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 ${\lambda }_1(G)\ge {\lambda }_2(G)\ge \cdots \ge {\lambda }_n(G)$ be its eigenvalues. For an integer $p\ge 2$, the positive $p$-energy and negative $p$-energy of $G$, denoted ${\mathcal{E}}_p^{+}(G)$ and ${\mathcal{E}}_p^{-}(G)$, are defined as follows: ${\mathcal{E}}_p^{+}(G)={\sum }_{{\lambda }_i(G)>0}\mathrm{\mid }{\lambda }_i(G){\mathrm{\mid }}^p$ and ${\mathcal{E}}_p^{-}(G)={\sum }_{{\lambda }_i(G)<0}\mathrm{\mid }{\lambda }_i(G){\mathrm{\mid }}^p,$ respectively. Tang, Liu, and Wang proposed a conjecture that, for any integer $p\ge 2$, every connected $n$-vertex graph $G$ satisfies ${\mathcal{E}}_p^{+}(G)\ge {\mathcal{E}}_p^{+}(P_n)$. Akbari, Kumar, Mohar, and Pragada conjectured that, for any $p\ge 2$, every connected $n$-vertex graph $G$ satisfies ${\mathcal{E}}_p^{-}(G)\ge {\mathcal{E}}_p^{-}(K_n)$, and they proved this conjecture for $p\ge 4$. In this paper, we prove that every connected $n$-vertex graph, except for $K_1$, $K_2$, and $P_3$, satisfies ${\mathcal{E}}_3^{+}(G)\ge \frac{\sqrt{5}}{2}n$. Moreover, we show that for any integer $p\ge 3$, every connected $n$-vertex graph $G$ satisfies ${\mathcal{E}}_p^{-}(G)\ge {\mathcal{E}}_p^{-}(K_n)$, which improves upon the previously known result.