Extremal graphs for a conjecture on the square energy of graphs

For a graph $G$, let $s^{+}(G)$ and $s^{-}(G)$ denote the sums of the squares of its positive and negative adjacency eigenvalues. We determine all equality cases in the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ on $n$ vertices satisfies $$\min \{s^{+}(G),s^{-}(G)\}\ge n-1.$$ Namely, equality for $s^{+}$ holds exactly for trees, whereas equality for $s^{-}$ holds exactly for trees and complete graphs. The proof combines the $P_3$-removal lemma in the no-cut-vertex case with a detailed equality analysis of the underlying doubly nonnegative matrix inequality. Every block is forced to be complete, and a minimal-counterexample argument gives an exact rank-one decomposition of the folded matrix $M^c$. The resulting non-edge vanishings, together with $AX=XA$, rule out an interface between a bridge and a nontrivial block.