A reciprocal eigenvalue property for unicyclic weighted directed graphs with weights from [formula omitted

Abstract

A weighted directed graph is a directed graph G whose underlying undirected graph is simple and whose edges have nonzero (directional) complex weights, that is, the presence of an edge (u,v) of weight w is as good as the presence of the edge (v,u) with weight w¯, the complex conjugate of w. Let G be a weighted directed graph on vertices 1,2,…,n. Denote by wuv the weight of an edge (u,v)∈E(G). The adjacency matrix A(G) of G is an n×n matrix with entries aij=wij or w¯ji or 0, depending on whether (i,j)∈E(G) or (j,i)∈E(G) or otherwise, respectively. We supply a characterization of those unicyclic weighted directed graphs G whose edges have weights from the set {±1,±i} and whose adjacency matrix A(G) satisfies the following property: ‘λ is an eigenvalue of A(G) with multiplicity k if and only if 1/λ is an eigenvalue of A(G) with the same multiplicity’.