Positive and negative 3-energies of graphs

For a simple graph $G$ with $n$ vertices, let $A_G$ denote the adjacency matrix of $G$, and let $\lambda_1(G) \geq \lambda_2(G) \geq \dots \geq \lambda_n(G)$ be its eigenvalues. For an integer $p \geq 2$, the positive $p$-energy and negative $p$-energy of $G$, denoted $\mathcal{E}^+_p(G)$ and $\mathcal{E}^-_p(G)$, are defined as follows: $\mathcal{E}^+_p(G) = \sum_{\lambda_i(G) > 0} |\lambda_i(G)|^p$ and $\mathcal{E}^-_p(G) = \sum_{\lambda_i(G) < 0} |\lambda_i(G)|^p,$ respectively. Tang, Liu, and Wang proposed a conjecture that, for any integer $p \geq 2$, every connected $n$-vertex graph $G$ satisfies $\mathcal{E}^+_p(G) \geq \mathcal{E}^+_p(P_n)$. Akbari, Kumar, Mohar, and Pragada conjectured that, for any $p \geq 2$, every connected $n$-vertex graph $G$ satisfies $\mathcal{E}^-_p(G) \geq \mathcal{E}^-_p(K_n)$, and they proved this conjecture for $p \geq 4$. In this paper, we prove that every connected $n$-vertex graph, except for $K_1$, $K_2$, and $P_3$, satisfies $\mathcal{E}^+_3(G) \geq \frac{\sqrt{5}}{2}n$. Moreover, we show that for any integer $p \geq 3$, every connected $n$-vertex graph $G$ satisfies $\mathcal{E}^-_p(G) \geq \mathcal{E}^-_p(K_n)$, which improves upon the previously known result.