Counting rooted forests in a network

Abstract: We use a recently found generalization of the Cauchy-Binet theorem to give a new proof of the Chebotarev-Shamis forest theorem telling that det(1+L) is the number of rooted spanning forests in a finite simple graph G with Laplacian L. More generally, we show that det(1+k L) is the number of rooted edge-k-colored spanning forests in G. If a forest with an even number of edges is called even, then det(1-L) is the difference between even and odd rooted spanning forests in G.