Let G be a graph of order n with eigenvalues ₁ ₙ. Let ^+ (G) =㶁>₀ ᵢ², s^- (G) =㶁<₀ ᵢ². \ The smaller value, s (G) =\s^+ (G), s^- (G) \ is called the square energy of G. In 2016, Elphick, Farber, Goldberg, and Wocjan conjectured that for every connected graph G of order n, s (G) n-1. No linear bound for s (G) in terms of n is known. Let H₁, , Hₖ be disjoint induced subgraphs of G. In this note, we prove that ^+ (G) ₈=₁^k s^+ (Hᵢ) and s^- (G) ₈=₁^k s^- (Hᵢ), \ and then use this result to prove that s (G) 3n4 for every connected graph G of order n 4.
Akbari et al. (Fri,) studied this question.