A Linear Lower Bound for the Square Energy of Graphs

Abstract

Let $G$ be a graph of order $n$ with eigenvalues ${\lambda }_1\ge \cdots \ge {\lambda }_n$. Let \[s^+(G)=\sum_{λ_i>0} λ_i^2, \qquad s^-(G)=\sum_{λ_i<0} λ_i^2.\] 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 \[s^+(G)\geq\sum_{i=1}^{k} s^+(H_i) \quad \text{ and } \quad s^-(G)\geq\sum_{i=1}^{k} s^-(H_i),\] which implies that $s(G)\ge \frac{3n}{4}$ for every connected graph $G$ of order $n\ge 4$.

arXiv comment5 pages, 1 figure