Square-energy conjecture

Elphick, Farber, Goldberg, and Wocjan formulated the conjecture in 2016: for every connected graph, both square-energy quantities should satisfy $s^{+}(G)\ge n-1$ and $s^{-}(G)\ge n-1$.

Known results

• The conjecture was proved for several classes, including bipartite, regular, complete multipartite, hyper-energetic, and barbell graphs (Elphick, Farber, Goldberg, and Wocjan, 2016).
• A general bound $s(G)>3n/4$ was obtained for connected graphs of order $n>4$ (2025), improving the previously known order-$\sqrt n$ bound.
• The same $3n/4$-type lower bound was reported for connected graphs with $n\ge 4$ in 2024.

July–August 2026 claimed proofs

A July preprint claims the full bound and reports formal verification of a key theorem in Lean 4. On August 20, Hu and Han claimed the stronger signed-graph inequality $s^{+}(\Sigma)\le 2m-n+1$; applying it to the negated signing yields $s^{+}(\Sigma)\ge n-1$ and the ordinary conjecture. A related August preprint claims the equality cases. These are unreviewed preprint claims.

Current status (as of August 2026): The conjectured bounds are claimed in multiple unreviewed preprints, with the latest deriving them from a stronger signed-graph theorem; independent confirmation remains pending.