Relationship between the number of edges in the dual of graph with the degree of the original graph?

Question

Is the number of edges in the dual of a graph (not necessarily a true dual) related to the degree of the nodes of the (original) graph? If so, is there a generalized formula for this relationship?

Answer 1

To avoid bad cases like stars, I’ll consider only finite simple 3-connected planar graphs. (In particular, all triangulations are 3-connected). Due to Steinitz theorem such graphs are exactly the 1-skeletons of convex polyhedra. By Whitney's theorem, all plane embeddings of a polyhedral graph $G$ are equivalent, that is, obtainable from one another by a plane homeomorphism up to the choice of outer face. In particular, the set of facial cycles (i.e., boundaries of faces) of $G$ does not depend on a particular plane embedding. I recall that the dual of a polyhedral graph $G$ is a graph $G^*$ whose nodes are the faces of $G$ (represented by their facial cycles). So the number of edges of the graph $G^*$ equals the number of edges of the original graph $G$, which equals a half of the sum of degrees of its nodes.