Permanental Energy of Graphs

Abstract

For a simple graph G with adjacency matrix A(G), let π(G,x):=per(xI-A(G)) be its permanental polynomial with roots μ1,⋯,μn∈C, and define the permanental energyEper(G):=∑i=1n|μi|. We prove a sharp universal lower bound: for every m-edge graph G, Eper(G)≥2m, with equality if and only if G is a star together with isolated vertices. We also prove the general upper bound Eper(G)≤nρ(G), where ρ(G) is the spectral radius, and we study Eper(G) on several graph families. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2027.