Counting Rooted Spanning Forests in Complete Multipartite Graphs

Abstract

Jin and Liu discovered an elegant formula for the number of rooted spanning forests in the complete bipartite graph $K_{a_1,a_2}$, with $b_1$ roots in the first vertex class and $b_2$ roots in the second vertex class. We give a simple proof of their formula, and a generalization for complete $m$-partite graphs, using the multivariate Lagrange inverse.