A Beginner's Guide to Counting Spanning Trees in a Graph [article]

Saad Quader
2012 arXiv   pre-print
(DRAFT VERSION) In this article we present a proof of the famous Kirchoff's Matrix-Tree theorem, which relates the number of spanning trees in a connected graph with the cofactors (and eigenvalues) of its combinatorial Laplacian matrix. This is a 165 year old result in graph theory and the proof is conceptually simple. However, the elegance of this result is it connects many apparently unrelated concepts in linear algebra and graph theory. Our motivation behind this work was to make the proof
more » ... cessible to anyone with beginner\slash intermediate grasp of linear algebra. Therefore in this paper we present proof of every single argument leading to the final result. For example, we prove the elementary properties of determinants, relationship between the roots of characteristic polynomial (that is, eigenvalues) and the minors, the Cauchy-Binet formula, the Laplace expansion of determinant, etc.
arXiv:1207.7033v2 fatcat:oqootixv45gvvgwgfskgg5wzi4