Conjecture 2 of Chang–Li–Zhang on exceptional graphs for edge-disjoint spanning trees

Let $k\ge 2$ , $k\le \delta\le 2k-1$ , and set $h=\delta-k$ . For all sufficiently large $n$ , among the graphs $G\in\mathcal{L}_{\mathcal{H}}^{2}(n,k,\delta)$ , the graph maximizing the adjacency spectral radius $\rho(G)$ is the configuration proposed by Chang, Li, and Zhang in Conjecture 2: in the core--placement representation, the missing-edge graph of the bounded core satisfies $M\cong hK_{2}\cup 2K_{1}$ , with the exceptional edges placed according to their proposed matching configuration. The conjecture is refuted for $h\ge 2$ : Cao, Xuanyu, and Wang report that the unique maximizer instead has $M\cong K_{1,h}\cup(h+1)K_{1}$ , with the exceptional edges nested on the large-clique side.

References

Primary source

arXiv

Additional references

Maximum and Minimum Spectral Radii in an Exceptional Family for Edge-Disjoint Spanning Trees — arXiv — Xuanyu Cao, Chunxiang Wang

extremal-graph-theory spectral-graph-theory

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A September 2026 preprint reports that the conjectured graph is wrong and gives a replacement and sharp results, but independent verification is unavailable.

Conjecture 2 of Chang–Li–Zhang concerns the extremal structure of exceptional graphs related to edge-disjoint spanning trees. The latest work claims to correct the maximizing configuration and determine the remaining extremal data.

Known results

No classical partial results were identified in the retrieved sources.

September 2026 claimed resolution

Xuanyu Cao and Chunxiang Wang report that, for $h \ge 2$ , the proposed maximizer is replaced by a star-shaped missing-edge graph; they also determine the minimizer, all equality cases, and a sharp spectral threshold. The result is stated only asymptotically for sufficiently large $n$ with fixed admissible parameters, and has not been independently verified.

Current status (as of September 2026): The conjectured maximizing structure is claimed to be false and replaced, with the associated extremal problem claimed solved asymptotically; independent verification remains outstanding.

Sources

No solutions have been posted yet.