For a simple graph $G$ with adjacency matrix $A(G)$, let $π(G,x):=\mathrm{per}(xI-A(G))$ be its permanental polynomial with roots $μ_1,\ldots,μ_n \in \mathbb{C}$, and define the permanental energy $E_{\mathrm{per}}(G):=\sum_{i=1}^n |μ_i|$. We prove a sharp universal lower bound: for every $m$-edge graph $G$, $E_{\mathrm{per}}(G) \ge 2\sqrt{m}$, with equality if and only if $G$ is a star together with isolated vertices. We also prove the general upper bound $E_{\mathrm{per}}(G) \le nρ(G)$, where $ρ(G)$ is the spectral radius, and we study $E_{\mathrm{per}}(G)$ on several graph families.

翻译：对于邻接矩阵为$A(G)$的简单图$G$，设$\pi(G,x):=\mathrm{per}(xI-A(G))$为其具有根$\mu_1,\ldots,\mu_n \in \mathbb{C}$的永久多项式，并定义永久能量$E_{\mathrm{per}}(G):=\sum_{i=1}^n |\mu_i|$。我们证明了一个尖锐的通用下界：对于每个具有$m$条边的图$G$，有$E_{\mathrm{per}}(G) \ge 2\sqrt{m}$，当且仅当$G$为星图加上孤立顶点时取等。我们还证明了通用上界$E_{\mathrm{per}}(G) \le n\rho(G)$，其中$\rho(G)$为谱半径，并在若干图族上研究了$E_{\mathrm{per}}(G)$。