Non-isomorphic single-molecule conductors can have transmission functions that are identical at all electron energies, within the Huckel tight-binding approximation. These equiconducting cases follow from the graph-theoretical nature of the transmission function, and the existence of isospectral chemical graphs. Systematic constructions of equiconducting molecular graphs are presented, relying on the notions of isospectral pairs and isospectral vertices. These allow identification of non-isomorphic equiconducting connection patterns for a given molecule, and infinite families of equiconducting non-isomorphic molecules. (c) 2008 Elsevier B.V. All rights reserved.