Permanental polynomials of skew adjacency matrices of oriented graphs

Abstract

Let $G^\sigma$ be an orientation of a simple graph $G$. In this paper, the permanental polynomial of an oriented graph $G^\sigma$ is introduced. The coefficients of the permanental polynomial of $G^\sigma$ are interpreted in terms of the graph structure of $G^\sigma$, and it is proved that all orientations $G^\sigma$ of $G$ have the same permanental polynomial if and only if $G$ has no even cycles. Furthermore, the roots of the permanental polynomial of $G^\sigma$ are studied.