025) proved a related structural theorem for graphs with large spectral radius that must contain a wheel.

August 2026 claimed settlement

Fang, Zhai, and Zhang claim that, for sufficiently large order, every $W_5$ -free graph satisfies $\rho(G)^2-\rho(G)\leq m$ , with the stated equality classification, and that for fixed $k\geq 3$ , every $W_{2k+1}$ -free graph satisfies $\rho(G)^2-(k-1)\rho(G)\leq m-\binom{k}{2}$ . The preprint calls this a complete resolution, but says nonzero residue classes have only natural candidate extremal graphs.

Current status (as of August 2026): A preprint claims the sufficiently-large-order inequalities and equality cases are proved, but the claim is unrefereed and independent verification is absent; the exact status of nonzero residue classes remains unclear.

Sources

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Dense-core approach to the Brualdi--Hoffman--Turán problem on odd wheels — arXiv — Fang, Longfei, Zhai, Mingqing, Zhang, Yuhan

extremal-graph-theory spectral-graph-theory

2 days ago