Skew-signings of positive weighted digraphs

Abstract

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 u v of D a nonzero real number ω (u v) . Given an arc-weighted digraph (D , ω) with vertices v 1 , … , v n , the weighted adjacency matrix of (D , ω) is defined as the n × n matrix A (D , ω) = [ a i j ] where a i j = ω ( v i v j ) if v i v j is an arc of D , and 0 otherwise. 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 ′ (u v) = ± ω (u v) and ω ′ (u v) ω ′ (v u) < 0 for every arc u v 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. Keywords: Arc-weighted digraphs, Skew-signing of a digraph, Weighted adjacency matrix, Mathematics Subject Classification: 05C22, 05C31, 05C50