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.
Keywords
Cite
@article{arxiv.1307.3810,
title = {Counting rooted forests in a network},
author = {Oliver Knill},
journal= {arXiv preprint arXiv:1307.3810},
year = {2013}
}
Comments
13 pages, 6 figures, Since submitting the first version, we have learned that the forest theorem has already been proven by Chebotarev-Shamis. We prove a generalization of their theorem. The proof relies on a general new result in linear algebra and is different from the one given by Chebotarev and Shamis