Positive Square Energy of Graphs with Minimum Degree at Least Two

Abstract

Let $s^+(G)$ denote the sum of the squares of the positive adjacency eigenvalues of a graph $G$. The square-energy conjecture of Elphick, Farber, Goldberg, and Wocjan, proved by Liu, Tang, and Zhang, gives a lower bound of $n-1$ for any connected graph of order $n$. We strengthen this bound to $s^+(G)\ge n$ for every connected graph $G$ of order $n$ with minimum degree at least two, unless $G$ is a cycle.