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 $ \mi...
Let $\Sigma=(G,\sigma)$ be a connected signed graph of order $n$ and size $m$, and let $s^{+}(\Sigma)$ and $s^{-}(\Sigma)$ denote the sums of the squares of its positive and negative adjacency eigenvalues, respectively. The square-energy conjecture of Elphick, Farber, Goldberg, and Wocjan states that every connected gr...
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 c...
For a graph $G$, let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of its positive and negative adjacency eigenvalues. We determine all equality cases in the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ on $n$ vertices satisfies \[ \min \{s^+(G),s^-(G)\}\ge n-1. \] Namely, e...
Fu-Tao Hu, Ya-Yang Liu, Yi Wang· 1 citation
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