Skew-signings of positive weighted digraphs

An arc-weighted digraph is a pair (D,ω) where D is a digraph and ω is an arc-weight function that assigns to each arc uv of D a nonzero real number ω(uv). Given an arc-weighted digraph (D,ω) with vertices v1,…,vn, the weighted adjacency matrix of (D,ω) is defined as the n×n matrix A(D,ω)=[aij] where aij=ω(vivj) if vivj is an arc of D, and 0otherwise. Let (D,ω) be a positive arc-weighted digraph and assume that D is loopless and symmetric. A skew-signing of (D,ω) is an arc-weight function ω′ such that ω′(uv)=±ω(uv) and ω′(uv)ω′(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,ω′) is the same for all skew-signings ω′ of (D,ω). Our main theorem generalizes a result of Cavers et al. (2012) about skew-adjacency matrices of graphs.