From the Square-Energy Conjecture to Signed Graphs: Sharp Bounds for Positive Square Energy

Let $\mathrm{\Sigma }=(G,\sigma )$ be a connected signed graph of order $n$ and size $m$, and let $s^{+}(\mathrm{\Sigma })$ and $s^{-}(\mathrm{\Sigma })$ denote the sums of the squares of its positive and negative adjacency eigenvalues, respectively. The square-energy conjecture of Elphick, Farber, Goldberg, and Wocjan states that every connected graph $G$ of order $n$ satisfies $$\min \{s^{+}(G),s^{-}(G)\}\ge n-1.$$ Liu and Ning~\cite{LiuNing2023} published a wide-ranging paper entitled ``Unsolved Problems in spectral graph theory", and this conjectures were placed first in their list of such problems. We prove that every signature $\sigma$ of a connected graph $G$ satisfies the sharp bound $$s^{+}(\mathrm{\Sigma })\le 2m-n+1.$$ For the all-positive signing this gives $s^{+}(G)\le 2m-n+1$, whereas for the all-negative signing it gives $s^{-}(G)\le 2m-n+1$. Since $s^{+}(G)+s^{-}(G)=2m$, these two special cases imply the square-energy conjecture; the present theorem is stronger in scope because the same bound holds for every signing of $G$. Applying the theorem to the negation $-\mathrm{\Sigma }$ also yields $$s^{+}(\mathrm{\Sigma })\ge n-1.$$ Both bounds are sharp. The proof is based on a doubly nonnegative matrix inequality. We also shorten the proof of that inequality by replacing its final case distinction with a fixed convex combination.