Skew-signings of positive weighted digraphs

Abstract

An arc-weighted digraph is a pair $(D,\omega)$ where $D$ is a digraph and $\omega$ is an \emph{arc-weight function} that assigns\ to each arc $uv$ of $D$ a nonzero real number $\omega(uv)$. Given an arc-weighted digraph $(D,\omega)$ with vertices $v_{1},\ldots,v_{n}$, the weighted adjacency matrix of $(D,\omega)$ is defined as the matrix $A(D,\omega)=[a_{ij}]$ where $a_{ij}=\omega(v_{i}v_{j})$, if $v_{i}v_{j}\ $an arc of $D$ and $0$ otherwise. Let $(D,\omega)$ be a positive arc-weighted digraphs and assume that $D$ is loopless and symmetric. A skew-signing of $(D,\omega)$ is an arc-weight function $\omega^{\prime}$ such that $\omega^{\prime}(uv)=\pm \omega(uv)$ and $\omega^{\prime}(uv)\omega^{\prime}(vu)<0$ for every arc $uv$ of $D$. In this paper, we give necessary and sufficient conditions under which the characteristic polynomial of $A(D,\omega^{\prime})$ is the same for every skew-signing $\omega^{\prime}$ of $(D,\omega)$. Our Main Theorem generalizes a result of Cavers et al (2012) about skew-adjacency matrices of graphs.