A reciprocal eigenvalue property for unicyclic weighted directed graphs with weights from {±1,±i}{±1,±i}

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)(u,v) of weight ww is as good as the presence of the edge (v,u)(v,u) with weight w¯, the complex conjugate of ww. Let G   be a weighted directed graph on vertices 1,2,…,n1,2,…,n. Denote by wuvwuv the weight of an edge (u,v)∈E(G)(u,v)∈E(G). The adjacency matrix A(G)A(G) of G   is an n×nn×n matrix with entries aij=wijaij=wij or w¯ji or 0, depending on whether (i,j)∈E(G)(i,j)∈E(G) or (j,i)∈E(G)(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}{±1,±i} and whose adjacency matrix A(G)A(G) satisfies the following property: ‘λ   is an eigenvalue of A(G)A(G) with multiplicity k   if and only if 1/λ1/λ is an eigenvalue of A(G)A(G) with the same multiplicity’.