A Linear Lower Bound for the Square Energy of Graphs

Let $G$ be a graph of order $n$ with eigenvalues ${\lambda }_1\ge \cdots \ge {\lambda }_n$. Let $$s^{+}(G)=\sum _{{\lambda }_i>0}{\lambda }_i^2,s^{-}(G)=\sum _{{\lambda }_i<0}{\lambda }_i^2\mathrm{.}$$ The smaller value, $s(G)=\min \{s^{+}(G),s^{-}(G)\}$ is called the \emph{square energy} of $G$. In 2016, Elphick, Farber, Goldberg and Wocjan conjectured that for every connected graph $G$ of order $n$, $s(G)\ge n-1.$ No linear bound for $s(G)$ in terms of $n$ is known. Let $H_1,\dots ,H_k$ be disjoint vertex-induced subgraphs of $G$. In this note, we prove that $$s^{+}(G)\ge \sum _{i=1}^k{s}^{+}(H_i)\text{ and }{s}^{-}(G)\ge \sum _{i=1}^k{s}^{-}(H_i),$$ which implies that $s(G)\ge \frac{3n}{4}$ for every connected graph $G$ of order $n\ge 4$.