Permanental Energy of Graphs

For a simple graph $G$ with adjacency matrix $A(G)$, let $\pi (G,x):=\mathrm{p}\mathrm{e}\mathrm{r}(xI-A(G))$ be its permanental polynomial with roots ${\mu }_1,\dots ,{\mu }_n\in \mathbb{C}$, and define the permanental energy $E_{\mathrm{p}\mathrm{e}\mathrm{r}}(G):={\sum }_{i=1}^n\mathrm{\mid }{\mu }_i\mathrm{\mid }$. We prove a sharp universal lower bound: for every $m$-edge graph $G$, $E_{\mathrm{p}\mathrm{e}\mathrm{r}}(G)\ge 2\sqrt{m}$, with equality if and only if $G$ is a star together with isolated vertices. We also prove the general upper bound $E_{\mathrm{p}\mathrm{e}\mathrm{r}}(G)\le n\rho (G)$, where $\rho (G)$ is the spectral radius, and we study $E_{\mathrm{p}\mathrm{e}\mathrm{r}}(G)$ on several graph families.