Linear algebraic techniques for spanning tree enumeration
Combinatorics
2020-08-20 v2 History and Overview
Abstract
Kirchhoff's Matrix-Tree Theorem asserts that the number of spanning trees in a finite graph can be computed from the determinant of any of its reduced Laplacian matrices. In many cases, even for well-studied families of graphs, this can be computationally or algebraically taxing. We show how two well-known results from linear algebra, the Matrix Determinant Lemma and the Schur complement, can be used to elegantly count the spanning trees in several significant families of graphs.
Keywords
Cite
@article{arxiv.1903.04973,
title = {Linear algebraic techniques for spanning tree enumeration},
author = {Steven Klee and Matthew T. Stamps},
journal= {arXiv preprint arXiv:1903.04973},
year = {2020}
}
Comments
This paper presents unweighted versions of the results in arXiv:1903.03575 with more concrete and concise proofs. It is intended for a broad audience and has extra emphasis on exposition. It will appear in the American Mathematical Monthly