Wang–Xu conjecture on spectral determination of complements of T-shape trees

For positive integers $\ell_1\leq \ell_2\leq \ell_3$ , let $T(\ell_1,\ell_2,\ell_3)$ be the tree with a unique degree-three vertex $v$ such that $T(\ell_1,\ell_2,\ell_3)-v=P_{\ell_1}\cup P_{\ell_2}\cup P_{\ell_3}$ . A graph is determined by its adjacency spectrum if every graph cospectral with it is isomorphic to it. The conjecture asserts that $\overline{T(\ell_1,\ell_2,\ell_3)}$ is determined by its adjacency spectrum if and only if $(\ell_1,\ell_2,\ell_3)\notin\{(\ell,\ell,2\ell-2):\ell\geq 2\}$ .

Progress summary

Solved

An August 2026 preprint claims to settle the conjecture, identifying exactly one infinite family of exceptions and its cospectral mates, but the proof has not been independently verified.

The Wang–Xu conjecture, posed by Wang and Xu in 2006, concerns when the complement of a T-shape tree is uniquely determined by its adjacency spectrum. The claimed classification excludes exactly the family

(\ell,\ell,2\ell-2),\quad \ell\geq 2.

August 2026 claimed classification

Feifan Gong, Kehua Wang, and Wei Wang claim that $\overline{T(\ell_{1},\ell_{2},\ell_{3})}$ is spectrally determined precisely when $(\ell_{1},\ell_{2},\ell_{3})\notin\{(\ell,\ell,2\ell-2):\ell\geq2\}$ . For the exceptional family, they claim the only matching auxiliary-polynomial graph besides $T(\ell,\ell,2\ell-2)$ is $R_{\ell}=P_{\ell-1}\cup L(2\ell+2,\ell-2)$ ; hence the complements have nonisomorphic cospectral mates, with $R_{2}=K_{1}\cup C_{6}$ . No verification, objection, withdrawal, or retraction was found.

Current status (as of August 2026): The conjecture is claimed settled by an unrefereed preprint, including a complete exceptional-family classification, but independent confirmation is absent.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

On the Spectral Determination of Complements of T-shape Trees — arXiv — Gong, Feifan, Wang, Kehua, Wang, Wei

algebraic-graph-theory spectral-graph-theory

2 days ago