The medial graph and voltage-current duality (Q1196985)

The author examines both current graphs and voltage graphs, utilizing the medial graph: a 4-regular graph derived from an imbedded graph encoding both the primal and the dual graphs. New insights into voltage-current duality, including wrapped coverings, are obtained. A method is developed for simultaneously giving a voltage and a current assignment on an imbedded graph, when the voltage-current group is abelian, and then applied to construct classes of self-dual imbeddings. Also constructed are orientable and nonorientable imbeddings of \(K_{p,q}\) with dual \(K_{r,s}\) whenever \(p,q,r\), and \(s\) are all even integers exceeding 2 with \(pq=rs\), except that there is no orientable self-dual imbedding of \(K_{6,6}\). This answers a question of the author and \textit{N. Hartsfield} [J. Comb. Theory, Ser. B 54, No. 2, 249-256 (1992; see the review below)]. In a useful conclusion, the author contrasts voltage assignments on the medial graph with current assignments on the radial graph.