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 $v_1,\dots ,v_n$, 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.