Refinement of a conjecture on positive square energy of graphs

Abstract

Let $G$ be a simple graph of order $n$ with eigenvalues ${\lambda }_1(G)\ge \cdots \ge {\lambda }_n(G)$. Define \[s^+(G)=\sum_{λ_i >0} λ_i^2(G), \quad s^-(G)=\sum_{λ_i<0} λ_i^2(G).\] 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.