For every connected graph G on n vertices, let lambda_1,…,lambda_n be the eigenvalues of its adjacency matrix, and define s^+(G)=∑_lambda_i>0lambda_i^2 and s^-(G)=∑_lambda_i<0lambda_i^2. Then mins^+(G),s^-(G)≥ n-1.

Elphick–Farber–Goldberg–Wocjan square-energy conjecture

The conjecture of Elphick, Farber, Goldberg, and Wocjan, posed in 2016, asserts that every connected graph $G$ on $n$ vertices satisfies $\min\{s^+(G),s^-(G)\}\ge n-1$ .

Known results

The conjecture was proved for regular and bipartite graphs in the original 2016 work.
By 2024, known cases included barbell graphs and all connected graphs with at most $10$ vertices; the best general lower bound reported there was $\sqrt{n}$ .
Earlier work also covered complete multipartite graphs and odd cycles.

July–August 2026 claimed resolution

A July preprint claims a full proof using doubly nonnegative matrix relaxations and reports Lean $4$ formalization. It also says ChatGPT assisted with exploratory code and prose, not that it produced the mathematics. An August preprint claims the equality cases: $s^+(G)=n-1$ exactly for trees, and $s^-(G)=n-1$ exactly for trees and complete graphs. Both manuscripts are unrefereed, and no independent verification was found.

Current status (as of August 2026): The conjectured bound and equality classification are claimed in unrefereed preprints, with author-reported Lean formalization, but remain unverified.

Elphick–Farber–Goldberg–Wocjan square-energy conjecture

Progress summary

Primary source

Additional references

Solutions 0