Permanental Energy of Graphs

TL;DR

This paper introduces bounds on the permanental energy of graphs, establishing a sharp lower bound and an upper bound related to spectral radius, with analysis on various graph families.

Contribution

It provides the first sharp universal lower bound and a general upper bound for the permanental energy of graphs, expanding understanding of graph spectral properties.

Findings

Lower bound: $E_{per}(G) \, \ge \, 2\sqrt{m}$ with equality for stars and isolated vertices.

Upper bound: $E_{per}(G) \, \le \, n\rho(G)$, linking permanental energy to spectral radius.

Analysis of $E_{per}(G)$ on multiple graph families.

Abstract

For a simple graph $G$ with adjacency matrix $A(G)$, let $\pi(G,x):=\mathrm{per}(xI-A(G))$ be its permanental polynomial with roots $\mu_1,\ldots,\mu_n \in \mathbb{C}$, and define the permanental energy $E_{\mathrm{per}}(G):=\sum_{i=1}^n |\mu_i|$. 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 $E_{\mathrm{per}}(G) \le n\rho(G)$, where $\rho(G)$ is the spectral radius, and we study $E_{\mathrm{per}}(G)$ on several graph families.