Refinement of a conjecture on positive square energy of graphs

Let $G$ be a simple graph of order $n$ with eigenvalues ${\lambda }_1(G)\ge \cdots \ge {\lambda }_n(G)$. Define $$s^{+}(G)=\sum _{{\lambda }_i>0}{\lambda }_i^2(G),s^{-}(G)=\sum _{{\lambda }_i<0}{\lambda }_i^2(G)\mathrm{.}$$ It was conjectured by Elphick, Farber, Goldberg and Wocjan that for every connected graph $G$ of order $n$, $s^{+}(G)\ge n-1.$ We verify this conjecture for graphs with domination number at most 2. We then strengthen the conjecture as follows: if $G$ is a connected graph of order $n$ and size $m\ge n+1$, then $s^{+}(G)\ge n$. We prove this conjecture for claw-free graphs and graphs with diameter 2.