Liu’s extremal spectral-radius conjecture

For integers $n\ge 3$ and $2\le \Delta\le n-1$ , let $\lambda_1(n,\Delta)$ denote the maximum adjacency spectral radius among all connected nonregular graphs $G$ of order $n$ with maximum degree $\Delta(G)=\Delta$ . Then, for each fixed integer $\Delta\ge 3$ , conjecture that

\lim_{n\to\infty} n^2\bigl(\Delta-\lambda_1(n,\Delta)\bigr)= \begin{cases} (\Delta-1)\pi^2/4, & \text{if $\Delta$ is odd},\\ (\Delta-2)\pi^2/2, & \text{if $\Delta$ is even}. \end{cases}

References

Primary source

arXiv

Additional references

Extremal spectral radius of nonregular graphs with a fixed odd maximum degree — arXiv — Zejun Huang, Chenxi Yang

spectral-graph-theory

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed progress

A new paper claims the conjecture for all odd maximum degrees at least five, but even degrees beyond four remain open.

Liu’s extremal spectral-radius conjecture concerns the conjectured asymptotic extremal value. A September 2026 paper claims the result for every odd maximum degree $\Delta\ge 5$ , extending the previously known $\Delta=3$ case.

Known results

The $\Delta=3$ case was previously known; the retrieved source gives no attribution or year.

September 2026 odd-degree advance

Zejun Huang and Chenxi Yang’s paper Extremal spectral radius of nonregular graphs with a fixed odd maximum degree claims the conjectured $n^2$ -scaled asymptotic for every odd $\Delta\ge 5$ . This covers the remaining odd cases in that parity class, while even degrees beyond $\Delta=4$ are not addressed; the claim is unverified.

Current status (as of September 2026): The conjectured asymptotic is claimed for all odd $\Delta\ge 3$ but remains unverified, while even cases beyond $\Delta=4$ remain open.

Sources

No solutions have been posted yet.