Positive and Negative Square Energies of $2$-Connected Graphs

Abstract

Let $G$ be a graph of order $n$, and let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of the positive and negative adjacency eigenvalues of $G$, respectively. Recently, Liu, Tang, and Zhang proved the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ of order $n$ satisfies $ \min\{s^+(G), s^-(G)\} \ge n-1. $ For positive square energy, we strengthen this result by showing that every $2$-connected graph $G$ of order $n$ which is not a cycle satisfies $s^+(G)\ge n$. The formally analogous assertion for $s^-$ is false: the complete graph $K_n$ satisfies $s^-(K_n)=n-1$. We prove a natural counterpart in the triangle-free class: every triangle-free $2$-connected noncycle $G$ satisfies $ \min\{s^+(G),s^-(G)\}>n. $ More generally, it is enough that some maximum-degree vertex of $G$ belongs to no triangle. Together with the exact square energies of cycles, this characterizes the triangle-free $2$-connected graphs for which $s^-(G)\ge n$; the only exceptions are the cycles $C_{4k+3}$ with $k\geq1$.